IEEE TRANSACTIONS ON INFORMATION THEORY 1 Shared Rate Process for Mobile Users in Poisson Networks and Applications

Pranav Madadi , François Baccelli, and Gustavo de Veciana, Fellow, IEEE

1 Abstract— This paper focuses on the modeling and analysis users moving through space. These are driven by geometric 35 2 of the temporal performance variation experienced by a mobile variations in the spatial and environmental relationships (asso- 36 3 user in a wireless network and its impact on system-level design. ciations) to the infrastructure, the presence of other users, 37 4 We consider a simple stochastic geometry model: the infrastruc- as well as random fluctuations intrinsic to wireless channels, 38 5 ture nodes are Poisson distributed while the user’s motion is the 6 simplest possible, i.e., constant velocity on a straight line. We first e.g., shadowing or fast fading. Our focus on temporal variabil- 39 7 characterize variations in the signal-to-noise ratio (SNR) process ity should be contrasted with the extensive work characterizing 40 8 and associated downlink Shannon rate, resulting from variations spatial variability as seen by randomly located users, e.g., 41 9 in the infrastructure geometry seen by the mobile. Specifically, through metrics such as coverage probability, spatial density of 42 10 by making a connection between stochastic geometry and queuing throughput, 90% quantile rate, “edge” capacity, spectral effi- 43 11 theory, the level crossings of the SNR process are shown to form 12 an alternating renewal process whose distribution is completely ciency, etc. Although space and time may be related through 44 13 characterized. For large/small SNR levels, and associated rare averages (when ergodicity holds), the temporal characteristics 45 14 events, we further derive simple distributional (exponential) of the stochastic processes modeling the capacity variations 46 15 models. We then characterize the second major contributor to of a typical mobile user are relatively unexplored beyond 47 16 such variations, namely, changes in the number of other users correlation analysis, and are of great practical interest. 48 17 sharing the infrastructure. Combining these two phenomena, 18 we study what are the dominant factors (infrastructure geom- Indeed, an increasing volume of data traffic is generated by 49 19 etry or sharing number) when a mobile experiences a very wireless devices on the move, e.g., 20-30% of cellular data 50 20 high/low shared rate. These results are then used to evaluate is generated during commute hours, [1]. In the future, with 51 21 and optimize the system-level quality of experience of the mobile increasing use of public transportation and/or the emergence 52 22 users sharing such a wireless infrastructure, including mobile of driverless cars, this volume could grow substantially. The 53 23 devices streaming video which proactively buffer content to 24 prevent rebuffering and mobiles which are downloading large Quality of Experience (QoE) seen by such users can be highly 54 25 files. Finally, we use simulation to assess the fidelity of this model dependent on the capacity variations they see as they move 55 26 and its robustness to factors which are presently not taken into through space. This is particularly the case for applications that 56 27 account. operate over longer time scales, e.g., video/audio streaming, 57 28 Index Terms— Poisson networks, wireless, signal-to-noise ratio, navigation/augmented reality, and real-time services, which 58 29 Shannon rate, mobile user, temporal variations, quality of may not be able to smooth substantial capacity variations 59 30 experience, stochastic geometry, queuing theory, level-crossings, through buffering, but are also for the opportunistic transfers 60 31 alternating renewal process, rarity and exponentiality. of large files over heterogeneous networks, e.g., WiFi offload- 61 ing. More broadly, increases in wireless network capacity 62 32 I. INTRODUCTION have been achieved through heterogeneous network densifi- 63

33 HE primary aim of this paper is to model and study cation leveraging technologies providing different coverage- 64 throughput trade offs, e.g., cellular (macro, pico, femto cells), 65 34 Tthe temporal capacity variations experienced by wireless WiFi and perhaps, in the future, mmWave access points. 66 Manuscript received September 12, 2016; revised July 5, 2017; accepted Characterizing and managing the capacity variations mobile 67 November 9, 2017. The work of P. Madadi was supported in part by the National Science Foundation, The University of Texas at Austin, under Grant users would see across such networks is a challenging but 68 NSF-CCF-1218338, in part by the Award from the Simons Foundation, The important problem towards understanding the efficiency and 69 University of Texas at Austin, under Grant 197982, and in part by NSF under performance of offloading/onloading based services. 70 Grant CNS-1343383 and Grant CNS-1731658. The work of F. Baccelli was supported in part by the National Science Foundation, The University of Texas at Austin, under Grant NSF-CCF-1218338, and in part by the Award from the A. Related Work 71 Simons Foundation, The University of Texas at Austin, under Grant 197982. There is a rich literature on modeling spatial capacity 72 The work of G. Veciana was supported in part by NSF under Grant CNS- 1343383 and Grant CNS-1731658. ThisIEEE paper was presented in part at the variabilityProof in wireless services experienced by a randomly 73 2016 Wi-Opt. located user. Of particular relevance is that based on stochastic 74 The authors are with the Department of Electrical and Computer Engi- geometry, which captures the effect of the variability in base 75 neering, The University of Texas at Austin, Austin, TX 78712 USA (e-mail: [email protected]; [email protected]; gus- station locations, as well as the variability in the environment 76 [email protected]). through shadowing, and in the channel through fading, see 77 Communicated by R. La, Associate Editor for Xxxx . e.g., [2] for a survey. In contrast, work studying the temporal 78 Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. capacity variations seen by a user on the move in a cellular 79 Digital Object Identifier 10.1109/TIT.2017.2781909 network is much more limited. 80 0018-9448 © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. 2 IEEE TRANSACTIONS ON INFORMATION THEORY

81 There is also significant related work on Delay-Tolerant Net- 4) Characterization of rare events. Conditional on a rare 137 82 works (DTN). This literature considers mobile nodes, where event, i.e., very poor or very good user rate, what is the 138 83 the contact duration and the inter-contact times are defined relative contribution of the user’s location vs network 139 84 and empirically measured from real traces [3], [4] as well as congestion? Can one characterize the time scales for rare 140 85 through underlying mobility models [5]. However, this work event occurrences? 141 86 is geared at studying opportunistic ad-hoc communication 5) Applications to QoE. What are the implications of 142 87 networks where the main question is the end to end delay of the temporal capacity variations mobiles see on their 143 88 a tagged packet, under the assumption that nodes are moving application-level QoE and system-level performance? 144 89 independently. Our focus is on studying the continuous-time For instance, what acceptable density of users streaming 145 90 stochastic process experienced by a tagged mobile user tra- video or downloading large files can be supported? 146

91 versing a static pattern of nodes e.g., base stations, modeling To the best of our knowledge the analysis of the SNR 147 92 the wireless infrastructure. and shared rate processes developed in this paper for Poisson 148 93 There certainly is a lot of interest in studying how to design wireless infrastructure is new and provides a first order answer 149 94 networks to better address the needs of mobile users or to to these basic questions. While our models are simplified, 150 95 leverage user mobility for the offloading/onloading traffic. For they should be viewed as a first and necessary step towards 151 96 example [6] studies how user mobility patterns and user per- characterizing temporal variability in more complex random 152 97 ceived QoE might drive the selection of macro-cell upgrades. structures, e.g., SINR variations which, in this paper, are 153 98 Theworkin[7]examinestheeffectiveness of algorithms for only explored via simulation for comparison to SNR process 154 99 optimizing offloading to a set of spatially distributed WiFi characteristics. Generalizations to SINR processes would be 155 100 APs. The work in [8] evaluates how proactive knowledge of desirable, particularly for systems operating in the interference 156 101 capacity variations could be used in designing new models limited regime. Such extensions may be tractable based on 157 102 for video delivery. These works exemplify applications and results known for shot noise fields (see [2, Volume 1]), yet 158 103 engineering problems which depend critically on the temporal are beyond the scope of this paper. 159 104 capacity variability that mobile users would experience, but C. Contributions and Organization 160 105 do not directly address the characteristics of such processes. Section II introduces the basic model studied in this paper. 161 We consider a tagged user moving at a fixed velocity along 162 106 B. Key Questions a straight line through a shared wireless network such that: 163 107 Two primary sources of temporal variation in a mobile’s (a) the access points/base stations form a realization of a 164 108 capacity are changes in the SNR which are associated with Poisson point process; (b) other users sharing the network 165 109 changes in the users geometric relationship to the infrastruc- also form a Poisson (or possibly Cox) point process; (c) all 166 110 ture, and the sharing number, i.e., the number of other mobiles users associate with the closest base station; (d) resources 167 111 sharing the resource. Our goal in this paper is to study the are shared equally with other users sharing the base station; 168 112 relative impacts these have on mobile users’ QoE. In this (e) a standard distance based path loss is used with neither 169 113 setting several basic questions arise: shadowing nor fading. Our goal is to characterize the shared 170 114 1) Characterization of the SNR process. As a mobile rate process seen by the tagged user by studying the SNR 171 115 user moves through a wireless network infrastructure and sharing number processes. 172 116 associating with the closest node, its SNR process and In Section III we characterize the SNR process, including 173 117 thus associated maximal rate (i.e., without accounting the intensity of peaks and valleys and, given a SNR threshold, 174 118 for sharing) will experience peaks and valleys. What is we provide a complete characterization of the on/off level 175 119 the intensity of peaks and valleys? Does it match the rate crossing process as an alternating-renewal process. Equiv- 176 120 of cell boundary crossings? Given an SNR threshold, alently, this characterizes the duration for coverage/outage 177 121 can one characterize the temporal characteristics of the events seen by the mobile. We then provide asymptotics for 178 122 on/off level crossing process associated with being above the likelihood of high and low SNR, and show that, after 179 123 and below the threshold, i.e., the coverage and outage appropriate rescaling, the time intervals between up crossings 180 124 durations? are exponential, which provides a new illustration of the “rarity 181 125 2) Characterization of the sharing number process. hence exponentiality” principle [9], [10]. 182 126 Assuming a population of other users sharing the net- Section IV provides a detailed discussion of the sharing 183 127 work, what are the characteristics of the sharing number number process, i.e., the number of other users which are 184

128 process seen by a mobile?IEEE If the network is shared by Proof 185 covered, i.e., meet an SNR threshold, and share infrastructure 129 heterogeneous users, i.e., static, pedestrian, and users on nodes with the tagged user. The process is somewhat complex 186 130 public transport and/or a road system, how will this bias and not independent of the SNR process; so we introduce 187 131 what a mobile user sees? a simpler bounding process which is used in Section V to 188 132 3) Smoothness of the effective rate process. How smooth characterize the shared rate process seen by our mobile. 189 133 is the bit rate obtained by the mobile seen as a function In particular, we highlight the contrast between the occurrence 190 134 of time? How often does this rate incur discontinuities, of rare events associated with high and low shared rates. 191 135 trend changes, large jumps or other phenomena nega- We also provide an asymptotic characterization for the 192 136 tively impacting real time applications? re-scaled inter-arrival times associated with up-crossings. 193 MADADI et al.: SHARED RATE PROCESS FOR MOBILE USERS IN POISSON NETWORKS AND APPLICATIONS 3

194 In Section VI we present a discussion of discrete event 195 simulation results which are used to assess the robustness 196 of this model to perturbations that were not yet taken into 197 account in the analysis. For example the relative impact of 198 variability associated with the changing geometry (proximity 199 of base stations) seen by a mobile versus that associated with 200 channel variability due to channel fades. We also consider the 201 robustness of our model to the changes in the direction of the 202 motion of the mobile user. 203 In Section VII we leverage our results to evaluate the QoE 204 experienced by mobile users in two concrete scenarios, which 205 give a system-level view on the performance that mobile users 206 would see. 207 Section VIII briefly discusses some important extensions of 208 our results to the case of heterogeneous wireless infrastructures 209 and the case where the locations of mobile users follow a Cox 210 process associated with a road network. The results show how Fig. 1. Mobile user motion in a sample of the Poisson cellular network. 211 heterogeneous technologies impact the temporal variations in 212 the mobile users SNR process. The results also show how a 213 mobile user on a roadway is likely to see poorer performance (L(t), t ≥ 0) be the process denoting the distance from the 250 214 than a static or pedestrian user. Section IX concludes the paper. mobile user to its closest node. 251 215 Here is the summary of our key contributions. We develop We consider downlink transmissions and assume that all 252 216 the framework to evaluate the temporal variations in the nodes transmit at a fixed power p. Based on the classical 253 217 shared rate seen by mobile users. This framework allows us power law path loss model and in the absence of fading 254 218 to derive the following new results on mobile user Quality and shadowing, the Signal-to-Noise Ratio (SNR) process, 255 219 of Experience (QoE): (1) a characterization of the outage (SNR(t), t ≥ 0), of the mobile user is 256 220 process seen by a mobile user requiring minimum rate, as an ( )−β 221 alternating renewal process, (2) an asymptotic characterization pL t SNR(t) = , for β>2, (1) 257 222 of the outage process for high rate requirements, and (3) the w 223 relative contributions of the two underlying factors, SNR and where w denotes the noise power. 258 224 sharing number process, towards the variability of the shared We assume that users are only served if their SNR exceeds 259 225 rate. We then illustrate the use of these results to study a given threshold γ . It follows from (1) that a user is served 260 226 the impact of such temporal variations on user’s QoE. This p 1/β if it is within distance rγ = ( wγ ) from the closest node. 261 227 includes a study of video streaming and file downloading for We refer to rγ as the radius of coverage. Note that a user may 262 228 mobile users. thus associate with a node but not be served. 263 (γ ) The Shannon rate process, (R (t), t ≥ 0), seen by the 264 229 II. SYSTEM MODEL mobile user is directly determined by the SNR process through 265 the relation: 266 230 Consider an infrastructure based wireless network consisting 231 of nodes e.g., base stations/WiFi hotspots, denoted through (γ ) a log (1 + SNR(t)) if L(t) ≤ rγ , 232 their locations on the Euclidean plane. The configuration of the R (t) = (2) 267 0otherwise, 233 nodes is assumed to be a realization of Poisson point process ={ , ,..} R2 λ 234 X1 X2 in with intensity . We consider a tagged where a is a constant depending on the available bandwidth. 268 235 v user moving at a fixed velocity along a straight line starting We assume that each infrastructure node shares time equally 269 236 = from the origin at time t 0. The mobile user shares the among the users it serves. We define the sharing number 270 (γ ) (γ ) 237 network with other (static) users which are spatially distributed process (N (t), t ≥ 0),whereN (t) is the number of 271 238 according to another independent Poisson point process of static users sharing the infrastructure with the mobile user if 272 ξ 239 intensity . it is served, and 0 if it is not served. In other words, if the 273 240 All users associate with the closest infrastructure node. For mobile user is served by its closest node X (t) at time t, 274 (γ ) 241 each X ∈ one can define a set of locations which are ( ) i IEEE it sharesProof the resource with N t static users. The sharing 275 \ (γ ) 242 closer to Xi than any other point in Xi .Thisisaconvex factor (F (t), t ≥ 0), is then defined by: 276 243 polygon known as the Voronoi cell associated with Xi [2]. (γ ) 1 244 The collection of such cells forms a tessellation of the plane F (t) = . (3) 277 + (γ )( ) 245 called the Voronoi tessellation see e.g., Fig.1. Thus, all the 1 N t

246 users located within the Voronoi cell of node Xi associate Finally, the shared rate process seen by the mobile user, 278 (γ ) 247 with Xi . (S (t), t ≥ 0),isgivenby 279 248 Let (X (t), t ≥ 0) denote the random process, where X (t) ∈ (γ )( ) = (γ )( ) × (γ )( ). 249 is the closest node to the mobile user at time t.Let S t R t F t (4) 280 4 IEEE TRANSACTIONS ON INFORMATION THEORY

281 Our aim is to characterize this process. In the next two 282 sections, we first study the two underlying processes namely: 283 (1) the SNR process (SNR(t), t ≥ 0) and its level sets, and (γ ) 284 (2) the sharing number process (N (t), t ≥ 0). 285 In the sequel, for any stationary random process, e.g., (γ ) (γ ) 286 (S (t), t ≥ 0),weletS represent a random variable whose 287 distribution is the stationary distribution of the associated 288 process. 289 A summary of the key notation is provided in Table I.

290 III. CHARACTERIZATION OF THE SNR PROCESS

291 This section is structured as follows: we start by considering 292 the on-off coverage structure of the SNR process and then 293 study in more detail the characteristics of its fluctuations. 294 We then analyze the scale of the inter-occurrence times of 295 certain rare events.

296 A. Analysis of the SNR Level Crossing Process

297 A first order question is whether the mobile is cov- 298 ered or not. To that end we define the SNR level crossing 299 process as follows: 300 Definition 1: Given an SNR threshold γ , the SNR level (γ ) (γ ) 301 crossing process (C (t), t ≥ 0) is defined as C (t) = 302 1(SNR(t) ≥ γ) as shown in Fig 2. 303 Clearly this is an on-off process, where the on and off Fig. 2. Level crossings of the SNR process as an on-off process (bottom) 304 periods correspond to coverage and outage periods respec- and the point process of its local maxima χ, denoted by “o” on the x axis, 305 tively. The process alternates between “on” intervals of length together with edge crossings, denoted by “×” (top). (γ ) (γ ) 306 (Bn , n ≥ 1) and “off” intervals of length (In , n ≥ 1) as We first show that the arrival process to the disc and thus the 332 307 depicted in Fig. 2. We also define the sequence of SNR up- (γ ) queue, is Poisson. We begin by proving that the arrival process 333 308 crossing times (Tn , n ≥ 1) for a given threshold γ .Let (γ ) (γ ) (γ ) has independent and stationary increments. The probability 334 309 ∼ − V Tn Tn−1 be a random variable whose distribution that there is an arrival in the next  seconds is the probability 335 (γ ) 310 is that associated with up-crossing inter arrivals as illustrated that there is a node in the set with area A = 2rγ v depicted 336 311 in Fig. 2. in Fig. 3. Since nodes are distributed according to a homo- 337 312 In order to characterize the SNR level crossing process, we geneous Poisson point process of intensity λ, the number of 338 313 establish a connection between the time-varying geometry seen nodes in any closed set of area b follows a Poisson distribution 339 314 by the mobile user and an associated queuing process. λ > (γ ) with parameter b. Thus, for any 0, the increments in the 340 315 Let D (t) denote the closed disc of radius rγ centered on arrival process have the same distribution. Also, the number 341 316 the mobile user’s location at time t. This closed disc follows (γ ) of nodes in any two disjoint closed sets are independent. 342 317 the mobile user’s motion along the straight line. Let K (t) = (γ ) Thus, the arrival process has independent stationary Poisson 343 318 |∩ D (t)| denote the number of infrastructure nodes in the increments. For a small value of , the probability that there 344 (γ ) 319 disc. is a single arrival is λA + o(). Thus, the arrival process is 345 320 The following theorem provides a simple characterization vλ (γ ) Poisson with rate 2rγ . 346 321 of (K (t), t ≥ 0) which in turn will enable the study of the Every node that enters the moving closed ball stays in it for 347 322 SNR level crossing process. (γ ) a time that depends on its entry locations and is proportional 348 323 Theorem 1: The process (K (t), t ≥ 0) is same as that to the chord length as shown in Fig. 3. This corresponds to 349 324 modeling the number of customers in an M/GI/∞ queue with (γ ) its service time in the queue. 350 325 arrival rate λ = 2rγ vλ and i.i.d. service times with density Since the mobile moves at constant velocity, the distribution 351 326 IEEE Proof ⎧ of the service times can be derived from the distribution of the 352 v2s ⎨  for s ∈[0, 2rγ /v], chord lengths, see e.g., 3. Without loss of generality suppose 353 2 2 2 2rγ 4rγ −v s 327 f (γ ) (s) = (5) the mobile moves along the x-axis. The y coordinate of a 354 W ⎩ (γ ) 0otherwise. typical node entering the disc, Y , is uniform on [−rγ , rγ ]. 355 (γ ) The random variable W representing the typical service 356 (γ )( ) 328 Proof: The entry of a node into the closed disc D t time can be represented as: 357 329 can be viewed as an arrival to the queue. The amount of time  (γ ) 2 − ( (γ ))2 330 spent by the node in D (t) corresponds to its service time 2 rγ Y (γ ) (γ ) ( ) W = . 358 331 and thus the exit from D t to its departure from the queue. v MADADI et al.: SHARED RATE PROCESS FOR MOBILE USERS IN POISSON NETWORKS AND APPLICATIONS 5

TABLE I TABLE OF NOTATION

Proof: Let us consider the M/GI/∞ queue defined in 384 Theorem 1. If the queue is idle there are no nodes within 385 a distance rγ of the tagged user. Thus, an off period of the 386 associated SNR level crossing process is equivalent to an idle 387 period of the queue. Similarly, the queue’s busy period is 388 equivalent to an on period. 389 Since the arrival process is Poisson, the inter arrival times 390 (γ ) are exponential with parameter λ = 2λvrγ . Because of 391 the memoryless property, all idle periods have the same 392 distribution. 393 ˆ (γ ) (γ ) Let B denote a random variable representing the forward 394 Fig. 3. The disc around the mobile user with area A and the chord along which the node moves. recurrence time associated with the on time defined as 395  ∞ P( ˆ (γ ) > ) = 1 P( (γ ) > ) . B x (γ ) B z dz (6) 396 E[B ] x (γ ) 359 The density of W is then given by (5), and has a mean Using the correspondence of the SNR level crossing process’s 397 E[ (γ )]= πrγ 360 W 2v . on times with the busy period of an M/GI/∞ queue, its 398 ( (γ )( ), ≥ ) 361 Thus, the process K t t 0 capturing the number distribution is that given in Theorem 12 in the Appendix A 399 (γ ) 362 of nodes in the moving disc follows the dynamics of the with service time S ∼ W and 400 363 number of customers in an M/GI/∞ queue with arrival rate (γ ) (γ ) (γ ) 2 (γ ) −ρ (γ ) ρ = rγ vλE[W ]=λπr ν = − e . 401 364 λ = 2rγ vλ and i.i.d service times following the distribution 2 γ , 1 (7) (γ ) (γ ) 365 of W . It follows that the stationary distribution for K (t) (γ ) (γ ) 2 The busy period B and idle period I are independent. 402 366 is Poisson with mean πrγ λ. Hence, the SNR level crossing process is an alternating- 403 367 Given this connection to an M/GI/∞ queuing model, renewal process. The mean of the busy period of the queue is 404 368 we can now characterize the SNR level crossing process as given by [11]: 405 369 an alternating renewal process defined as follows: ν(γ ) 370 Definition 2: A process alternating between successive E[ (γ )]= . B (γ ) (8) 406 371 on and off intervals is an alternating renewal process if λv γ ( − ν ) (γ ) 2 r 1 372 the sequences of on period (Bn , n ≥ 1) and off period (γ ) Thus, the probability that the SNR level crossing process is 407 ( , ≥ ) 2 373 In n 1 are independent sequences of i.i.d. non-negative −λπrγ “on” is 1 − e . 408 374 random variables. It is easy to see that all path loss functions which are 409 375 Theorem 2: For any γ>0, theIEEE SNR level crossing process, Proof (γ ) monotonic lead to an analogue of Theorem 2. 410 376 (C (t), t ≥ 0), is an alternating-renewal process. Further, its (γ ) (γ ) 377 typical on period, B , and off period, I , are distributed as

378 the busy and idle periods of an M/GI/∞ queue with arrival B. Fluctuations of the SNR Process 411 λ(γ ) = vλ 379 rate 2rγ and i.i.d. service times with distribution The SNR process is pathwise continuous and has continuous 412 (γ ) ∼ ( λv ) 380 given in (5). Thus, I exp 2 rγ and the busy period derivatives except at a countable set of times corresponding to 413 381 distribution can be explicitly characterized as in [11]. Also, Voronoi cell edge crossings where the mobile sees a hand- 414

382 in the stationary regime, the probability that the SNR level 415 2 off and where the first derivative of the SNR process is − −λπrγ 383 crossing process is “on” is 1 e . discontinuous. 416 6 IEEE TRANSACTIONS ON INFORMATION THEORY

The following theorem gives a characterization of the 454 asymptotic behavior of the distribution of the random variable 455 (γ ) V corresponding to the up-crossings as γ →∞and as 456 γ → 0, which can be seen as “good” and “bad” events 457 respectively. 458 Theorem 4: For all γ>0, the up-crossings of the SNR 459 (γ ) level crossing process, (Tn , n ≥ 1), constitute a renewal 460 (γ ) process. Let V be the typical inter arrival for this process, 461 and let f (γ ) = 2λvrγ ,thenasγ →∞, we have 462

(γ ) d f (γ )V −→ exp(1), 463

2 −λπrγ ⊥ ⊥ and for g(γ ) = λvrγ e ,asγ → , we have 464 Fig. 4. The disc B (Z ) around the projection point Z in two scenarios: 2 0 ⊥ h (a) where Z does not belong to the Voronoi cell of nucleus Z and (b) where (γ ) d it does. g(γ )V −→ exp(1), 465 d where −→ implies convergence in distribution. 466 v 417 Since the mobile is moving at a fixed velocity√ , the inten- Proof: The proof is given in the Appendix B. 467 418 sity of the cell-edge crossings is given by 4 v λ/π [12], [13]. The difference in the scales of the inter-event times of “good 468 419 As the mobile traverses a particular cell, there is also a time events” and “bad events” is due to the fact that the probability 469 420 where it is the closest to the associated node. The SNR process of occurrence of a bad event goes to zero faster than the 470 421 has a local maximum at these times. Perhaps surprisingly such probability of occurrence of good event. Since for bad events, 471 422 maxima may happen at the edge of the cell. Let us denote the as rγ tends to ∞, a disc of radius rγ should be empty, an event 472 −λπ 2 423 point process of times corresponding to interior maxima by rγ whose probability is e . Conversely, for good events, as rγ 473 424 χ. The intensity of χ is given in the following theorem. tends to 0, there should be at least one node in the disc of 474 2 425 Theorem 3: The intensity of the point process χ of the SNR −λπrγ √ radius rγ , an event whose probability is given by 1 − e . 475 426 maxima occurring in the interior of cells is v λ. 427 Proof: Let Z ∈ , be a typical associated node to the IV. SHARING NUMBER PROCESS 476 428 mobile user at any given time. Let us consider the projection ⊥ ⊥ 429 Z of Z onto the straight line (mobile user’s path). Then, Z , In contrast to the SNR or the Shannon rate process, which 477 ⊥ 430 is a point of χ if Z lies within the Voronoi cell of Z. Thus, take their values in the continuum, the sharing number process 478 431 the point process can be characterized by the fact that given is discrete valued and piece-wise constant. We start with 479 432 a node at a height h from the straight line, the closed ball evaluating the frequency of discontinuities and then introduce 480 ⊥ 433 Bh(Z ) of radius h centered at the projection point is empty an upper-bound process to be used in the sequel. We conclude 481 434 of nodes. To that extent, consider a rectangle of unit length this section with a study of rare events. 482 435 and height 2x, with the mobile user’s straight line path passing 436 through the center. From Mecke’s formula [14], the intensity A. Discontinuities 483 437 of the point process χ is   In the sequel, we will use the following simplified version 484 x −λπh2 1 of the Johnson-Mehl cell, see [12]. 485 438 μχ = v λ x e dh lim→∞ 2 ={ } x −x 2x Definition 3: Consider Xi the Poisson point process 486 √ √ √ 2 of nodes on R . For any threshold γ , the JohnsonÐMehl cell 487 439 = v lim λerf( λπx) = v λ. (9) →∞ (γ ) x J (Xi ) associated with Xi is the intersection of the Voronoi 488 440 cell of Xi w.r.t. with a disc of radius rγ centered at Xi ,see 489

441 Thus, the fraction of SNR maxima that happen in the Fig.5. 490 γ 442 interior of the cell is given by π/4. In other words, a proportion Thus for a fixed threshold , the mobile user is cov- 491 π − ered/served at time t if and only if it is within the Johnson– 492 443 1 4 of the SNR maxima seen by the mobile occur at the ( ) γ 493 444 cell edges. Mehl cell of its associated node X t for the radius r . We recall the following definition. 494 (γ ) Definition 4: The process (N (t), t ≥ 0) is defined as 495 445 C. Rare Events IEEE theProof number of static users present in the Johnson-Mehl cell 496 (γ )( ( )) (γ )( ( )) 446 In this subsection we consider certain rare events such as J X t if the mobile is in J X t and 0 otherwise. 497 ( (γ )( ), ≥ ) N 447 the occurrence of large SNR values. We show that the “rarity The process N t t 0 is an -valued piece-wise 498 448 hence exponentiality” principle [9], [10] applies here. We also constant process, with jumps at certain Johnson–Mehl cell 499 449 give the scale at which the inter-arrival times of high SNR edge crossings, see e.g., Fig 6. The value it assumes upon 500 450 are close to being exponentially distributed. As we shall see entering a cell is a conditionally Poisson random variable with 501 451 in Section VII, these asymptotic results can also be used a parameter that depends on the area of the cell in question. 502 452 for moderate values of SNR for settings typically found in The following theorem provides an upper bound for the jump 503 453 wireless networks. intensity: 504 MADADI et al.: SHARED RATE PROCESS FOR MOBILE USERS IN POISSON NETWORKS AND APPLICATIONS 7

(SNR(t), t ≥ 0), are not independent as is easily seen in 527 the case where γ = 0. For example, assume that the mobile 528 user experiences a low SNR, i.e., it is far from its associated 529 node. Then the area of the Voronoi cell is likely to be large. 530 This in turn implies that the mobile is more likely to share 531 its infrastructure node with a large number of users. 532 Fortunately, the sharing number process admits a simple 533 upper bound which is independent of the SNR process. This 534 upper bound is defined as follows: 535 Definition 5: We define an upper bound 536 ˆ (γ ) process,(N (t), t ≥ 0), on the sharing number process. Let 537 D(x, rγ ) denote a disc of radius rγ centered at location x. 538 ˆ (γ ) Let N (t), t ≥ 0, be the random process capturing the 539 number of static users which are in the disc D(X (t), rγ ) if 540 (γ ) the mobile is in J (X (t)),and0otherwise. 541 Fig. 5. Johnson-Mehl cells. ˆ (γ ) The stochastic process (N (t), t ≥ 0) enjoys most of the 542 (γ ) structural properties of (N (t), t ≥ 0); in particular, it is 543 piece-wise constant and one can derive natural bounds on the 544 intensity of its jumps. In addition and more importantly: 545 ˆ (γ ) (γ ) ˆ (γ ) • For all t ≥ 0, P(N (t) ≥ N (t)) = 1, i.e., N (t) 546 is an upper bound for the sharing number; this bound is 547 tight in the regime where the area of the typical Voronoi 548 1 cell ( λ ) is large compared to the area of the disc of radius 549 rγ ; 550 • When the mobile user is covered, the sharing number 551 ˆ (γ ) 2 N (t) is Poisson with parameter ξπrγ ; 552 ˆ (γ ) • The stochastic processes (N (t), t ≥ 0) and 553 (SNR(t), t ≥ 0) are independent. 554

V. S HARED RATE PROCESS 555 Fig. 6. Trace of the sharing number and shared rate processes. Recall that the shared rate is given by 556

(γ ) (γ ) (γ ) (γ ) 1 505 Theorem 5: An upper bound for the intensity of disconti- S (t) = R (t)F (t) = R (t) , (10) 557 N(γ )(t) + 506 nuities of the sharing number process is the intensity of the 1 (γ ) (γ ) 507 Johnson-Mehl cell edge crossings, which is given by: where R (t) is the Shannon rate defined in (2) and F (t) 558 508 √ the sharing factor at time t. A realization of this process is 559 v λ √ √ 2 illustrated in Fig.6. In the sequel, we use our upper bound 560 4 −λπrγ 509 (erf( λπrγ − 2 λrγ e ), (γ ) π process on N (t) to obtain a lower bound on the shared 561 rate: 562 ( ) = √2 z −t2 510 where, erf z π 0 e dt. ˆ(γ )( ) = (γ )( ) ˆ (γ )( ) = (γ )( ) 1 , 511 Proof: The intensity of cell edge crossings in a Johnson- S t R t F t R t (11) 563 Nˆ (γ )(t) + 1 512 Mehl tessellation is given in [15]. Not all cell edge crossings 513 lead to jumps as the number of static users can be the which is now a product of two independent random variables. 564 514 same across two adjacent Johnson-Mehl cells, thus the given 515 intensity is an upper bound. (γ ) A. Shared Rate Variability 565 516 Given that the value of N (t) is a non-zero constant, its The shared rate process is continuous and differentiable 566 517 time of constancy is defined as the typical amount of time it except for a countable number of points and a countable set 567 518 remains at that same constant. This time of constancy is lower of points of discontinuity for the first derivative. It is equal 568 519 bounded by the distribution of intersection of the motion’s line IEEE Proof 569 to zero when the mobile is not covered. The point process 520 with a typical Johnson–Mehl cell [15]. It is a lower bound only of discontinuities is upper bounded by the Johnson-Mehl cell 570 521 since the number of static users can be the same across two edge crossings (see Theorem 5). The point process of its local 571 522 adjacent Johnson-Mehl cells. maxima where the derivative is zero is the same as that of the 572 SNR process given in Theorem 3. 573 523 B. Upper Bound for the Sharing Number Process As can be seen from (11), the variability in the mobile user’s 574 524 Note that the two underlying processes that are used shared rate is driven by two processes: the Shannon rate and 575 525 to define the shared rate process, namely the sharing the sharing factor process. It is of interest to understand their 576 (γ ) 526 number process (N (t), t ≥ 0) and the SNR process relative contributions. To answer this question, let us consider 577 8 IEEE TRANSACTIONS ON INFORMATION THEORY

TABLE II Theorem 6: The likelihood of the rare events associated 609 ξ VALUES OF VARIANCE WITH INCREASING USER DENSITY, , with high shared rate is the same (up to a logarithmic 610 CONSTANT DENSITY OF NODES, λ = 25/π AND CONSTANT = equivalence) as that for the high SNR in the sense that 611 RADIUS OF COVERAGE rγ 500m 1 (γ ) lim − s log P(S > s) 612 s→∞ 1 = lim − s log (P(log(1 + SNR)>s)) 613 s→∞ = 2 . β (13) 614

Notice that as a direct corollary of the last theorem, 615

1 (γ ) TABLE III lim − s log P(S > s) 616 s→∞ VALUES OF VARIANCE WITH INCREASING NODE DENSITY, λ,CONSTANT DENSITY OF USERS, ξ = 50 AND CONSTANT 1 (γ ) (γ ) = lim − s log P(S > s|N = 0) 617 RADIUS OF COVERAGE rγ = 200m →∞ s 1 (γ ) (γ ) = lim − s log P(S > s, N = 0) 618 s→∞ = 2 . β (14) 619

For the rare events associated with a low shared rate, we will 620 ˆ(γ ) consider the lower bound process (S (t), t ≥ 0): 621 Theorem 7: The rare events associated with achieving low 622 shared rates are predominantly the same as the rare events 623 ˆ(γ ) associated with high sharing number given that the mobile is 624 578 the variance of S which can be written using the conditional at the covering cell edge, in the sense that for some sequence 625 579 variance formula as: −β {sn} such that sn log (1 + Krγ ) ∈ Z ∀n and limn→∞ sn → 626 ˆ(γ ) (γ ) ˆ (γ ) 2 ˆ (γ ) (γ ) 2 580 var(S )=var(R )E[F ] +var(F )(E[(R ) ]). (12) ∞, we have 627   1 (γ ) 1 581 Note that if the density of users increases, the variance of the ˆ lim − log P S < sn 628 n→∞ ( ) 582 shared factor decreases whereas the variance of the Shannon sn log sn  −β 583 rate remains constant. Thus, the variance of the shared rate 1 log(1 + Krγ ) 1 = lim − log P < 629 584 varies approximately linearly with a slope equal to the second →∞ ˆ (γ ) n sn log(sn) N + 1 s n 585 moment of the Shannon rate. By contrast if the density of −β = ( + ). 630 586 nodes increases, the variance of both the sharing factor and log 1 Krγ (15) 587 the Shannon rate vary making their relative contributions more Theorem 8: Conditioned on a very good or very bad shared 631 588 complex. rate, the relative contribution of the mobile user’s location and 632 589 Let us evaluate empirically what are the contributions of the network congestion is as follows: 633 590 the user and node density to the variability of shared rate. P( (γ ) > , (γ ) = | (γ ) > ) = , 591 For a given node density λ = 25/π and radius of coverage lim R s N 0 S s 1 (16) 634 s→∞ 592 rγ = 500m, by increasing the density of static users, the (γ ) (γ ) P(N > as log(1 + γ)− 1 | 0 < S < 1/s) = 1. (17) 635 593 variance of the shared rate decreases linearly with the variance

594 of the sharing factor. Table II gives numerical values evaluated Conditioned on a very high shared rate, the mobile user has 636 ξ = 595 via simulation. For a given static user density 50 to be close to the associated node with no other static users 637 596 and radius of coverage of 200m, by increasing the density sharing its resources. By contrast, conditioned on the mobile 638 597 of nodes the variance of the sharing factor monotonically being served and experiencing a very low shared rate, it has 639 598 increases, whereas the changes in the variance of SNR process to share its associated node with a number of users that is 640 599 is complicated due to the way it is defined. Thus, the changes inversely proportional to the desired shared rate. 641 600 in the variance of the sharing number process is difficult to From the previous theorem, we know that high shared rates 642 601 characterize with densification of nodes due to the complex IEEE are predominantlyProof the same as the event where the SNR is high 643 602 relation given in (12). Table III gives numerical values evalu- and the sharing number is zero. Thus, for a given threshold 644 603 ated by simulations. δ for the shared rate process, we can study the asymptotic 645 behavior of the distribution of the inter arrival time of shared 646 rate up-crossings as δ →∞. 647 604 B. Rare Events ˆ Corollary 1: Let Zδ denote the inter arrival time for the 648 605 Since the mobile user’s shared rate depends on two factors it ˆ(γ ) good events of the lower bound process (S (t), t ≥ 0) and 649 606 is of interest to understand their relative contributions towards let f (rδ) = 2λvrδ , then as δ → 0, we have 650 607 the events of high/low shared rate. The proofs of the following 2 −ξπrγ ˆ d 608 theorems are given in Appendix C and D respectively. e Zδ −→ exp(1). 651 MADADI et al.: SHARED RATE PROCESS FOR MOBILE USERS IN POISSON NETWORKS AND APPLICATIONS 9

ˆ = 652  Proof: The result follows from the fact that Zδ X rδ 653 i=1 V ,whereX is a geometric random variable with 2 −ξπrγ (rδ ) 654 parameter e and V the typical interval time of the 655 renewal process of up-crossings associated with the SNR 656 process of threshold δ.

657 VI. SIMULATION AND VALIDATION

658 In this section we evaluate the robustness of our mathemat- 659 ical model and associated asymptotic results to more realistic 660 settings. We use simulation to study the temporal variations of 661 the SNR process experienced by a mobile user under various 662 scenarios which are not captured by our analytical framework. 663 Our model is challenged in various complementary ways: e.g., 664 by adding fading, accounting for interference from other base 665 stations and relaxing the straight line mobility model. In each 666 case the objective is to determine to what degree our simplified Fig. 7. Q-Q plot for various thresholds. 667 mathematical model is still approximately valid, providing 668 robust engineering rules of thumb to predict what mobile users

669 see in practice. In the sequel we evaluate how quickly the convergence to 705 670 In particular, we will answer the following questions: exponential stated in Theorem 4 arises. To that end we com- 706

671 • How quickly do the SNR up crossings converge to pare the re-normalized distributions obtained via simulation to 707 672 exponential asymptotics as a function of the associated a reference exponential distribution with parameter 1 using the 708 673 thresholds? Kolmogorov-Smirnov (K-S) test. The K-S test finds the great- 709 674 • Are the results obtained robust to the presence of fast est discrepancy between the observed and expected cumulative 710 675 fading? frequencies– called the “D-statistic”. This is compared against 711 676 • Are there regimes where the temporal characteristics of the critical D-statistic for that sample size with 5% significance 712 677 the SNR process are good proxies for those of the SINR level. If the calculated D-statistic is less than the critical one, 713 678 process, e.g., high path loss? we conclude that the distribution is of the expected form, see 714 679 • Does the characterization of the SNR level crossing e.g. [16]. 715 680 process valid for other mobility models?

681 We begin by introducing our simulation methodology and the B. Convergence of Level-Crossing Asymptotics 716

682 default parameters used throughout this section. Theorem 4 shows that as the SNR threshold γ in dB 717 increases, the rescaled distribution for up-crossings of the SNR 718

γ 719 683 A. Simulation Methodology process becomes exponential. The question is how large needs to be for this result to hold. To that end we simulated 720 684 We initially consider a user moving on a straight line (road) the level crossing process for various γ and subjected it to the 721 685 at a fixed velocity of 16 m/s. The base stations are randomly K-S test as mentioned above. 722 686 placed according to a Poisson point process with intensity λ The Q-Q plots for up-crossing inter-arrivals rescaled by 723 687 such that the mean coverage area per base station is equal to f (rγ ) as introduced in the theorem are depicted in Fig. 7. 724 688 that of a disc with radius 200m. Unless otherwise specified, we As expected, as the threshold increases, the points become 725 689 consider the path loss function given by Eq. (1) with exponent linear, and for a threshold value of γ = 50 or more, it is 726 690 β = 4 and assume that all base stations transmit with equal accepted as an exponential with unit mean by the K-S test. 727 691 power p = 2W. The signal strength received by the mobile −2 In practice a SNR of 50 dB is not realistic for wireless users. 728 692 user is recomputed every 10 seconds. The total distance However, as seen from the Q-Q plots in Fig.7, for moderate 729 693 traveled by the mobile is 750 km along a straight line. The values of γ such as 0.1, the up-crossing inter-arrivals can be 730 694 number of simulation runs considered to evaluate the means well approximated by an exponential with parameter 1/f (rγ ). 731 695 is 1000. The number of handovers experienced by the mobile Further, the absolute error in the mean of the inter-arrival of 732 696 while traveling this distance is on average 4269. up-crossings for moderate value of γ = 1 is approximately 733 697 IEEE Proof We calibrate the thermal noise power to the cell-edge only 9.3 s. 734 698 user. Let L be a random variable denoting the distance of a 699 typical user to the closest base station. Define dedge by the C. Robustness of Level-Crossing Asymptotics to Fading 735 700 relation P(L ≤ dedge) = 0.9. Since in our simulation setting 2 − ln(0.1) Next we study the effect that channel fading might have 736 701 P(L ≤ l) = 1 − exp(λπl ),wehavededge = πλ .Ifwe on the level-crossing asymptotics. We consider channels with 737 702 fix the desired SNR at the cell edge to be SNRedge, this then −β pd Rayleigh fading with unit mean, so that the SNR experienced 738 w = edge 703 determines the noise power to be .Wesetthe by the tagged mobile user at a distance d from the base station 739 SNRedge −β 704 SNRedge to be 1 dB. is given by Hpd /w,whereH is a fading random variable 740 10 IEEE TRANSACTIONS ON INFORMATION THEORY

Fig. 8. SNR process in presence of fading with mean 1. Fig. 9. Threshold above which the inter-arrival of up-crossing converges to an exponential with parameter 1, for different variance of fading.

741 which is exponential with unit mean. The coherence time is 742 set to tc = 0.423/fd,where fd is the Doppler shift given by = v v 743 fd c fo where is the vehicle velocity, c is the speed of 744 light and fo = 900MHz is the operating frequency. This gives 745 a coherence time tc = 0.007 s. Thus, fading (power) changes 746 every 0.007 seconds. A realization of the SNR process with 747 fading is exhibited in Fig. 8. 748 If we assume no fast fading, the realizations of the SNR 749 process seen by a mobile are continuous and differentiable 750 and level crossings are well defined. However when the 751 fluctuations associated with fast fading are added to the model, 752 level crossings will exhibit bursts of up-crossings associated 753 with fast fading when the mean channel gains reach the 754 thresholds, see Fig. 8. Thus, in our simulations, we avoid these 755 bursts of up-crossings by considering the first up-crossing 756 and by suppressing all subsequent up crossings (if any) for 757 an appropriate time period. We take a time period for the Fig. 10. Threshold above which the inter-arrival of up-crossing converges 758 suppression of up-crossings equal to twice the expected on to an exponential with parameter 1, for different path loss exponent. (rγ ) 759 time of 2E[B ] [11]. 760 In order to vary the variance while keeping the mean of the To that end we simulated the SINR process for a setting 779 761 fading equal to 1, we now consider fading which is a mixture with a high path loss exponent of β = 4 and found once again 780 762 of exponentials. For this process, we would expect that for that the rescaled distribution for the up-crossing inter-arrivals 781 763 fading with mean one, if variance is small, the appropriately converges to an exponential with parameter 1. The test requires 782 764 rescaled inter-arrival distribution for up-crossings which are a threshold γ = 31.7dB. We then evaluated, for different 783 765 suppressed would once again be asymptotically exponen- path loss β, what threshold values are needed to obtain a 784 766 tial with parameter 1. In other words we expect geometric similar convergence. As shown in Fig. 10, the threshold in 785 767 variations associated with base station locations to dominate question increases as β decreases. Further we found that for 786 768 channel variations. Whereas, if the fading variance is high, one β<3.5, we no longer have the desired convergence property. 787 769 might expect the SNR threshold required to see convergence In summary, for high path-loss exponents β ∈[3.5, 4], the up- 788 770 to an exponential to increase. Fig. 9 exhibits such thresholds as crossing asymptotics for the SNR and SINR processes are 789 771 a function of the fading variance.IEEE As can be seen for fading Proof similar. 790 772 variances exceeding 8, the channel variations dominate the 773 geometric variations and thus up-crossing asymptotics differ

774 from those of Theorem 4. E. Robustness to Generalized Mobility Models 791

So far we have focused on a mobile moving strictly along 792 775 D. Robustness of Level-Crossing Asymptotics to Interference a straight line. Next we evaluate to what extent one can relax 793 776 So far we have focused on the SNR process. One might this requirement. To that end, we consider a variant of the 794 777 ask to what degree the Signal-to-Interference-plus-Noise random way point model, [17], where a user moves at a 795 778 Ratio (SINR) process, share similar characteristics. constant velocity, v meters per sec, but every t0 secs, with 796 MADADI et al.: SHARED RATE PROCESS FOR MOBILE USERS IN POISSON NETWORKS AND APPLICATIONS 11

cellular or WiFi network and (2) large file downloads using 835 WiFi. In both applications, we consider a service model where 836 the users are served only if the SNR experienced by them 837 is above a certain threshold. Thus, the users are served only 838 during the on period of the SNR process. We illustrate how the 839 characterization of the level-crossing process can be leveraged 840 to improve the QoE of the tagged user. 841

A. Stored Video Streaming to Mobile Users 842

Let us consider a scenario where mobile users are viewing 843 stored videos which are streamed over a sequence of wireless 844 downlinks. The users are distributed according to a Poisson 845 point process of density ξ and are moving independently of 846 each other. Consider a policy where a mobile user is served 847 Fig. 11. Three specific mobile user motion paths. by a node only if the SNR experienced is greater than a 848 threshold γ . Thus, the nodes serve the mobile users within 849 1/β the radius of coverage rγ = (p/(γ w)) . A lower threshold 850 797 probability p0, selects a new independent direction (angle) γ corresponds to a lower transmission rate (when served), 851 (−θ ,θ ) 798 uniformly distributed in 0 0 . a higher probability of coverage and sharing with a large 852 799 We conducted simulations where for a given SNR threshold, number of other mobile users. Conversely a higher threshold 853 γ θ v 800 , various values for the parameters, p0, 0, and t0 were implies higher transmission rate and sharing with fewer other 854 801 varied and the following quantities were measured: mobile users. For simplicity we consider rebuffering as the 855 802 • The means of the on-off periods; primary metric for user’s video QoE [8]. 856 803 • The independence of the on-off periods; The playback buffer state of the tagged mobile user moving 857 804 • The distribution of the up-crossing inter-arrivals. at a fixed velocity v on a straight line, can be modeled as a 858 fluid queue. The arrival rate to the queue alternates between 859 805 We considered three particular sets of values for the parameters (γ ) an average ergodic rate, h(γ ) = E[R |SNR >γ] and zero 860 806 (v, t0, p0,θ0), each reflecting a specific kind of traffic: (1) Car 861 807 on a highway: (20, 10, 0.001,π/45), (2) Car in an urban area: depending on whether the mobile is being served or not. Let η 862 808 (10, 1, 0.1,π/4) and (3) Wanderer: a pedestrian moving in a denote the video playback rate in bits. Hence, as long as the η 863 809 city: (1, 0.1, 0.6,π). Fig.11 illustrates a snapshot of the paths buffer is non-empty, the fluid depletion rate of the queue is . 864 810 of the three cases. Re-buffering of the video is directly linked to the proportion − 865 811 The mean of the on-off periods of the SNR process is of time the playback buffer is empty, which is given by 1 ρ(γ,ξ) ρ(γ,ξ) 866 812 within a 5 percent error margin in the case (1) but they do ,wheretheload factor, , [14] of the queue is:   813 not match for the other two cases for moderate values of the (γ ) ν h(γ ) 1 814 SNR threshold. However, for certain high threshold values, ρ(γ,ξ) = E |SNR >γ . (18) 867 η ˆ (γ ) + 815 γ ∈[20dB, 60dB] the mean coverage time for the case (2), N 1 816 is within an acceptable error margin of 10 percent. Further, (γ ) where ν is the probability that the alternating renewal 868 817 the mean on time for both the second and third cases is larger process associated with the arrival rate to the fluid queue is 869 818 than the mean on time in the case where the mobile is moving “on”. 870 819 along a straight line. The first natural question one can ask is whether there is a 871 820 The on-off periods are also highly correlated for the choice of γ such that the fluid queue is unstable, thus ensuring 872 821 later two cases, while independence approximately holds in no rebuffering in the long term. In other words, does there exist 873 822 case (1). The distribution of the up-crossing inter-arrivals for a a γ>0 such that ρ(γ,ξ) > 1? Given our policy and the QoE 874 823 specific high threshold value of γ = 50 dB is now considered. metric, the network provider may choose a lower value of 875 824 The convergence to an exponential of unit mean is only threshold,γ , as long as the typical mobile user in the long run 876 825 observed for the first case. In summary, the robustness of our experiences no rebuffering. 877 826 model is only guaranteed for slight changes in the direction. Now, for simplicity let us consider a constant mean trans- 878 827 Major changes in the direction have a strong effect on the 1 IEEE missionProof rate κ = a log (1 + γ ) E instead of the 879 (γ ) 828 characterization of on-off periods as an alternating renewal N +1 829 process with the proposed distribution. average ergodic rate. This constant bit rate is when the network 880 does not rely on adaptive coding/decoding. For additional 881 motivation for this scenario, see [8]. The load factor ρ(γ,ξ) 882 830 VII. APPLICATIONS is then given by 883 831 In this section we use our framework to evaluate  −b 832 application-level performance of mobiles in a shared wire- a log(1 + γ) 1 − e γ 2/β   833 less network. In particular we consider two very different ρ(γ,ξ) = E 1 , (γ ) (19) 884 834 wireless scenarios: (1) video streaming to mobiles sharing a η N + 1 12 IEEE TRANSACTIONS ON INFORMATION THEORY

and κ>η, the fluid queue alternates between busy and 901

idle period representing the periods when the video is un- 902 interrupted or frozen, respectively. The distribution of the on 903 (γ ) (γ ) periods B and that of the off periods I of the M/GI/∞ 904

queue discussed in Section determine the distribution of the 905 busy period B f and that of the idle period I f of the fluid 906 queue. When denoting by κ the constant input rate during on 907

period and by η the constant output rate when the queue is 908 non-empty, the Laplace transform of B f is given by [18] 909

L ( ) = L (γ ) ( σ + λ(σ − )( − L ( ))), B f s B s 1 1 B f s (20) 910 (γ ) where σ = κ/η > 1, and the idle period I f has an exp(λ ) 911 distribution. 912

B. WiFi Offloading 913

WiFi offloading helps to improve spectrum efficiency and 914 Fig. 12. The load factor of the fluid queue for a single user (top curve) and reduce cellular network congestion. One version of this 915 for a positive density of users as a function of γ (curves below the top curve). scheme is to have mobile users opportunistically obtain 916 For the latter curves, the number of users per base station are ξ/λ = 1, 4and 10 from top to bottom. Here b = 1. All functions can be multiplied by an data through WiFi rather than through the cellular network. 917 arbitrary positive constant when playing with a and η. Offloading traffic through WiFi has been shown to be an 918 effective way to reduce the traffic on the cellular network, 919 when available WiFi is faster and uses less energy to transmit 920 the data. 921 Let us consider a scenario where the mobile users download 922 a large file from the service provider, relying on WiFi hotspots, 923 distributed according to a Poisson point process of intensity 924 λ, rather than from the cellular base stations. The users are 925 distributed according to a Poisson point process of density ξ 926 and are moving independently of each other. Assume that the 927 WiFi hotspots have a fixed coverage area, i.e., the mobile user 928 connects to WiFi only if it is within a certain distance r from 929 the hotspot. Thus, higher the density of hotspots, λ,deployed 930 by the provider the better the performance experienced by 931 mobile users which rely only on them. We consider the time 932 it takes to complete the download as the primary metric for 933 user’s quality of experience. 934 Consider again the case without adaptive coding/decoding 935 ∗ Fig. 13. Level set curve of ρ(γ ,ξ)= 1 for an arbitrary positive constants and the tagged mobile user moving on a straight line at 936 a and η. constant velocity v. Then, the shared rate experienced by the 937 1 mobile user is the constant κ = a log (1 + γ ) E as 938 N(γ )+1   2 p β (γ ) defined above. In addition, the mobile user experiences an 939 885 where b = λπ w and E[1/(N + 1)] can be calculated alternating on and off process, as characterized in Theorem 2. 940 886 by numerical integration as described in the section VIII A. Below, for the sake of mathematical simplicity, we assume 941 887 It is easy to check that the function ρ(γ,ξ) has a unique ∗ ∗ that the file size F is exponential with parameter δ and that the 942 888 maximum γ on (0, ∞). A plot of (19) and the value of γ mobile user starts to download the file at the beginning of an 943 889 are exhibited in Fig. 12. Notice that for these parameters, the ∗ on period. Let T be a random variable denoting the time taken 944 890 value of γ increases with ξ. (λ) to download the file. Consider the event J ={F >κB }, 945 891 For a given base station density λ and density of users ξ, (λ) ∗ where B is a random variable distributed like a typical on 946 892 IEEEγ Proof one can evaluate the SNR threshold value for which the period seen by the mobile (see Theorem 2) independent of F 947 893 load factor ρ is maximum. Fig.13 illustrates the level set curve ∗ and let 948 894 of ρ(γ ,ξ) = 1 for various values of λ and ξ. In this setup, η α = P( ) = P( >κ (λ)) = L (δκ). 895 given the video consumption rate , it is possible to answer J F B B(λ) (21) 949 896 questions like what is the minimum density of base stations Now, define the non-negative random variable X as the busy 950 897 required to serve a certain density of users such that in the long (λ) period, B , conditioned on event J, whose c.d.f. is given by: 951 898 term the video streaming is uninterrupted for all the users.  γ x 899 Remark 1: In the case where there exists no threshold for P( < ) = 1 −δκz ( ) X x e fB(λ) z dz (22) 952 900 which the condition for no long term rebuffering is satisfied α 0 MADADI et al.: SHARED RATE PROCESS FOR MOBILE USERS IN POISSON NETWORKS AND APPLICATIONS 13

(λ) 953 Conditioned on the event, {F ≤ κ B },letY denote the straight line. Let us define the mobile sharing number process 1002 F (γ ) 954 time taken to download the file, i.e., κ . The c.d.f. of the non- (Np (t), t ≥ 0) as the number of users sharing the node 1003 955 negative random variable Y is then given by: associated with the tagged user when it is served and zero 1004  yκ otherwise. Thus, at any given time t, the tagged user shares 1005 P( < ) = 1 δ −δzP( (λ) > z ) . (γ ) 956 Y y e B dz (23) its resources with a random number of users Np (t) which 1006 1 − α 0 κ is Poisson with a parameter depending on the area of the 1007 957 Notice that Johnson-Mehl cell. 1008 1 L ( ) = L ( + δκ), For simplicity, we define the shared rate process 1009 958 X s B(λ) s (24) (γ ) α (S (t), t ≥ ) 1010  ∞ c 0 as 1 −y (δκ+s) (λ) ⎧   959 L (s) = δκe P(B > y)dy. (25) Y − α ⎨ 1 1 0 (γ ) a log (1 + γ ) E (γ ) if L(t) ≤ rγ , + Sc (t) = Np 1 (27) 1011 960 The following representation of the Laplace transform of T is ⎩ 0otherwise. 961 an immediate corollary of the on-off structure:

962 Theorem 9: Consider a network of WiFi hotspots distrib- Since the distribution of the area of the Voronoi cell is 1012 λ (γ ) 963 uted according to a Poisson point process of intensity and unknown, we approximate Np to be Poisson with parameter 1013 ˆ(γ ) ˆ(γ ) 964 radius of coverage r shared by mobile users distributed inde- ξE[J ],whereJ denotes the area of the Johnson-Mehl 1014 ξ 965 pendently according to a Poisson point process of intensity . cell of radius rγ , conditioned on the fact that the tagged 1015 966 Assuming that the tagged mobile user starts to download the user is within a distance rγ from the associated node, which 1016 967 file at the beginning of an on period, the Laplace transform of introduces an additional bias. We can compute the expectation 1017 ∼ (δ) 968 the time taken to download a file of size F exp is given with the help of integral geometry as shown in Appendix E. 1018 969 by We found the value of the expectation using numerical inte- 1019 (γ ) ( − α)L ( ) gration and compared the mean number of users E[N ]= 1020 L ( ) = 1 Y s . p 970 T s (26) ξE[ ˆ(γ )] − αL ( ) 2λvr J with the sample mean obtained from simulation for 1021 1 X s 2λvr+s various parameters λ,rγ . To validate the approximation of 1022 (γ ) 971 Proof: Proof is given in the Appendix G. ˆ(γ ) Np by a Poisson random variable with parameter ξE[J ], 1023 972 It is remarkable that the Laplace transform of T admits a (γ ) (λ) we compared the value of E[1/(Np + 1)] calculated using 1024 973 quite simple expression in terms of that of B . Other and numerical integration with that from simulations. We found 1025 974 more general file distributions can be handled as well when that the calculated value of the expectation is within the 95% 1026 975 using classical tools of Laplace transform theory. Note that this confidence interval of the simulated mean. 1027 976 setting also leads to interesting optimization questions such as 2) Cox Process: Let us consider a population model where 1028 977 the optimal density of WiFi hotspots needed to be deployed roads are distributed according to a Poisson line process of 1029 978 for lower expected download times. 2 intensity λr on R [19]. Then independently on each road, 1030 we consider users distributed according to a stationary Poisson 1031 979 VIII. VARIANTS point process of intensity λt . This is known as a Cox process 1032 980 In this section, we consider two generalizations of our and we denote it by u [19]. This model can be used to 1033 981 framework: (1) we move from sharing with static users to represent a car motion on a road network. 1034 982 sharing with mobile users, and (2) we move from homoge- For a line L of the line process, let us denote the orthogonal 1035 983 neous to heterogeneous wireless infrastructures. projection of the origin O on L by (θ, r) in polar coordinates. 1036 For θ ∈[0,π) and r ∈ R, (θ, r) is unique. Thus, a Poisson 1037 984 A. Sharing the Network With Mobile Users line process with intensity λr is the image of a Poisson point 1038 985 Until here we have considered a tagged mobile user sharing process with the same intensity on half-cylinder [0,π)× R. 1039 986 the network with static users. We now consider two scenarios: Suppose all users on the roads are moving arbitrarily 1040 987 (1) that where the other users are mobile and are initially but independently from each other. Thus, at any instant the 1041 988 distributed according to a homogeneous Poisson point process, distribution of users on a given road remains Poisson. Consider 1042 989 and (2) that where they are mobile but restricted to a random a tagged user moving along a given road, then we have the 1043 990 road network, i.e., form a Cox process. following theorem by [20]. 1044 991 1) Homogeneous Poisson Case: Consider the case where Theorem 10: u is stationary, isotropic, with intensity 1045 992 other users sharing the network are initially located according πλr λt . From the point of view of the tagged user i.e., under 1046

993 IEEEξ Proof 1047 to a homogeneous Poisson point process of intensity ,and its Palm distribution, the point process is the union of three 994 subsequently exhibit arbitrary independent motion. This is a counting measures: (1) the atom at the origin, O, (2) an 1048 995 possible model for pedestrian motion. It follows from the independent λt -Poisson point process on a line through O with 1049 996 displacement theorem for Poisson point processes [2] that the a uniform independent angle and (3) the stationary counting 1050 997 other users at any time instant will remain a Poisson point measure u. 1051 998 process of intensity ξ. Following our previous framework, let us define an another 1052 ( (γ )( ), ≥ ) 999 As considered before, the users are served only if they are sharing number process Nd t t 0 as the number of 1053 1000 at a distance less than rγ from the closest node. Consider users sharing the node associated with the tagged user when 1054 1001 a tagged mobile user moving at a fixed velocity along a it is served and zero otherwise. 1055 14 IEEE TRANSACTIONS ON INFORMATION THEORY

(γ ) 1056 Evaluation of the mean of Nd . Suppose the tagged user Note that the SNR level crossing process as previously 1109 1057 is at the origin O. Let the associated node X (t) be at a defined is again an alternating renewal process which now 1110 1058 distance x from the origin. Let d(0,θ) be a line through the depends on the heterogeneous resource deployment. 1111 1059 origin with θ uniform from [0,π). From the aforementioned Theorem 11: For heterogeneous networks with preferential 1112 (γ ) 1060 theorem, the number of sharing users Nd can be split into association to micro base stations, the probability that the 1113 1061 two terms: Ns denoting the number of sharing users from stationary SNR level crossing process seen by a tagged user 1114 2 ˆ 2 −π(λrγ +λrˆγ ) 1062 stationary u and Nl denoting the number of sharing users is “on” is 1 − e . Also, the mean time for which 1115 ( ,θ) 1063 on the line d 0 . the process is “on” is given by 1116 1064 Given any convex body Z, N (Z) denotes the number of s 2 ˆ 2 π(λrγ +λrˆγ ) − 1065 sharing users from stationary u present in Z,andE[Ns (Z)] e 1 ˆ(γ ) . (29) 1117 1066 πλ λ ( ) ˆ is given by r t area Z [21]. Let J denote the area as 2v(λrγ + λrˆγ ) 1067 defined before. Thus, ˆ(γ ) Proof: In order to characterize the SNR level crossing 1118 1068 E[N ]=πλ λ E[J ], s r t process, we establish a connection to a Boolean model. 1119 ˆ(γ ) 1069 where, E[J ] is evaluated using integral geometry (see Assume that the mobile user is moving with unit velocity. 1120 ( , ) 1070 Appendix E). Let B Xi rγ denote the closed ball of radius rγ centered at 1121 ( ˆ , ˆ ) ˆ ˆ 1071 Now, let l(0,θ) denote the length of the line d(0,θ) in the Xi and B Xi rγ a closed ball of radius rγ centered at Xi . 1122 ˆ(γ ) 1072 area J . Then, N , the number of sharing users on the line The union of all these closed balls forms a Boolean model 1123 l   1073 d(0,θ), is Poisson with parameter λ l(0,θ). Thus, E[N ]= t l ˆ λ E[ ( ,θ)] E = ∪ ∈ B(X , rγ ) ∪ ∪ ˆ ˆ B(X , rˆγ ) . (30) 1124 1074 t l 0 . Xi i Xi ∈ i 1075 One can evaluate E[l(0,θ)] using integral geometry (see  1076 Appendix F). We have, Now, assume that E is intersected by the directed line l.Note 1125 (γ ) E[ ]=E[ ]+E[ ]=πλ λ E[ ˆ(γ )]+λ E[ ( ,θ)]. that the Boolean model under consideration has independent 1126 1077 Nd Ns Nl r t J t l 0  convex grains and thus the intersection E ∩ l yields an 1127 1078 (28) alternating sequence of “on” and “off” periods which are 1128 (γ ) (γ ) 1079 Thus, the mean number of users sharing the tagged user’s independent. Let Bh and Ih be random variables denoting 1129 1080 association node is larger when the users are distributed the length of a typical on and off periods respectively. 1130 (γ ) 1081 according to a Cox process than when the users are distributed The distribution of the length of off period Ih is easy 1131 1082 according to a Poisson point process, assuming that both have to establish using the contact distribution functions and is 1132 ∗ 1083 the same mean spatial intensity. exponential with parameter λ [12]: 1133

∗ −λ∗l f (γ ) (l) = λ e , 1134 1084 B. Mixture of Pedestrian and Road Network Ih 1085 Suppose now we have two types of users : drivers who stay ∗ ˆ (γ ) (γ ) where, λ is 2(λ+λ)E[R ].HereR is the random variable 1135 1086 on roads, and pedestrians which are unconstrained. If pedestri- h h denoting the radius of the closed ball and is given by 1136 1087 ans are supposed to follow a Poisson point process, from their ⎧ 1088 point of view, the number of sharing users corresponds to the λ ⎨ γ , (γ ) r w.p. λ+λˆ 1089 sum of a stationary Poisson point process and a stationary = Rh λˆ (31) 1137 ⎩ˆγ . 1090 Poisson line process. On the other hand, from a driver’s point r w.p. λ+λˆ 1091 of view, the number of sharing users corresponds to the same Thus, the mean off period under the assumption of unit 1138 1092 sum, but in addition with a Poisson point process on a road velocity is 1139 1093 the driver is taking. Thus, the mean number of sharing users 1094 is always greater for drivers. In other words, pedestrians are (γ ) 1 1 E[I ]= = . 1140 1095 likely to share their infrastructure node with fewer users than h ∗ ˆ λ 2(λrγ + λrˆγ ) 1096 drivers.

Now, in the stationary regime, the probability that the 1141

1097 C. Heterogeneous Networks mobile user is either within a distance rˆγ from a micro-BS or a 1142 ˆ distance rγ from a macro-BS i.e., it’s SNR level crossing 1143 1098 Let us consider a deployment of micro-base stations = ˆ ˆ process is “on”, is given by the volume fraction [12]: 1144 1099 {X1, X2,..} distributed according to some homogeneous Pois- ˆ 1100 λ IEEE Proofˆ ¯ son process of intensity independent of the existing macro- (−(λ+λ)V ) c = 1 − e , (32) 1145 1101 base stations ={X1, X2,..}. Assume that all micro-BS 1102 transmit at a fixed power pˆ. where 1146 1103 Let us consider a mobile user moving at a fixed velocity v γ π(λr 2 + λˆrˆ2) 1104 along a straight line. For a given SNR threshold , the mobile ¯ = πE[( (γ ))2]= γ γ . V Rh (33) 1147 1105 is served by a micro-base station if its distance from its closest λ + λˆ 1/β 1106 micro-BS is less than rˆγ = (pˆ/wγ ) . Otherwise it is served 2 ˆ 2 −π(λrγ +λrˆγ ) 1107 by a macro-base station provided its distance from the closest Thus, c = 1 − e . The probability evaluated above 1148 1108 macro-BS is less than rγ . does not depend on the velocity of the mobile user and thus 1149 MADADI et al.: SHARED RATE PROCESS FOR MOBILE USERS IN POISSON NETWORKS AND APPLICATIONS 15

Fig. 14. Comparing mean-on time for heterogeneous and homogeneous Fig. 15. Comparing volume fraction for heterogeneous and homogeneous networks. networks.

v E[ (γ )] 1150 holds for any constant velocity . The mean on period Bh no rebuffering in the long term. Let the event Q denote that 1182 1151 can be evaluated using the following relation the tagged user is served, then 1183  (γ )   E[B ] a log(1 + γ)P(Q) 1 = h , ρ(γ,ξ) = E |Q . (34) 1184 1152 c (γ ) (γ ) η (γ ) E[ ]+E[ ] Np + 1 Bh Ih Assuming our approximation is valid, we need to find 1185 1153 which results in 1 2 ˆ 2 1186 π(λrγ +λrˆγ ) E (γ ) in the case of heterogeneous network. Let G and (γ ) e − 1 Np +1 1154 E[B ]= . h ˆ H be the events denoting that the tagged user is served by 1187 2(λrγ + λrˆγ ) micro BS and macro BS respectively, then 1188   1155 Since, the mobile user is moving with a fixed velocity v, E 1 | 1156 the mean time for which the mobile user is “on” is given by (γ ) Q 1189 E[ (γ )] N + 1 Bh p      1157 v . 1 1 1158 E (γ ) |G P(G) + E (γ ) |H P(H ) Np +1 Np +1 1159 These results provide an analytical characterization of the = . P( ) (35) 1190 1160 impact of heterogeneous densification on the mobile user’s Q

1161 temporal performance. Let us now compare the performance Since, the micro and macro base stations are distributed 1191 1162 improvement seen by the mobile user in a heterogeneous independently, the mobile user experiences two independent 1192 1163 network as compared to that of a homogeneous network. alternating renewal processes. Thus, in the stationary regime, 1193 1164 The graphs in Fig. 14 and Fig. 15 illustrate the difference the probability that mobile is served by macro BS is the 1194 1165 in the expected on-times and volume fraction of the networks product of the probabilities that the mobile is in “on” period of 1195 1166 respectively. alternating renewal process of macro BS and in “off” period 1196 1167 Notice that as micro-base stations are added, the expected of that of micro BS which gives: 1197 1168 on time decreases and later increases. The initial decrease ˆ 2 ˆ 2 2 −λπrˆγ −λπrˆγ −λπrγ 1169 is due to the inclusion of many relatively small length P(G) = 1 − e , P(H ) = e (1 − e ). (36) 1198 1170 on-times resulting from the micro-base stations in the voids ˆ(γ ) ˆ(γ ) Let J and J denote the area similar to that of what 1199 1171 of the homogeneous network. However, the volume fraction 1 2 we considered before. Given the mobile is served by macro 1200 1172 increases monotonically with the addition of micro-base BS, we need to consider the users which are within the area 1201 1173 stations. ˆ(γ ) J1 excluding the area covered by micro BS in this area. The 1202 ˆ(γ ) IEEE areaProof covered by the micro BS in J is approximated to be 1203 1 (ˆ ) γ (ˆ ) 1174 D. Stored Video Streaming in Heterogeneous Networks ν rγ × E[ ˆ ] ν rγ J1 ,where is the fraction of space associated 1204 1175 Consider a scenario as introduced earlier where mobile users with micro-BS as given by (7). Then 1205   (γ ) 1176 are viewing a video being streamed over a sequence of wireless ξE[Jˆ ] 1 1 − e 2 1177 downlinks, but now served by a heterogeneous network with E |G = , 1206 (γ ) + ξE[ ˆ(γ )] 1178 micro and macro BS. In this setting we once again consider Np 1 J2   (γ ) 1179 a fluid queue representing the tagged mobile user’s playback ξE[Jˆ ](λπˆ rˆ2 ) 1 1 − e 1 γ 1180 buffer state similar to Section VII A, and ask whether there is a E | = . (γ ) H (γ ) 1207 γ + ξE[ ˆ ](λπˆ ˆ2) 1181 choice of such that the fluid queue is unstable, thus ensuring Np 1 J1 rγ 16 IEEE TRANSACTIONS ON INFORMATION THEORY

deployed by the operator to serve a certain density of users, 1227 given the density of macro-BS λ such that the video streaming 1228 is uninterrupted for all the users. Thus, operators evaluate the 1229 cost incurred to serve a higher density of users by deploying 1230 micro-BS and in case the cost incurred is higher, an operator 1231 might consider other technologies. 1232 Remark 2: In heterogeneous networks, the interference 1233

from macro-BS to micro-BS is of major concern. One can 1234 introduce blanking, to reduce the effect of such interference 1235

i.e., assume that with certain probability f micro-BS transmit 1236 and with probability 1 − f macro BS transmit. 1237 Also, by considering heterogeneous networks with cellular 1238

BS and WiFi hotspots, there is no such problem of interference 1239 since both cellular BS and WiFi hotspots operate at different 1240

frequencies. 1241

∗ Fig. 16. γ with increase in micro- BS density for a certain fixed density of IX. CONCLUSION 1242 macro-BS for an arbitrary positive constants a and η in heterogeneous case. As explained in the introduction, the analysis of temporal 1243 variations of the shared rate experienced by a mobile user 1244 requires the characterization of both the functional distribution 1245 of a continuous time, continuous space stochastic process (rate 1246 process) constructed on a random spatial structure (e.g., 1247 the Poisson Voronoi tessellation) and of another stochastic 1248 process (sharing number process) constructed on a random 1249 distribution of static users. This paper addressed the simplest 1250 question of this type by focusing on the underlying SNR 1251 process and the sharing number process in the absence of 1252 fading. This allowed us to derive an exact representation of 1253 the level crossings of the stochastic process of interest as 1254 an alternating renewal process with a full characterization of 1255 the involved distributions and of the asymptotic behavior of 1256 rare events. The simplicity and the closed form nature of 1257 this mathematical picture are probably the most important 1258 observations of the paper. We also showed by simulation 1259 ∗ that this very simple model provides a good representation 1260 Fig. 17. Level set curve of ρ(γ ,ξ)= 1 for an arbitrary positive constants a and η in heterogeneous case. of the salient characteristics which happen for more complex 1261 systems, such as those with fading when the fading variance is 1262 small enough, or those based on SINR rather than SNR when 1263 the path loss exponent is large enough or those with changes 1264 1208 For a given density of macro-BS λ, the density of micro- ˆ in the direction of motion, when the change in direction is 1265 1209 BS λ and the density of users ξ, we can evaluate the SNR ∗ small. This model is hence of potential practical use as is, 1266 1210 value γ for which the load factor ρ(γ,ξ) given in (34) is ∗ in addition to being a first glimpse at a set of new research 1267 1211 maximum. Fig.16 illustrates the optimal SNR value (γ ) with ˆ questions. 1268 1212 varying density of micro BS λ. Notice that the optimal gamma 1213 value initially decreases with the density of micro-BS. X. FUTURE WORK 1269 1214 Since in our setting the micro BS have smaller transmission 1215 power, for a given threshold γ , the radius of coverage for The most challenging questions on the mathematical side 1270 1216 micro BS is smaller than that of the macro BS i.e., rˆγ < are as follows: (1) The understanding of the tension between 1271 1217 rγ . Since the micro-BS are given higher preference, with the randomness coming from geometry (studied in the present 1272

1218 IEEE Proof 1273 an increase in their density, the optimal threshold decreases paper), from sharing the network with other users (also studied 1219 in order to increase their coverage. Also, the cost incurred in the present paper) and that coming from propagation (only 1274 1220 by increasing coverage of macro-BS is compensated by the studied by simulation here): it would be nice to analytically 1275 1221 increased density and coverage of micro-BS till a certain quantify when one dominates the other. (2) The extension 1276 1222 density. of the analysis to SINR processes, which is our long term 1277 ∗ 1223 Fig.17 illustrates the level set curve of ρ(γ ,ξ) = 1for aim and will require significantly more sophisticated mathe- 1278 ˆ 1224 various values of λ and ξ for a given density of macro-BS λ. matical tools (e.g. based on functional distributions of shot 1279 1225 Given this setup, it is possible to answer questions like what is noise fields), than those used in the present paper. (3) The 1280 1226 the minimum density of micro-base stations that needs to be characterization of the distribution of the shared rate process. 1281 MADADI et al.: SHARED RATE PROCESS FOR MOBILE USERS IN POISSON NETWORKS AND APPLICATIONS 17

1282 (4) The extension the case where all other users are mobile. and 1328  1283 To begin with, we might consider a stationary tagged user, with 2 rγ /v E[ (γ )]= P( (γ ) > ) 1284 an independent homogeneous Poisson point process of mobile U U u du 1329 0 1285 users moving on independent straight lines. Considering the (γ )  −ρ 2 rγ /v e (γ ) ˆ (γ ) 1286 Voronoi cell of the associated base station of the tagged user, ρ P(W >u) = (e − 1)du. (42) 1330 ν(γ ) 1287 we might try to characterize the number of mobile users 0 / /∞ 1288 entering and leaving the cell as an M GI queue. Thus, Thus, 1331 1289 the sharing number process associated with the stationary user   −ρ(γ ) 2r/v e (γ ) ˆ (γ ) 2rγ 1290 will be equivalent to the number of customers in a queue. (γ ) ρ P(W >u) E[U ]= e du − . (43) 1332 ν(γ ) v 1291 Later, one could extend this to the case where the tagged user 0 1292 is mobile. ˆ (γ ) Since W is the forward recurrence time associated with 1333 1293 On the practical side, the main future challenges are linked (γ ) the service time W defined in (6), 1334 1294 to the initial motivations of this work, namely in the prediction  ∞ 1295 and optimization of the user quality of experience. Many sce- ˆ (γ ) 1 (γ ) P(W > u) = P(W > t)dt, 1335 E[ (γ )] 1296 narios refining those studied can be considered. For instance, W u 1297 the stationary analysis of the fluid queue representing video (γ ) ρ(γ ) (γ ) where, E[W ]= vλ from (7) and P(W > t) = 1336 1298 streaming should be completed by a transient analysis and by  2rγ 2 2 2 1299 a discrete time analysis. This alone opens an interesting and 1/2rγ 4rγ − v t from (5). Thus, 1337 1300 apparently unexplored connection between stochastic geome- (γ ) 1301 try and queuing theory with direct implications to video QoE. E[U ] 1338 −ρ(γ )   2rγ 2rγ  e v λv v r2 −v2t2dt 2rγ = u 4 γ − 1302 APPENDIX (γ ) e du 1339 ν 0 v 1303 ∞  A. Busy Period of the M/GI/ Queue −ρ(γ )   2rγ e v λ 2rγ 2 − 2 2rγ / /∞ u 4rγ x dx 1304 Theorem 12 (Makowski [11]): Consider an M GI = e du − 1340 ν(γ ) v 1305 queue with arrival rate λ and generic service time W. Let M 0     γ  2r −λ 1 2 −v2 2+ 2 ( √ uv ) 1306 denote an N-valued random variable which is geometrically v 2 u 4rγ u 2rγ arctan 1 4rγ2 −v2u2 = e du 1341 1307 distributed according to ν(γ ) 0 − P( = ) = ( − ν)(ν)l 1, = , ,... 2 γ 1308 M l 1 l 1 2 (37) −λπrγ 2r − e 1342 v 1309 with   2 2 −λπrγ γ 2 γ −ρ 1 r −λrγ q(z) 2e r 1310 ν = 1 − e and ρ = λE[W]. (38) = e dz − . (44) 1343 −λπr2 − γ v 0 v + 1 e 1311 Consider the R -valued random variable U distributed The second equality is by the change of variables vt = x 1344 1312 according to and the last equality√ is by the change of variables z = 1345 ˆ 1 2 √ z 1 −ρP(W≤u) uv/rγ and q(z) = z 4 − z + 2arctan( ).Now,from 1346 1313 P(U ≤ u) = ( − e ), u ≥ , 2 − 2 ν 1 0 (39) 4 z Equation (44) we have, 1347 ˆ  1314 2 where, W is the forward recurrence time associated with the γ 2 (γ ) r −λrγ q(z) { , ≥ } lim E[U ]∼ e dz. 1348 1315 generic service time W. Let Un n 1 be an i.i.d. sequence →∞ rγ v 0 1316 independent of the random variable M. Let B denote a typical ˆ rγ 2 −λr2 q(z) rγ 2 −λr2 2z 1317 busy period. Then the forward recurrence time B associated γ γ Now, the integral v 0 e and v 0 e dz are 1349 1318 with B admits the following random sum representation: asymptotically equal as rγ goes to ∞. 1350 M The asymptotic equality of the two functions as rγ goes to 1351 ˆ ∞ 1319 B =d Ui , (40) means, that the relative error of the approximate equality 1352 i=1 goes to 0 as rγ goes to ∞ i.e., 1353 = 2 2 1320 where d denotes equality in distribution. 2 −λrγ q(z) − −λrγ 2z 0 e e dz = . 1354 lim −λ 2 0 rγ →∞ 2 e rγ 2z 1321 B. Proof of Theorem 4 0 + (γ ) IEEE Proof 1355 1322 Lemma 1: For a R -valued random variable U defined With the help of the Taylor series expansion, we get that 2 1323 by (39) and service time distribution by (5). We have the C/rγ −λ 2 ( ) −λ 2 e rγ q z − e rγ 2zdz 1324 following limit: 0 lim = 0, 1356 →∞ 2 −λ 2 (γ ) rγ e rγ 2z 1325 lim 2λrγ vE[U ]=1. 0 rγ →∞ for some constant C. Thus, we need to prove that 1357 1326 Proof: From Equations (7) and (39): 2 2 2 −λrγ q(z) −λrγ 2z 2 e − e dz −ρ(γ ) C/rγ (γ ) (γ ) (γ ) e ρ P( ˆ > ) lim = 0. 1358 P( > ) = ( W u − ), ≥ , →∞ 2 −λ 2 1327 U u (γ ) e 1 u 0 (41) rγ rγ 2z ν 0 e 18 IEEE TRANSACTIONS ON INFORMATION THEORY

2 2 −λrγ q(z) −λrγ 2z (γ ) 1359 Now let us observe the function h(z) = e − e . U is also [0, 2rγ /v] and all its moments are bounded 1402 (γ ) k k 1360 The derivative of the function h(z) is E[(U ) ] <(2rγ /v) . Thus, for all values of s, we get that 1403   2 2  lim → l (s, rγ ) = 0. 1404 ( ) = − − 2 −λrγ q(z) + −λrγ 2z 2 rγ 0 k 1361 h z 4 z e 2e rγ = , ..    For all k 2 3 we also have that 1405 2 2 −λrγ 2z 2 2 −λr [q(z)−2z] 2 k 1362 = e rγ 2 − 4 − z e . (45) (2λvsrγ ) |l (s, rγ )|≤ . 1406 k k! = / 2  1363 Thus by substituting z C √rγ and using the Taylor series ∞ Therefore, for all values of s, limrγ →0 = lk(s, rγ ) = 0. 1407 2 k 2 1364 expansion of [q(z) − 2z] and 4 − z ,weget Thus, Eq (49) can be written as 1408  2 −2Cλz (γ ) 1365 lim h (C/rγ ) = e [o(rγ )]=0. −sf(γ )U (γ ) rγ →∞ E[e ]=1 − s2λrγ vE[U ]+o(rγ ). 1409 2 1366 The derivative of function h(z) is 0 for z = C/rγ as rγ goes Now, substituting into Eq (48) we have 1410 ∞ ( ) / 2 (γ ) 1367 to i.e., the function h z has a local maximum at C rγ for −sf(γ )Bˆ E[e ] 1411 1368 some constant C and it can be observed that it is decreasing (γ ) (γ ) (γ ) 1 − ν + 2sλvrγ E[U ](1 − ν ) + so(rγ ) 1369 = for C 3. = , (50) 1412 − ν(γ ) + ν(γ ) λv E[ (γ )]+ ( ) 1370 Therefore, the function h(z) is a decreasing function from 1 2 s rγ U so rγ / 2 1371 3 rγ to 2 and the area under its curve has an upper bound from which it follows that 1413 1372 that goes to 0. (γ ) −sf(γ )Bˆ E[ ]= . 1414 1373 Thus, the two integrals are asymptotically equal as rγ goes lim e 1 (51) rγ →0 (γ ) 1 1374 ∞ E[ ]∼ γ ∞ to , which implies that U 2vrγ λ as r goes to . 1415 1375 From the results in [11] we have that (γ ) (γ ) (γ ) ˆ (γ ) 1376 Proof: V is the sum of a busy period B and E[ −sf(γ )B ]= − (γ )E[ (γ )]E[ −sf(γ )B ], (γ ) e 1 sf B e (52) 1416 1377 idle period I of an M/GI/∞ queue. We know that the 1378 distribution of the idle period is exponential with parameter and also that 1417 λv (γ ) (γ ) (γ ) 1379 2 rγ and that the busy period B and idle period I are (γ ) ν E[B ]= . (53) 1418 1380 independent. λv ( − ν(γ )) (γ ) 2 rγ 1 1381 The Laplace transform of the random variable V scaled Thus, we get 1419 1382 by 2λvrγ is given by 2 −λπrγ (γ ) ( (γ ) ) = (γ ) ( (γ ) ) (γ ) ( (γ ) ), (γ ) 1 − e ˆ (γ ) 1383 V f s B f s I f s (46) E[ −sf(γ )B ]= − E[ −sf(γ )B ]. e 1 s −λπ 2 e 1420 e rγ 1384 where, 1 Using the limit in (51), we get 1421 1385 (γ ) ( f (γ )s) = . (47) (γ ) I + −sf(γ )B 1 s lim E[e ]=1. (54) 1422 ( (γ ) ) rγ →0 1386 So, asymptotically, as rγ goes to 0, I (γ ) f s converges (γ ) 1387 in distribution to an exponential random variable with unit Thus, the distribution of the random variable V scaled 1423 1388 mean. by f (γ ) = 2λvrγ converges in distribution to an exponential 1424 1389 Next, we need to prove that asymptotically, the busy period random variable with unit mean. 1425 (γ ) (γ ) 1390 B scaled by f goes to one. Let us consider the forward Now, consider a continuous function g(γ ) such that 1426 ˆ (γ ) 1391 recurrence time of the busy period, B whichisdefinedin limrγ →∞ g(γ ) = 0. The Laplace transform of the random 1427 (γ ) 1392 (6). From Theorem 2 we get, variable V scaled by g(γ ) is given by 1428 (γ ) − (γ ) (γ )  (γ ) (γ ) ( − ν )E[ sU ] − ˆ − M 1 e (γ ) = (γ )(g(γ )s) (γ ) (g(γ )s), (55) 1429 E[ s B ]=E[ s i=1 Ui ]= , V (s) B I 1393 e e (γ ) 1 − ν(γ )E[e−sU ] where, 1430 1394 and it follows that 1 (γ ) (γ ) − (γ ) (γ ) (g(γ )s) = . (56) 1431 ˆ (γ ) ( − ν )E[ sf U ] I + (γ )( / v λ) E[ −sf(γ )B ]= 1 e . 1 sg 1 2 rγ 1395 e (γ ) (48) 1 − ν(γ )E[e−sf(γ )U ] ∞ ( (γ ) ) Asymptotically, as rγ goes to , I (γ ) g s goes to 1. 1432 1396 Further, we have that By the same arguments as those for the up-crossing case 1433

IEEE∞ Proof 1434 we have that (γ )  E[ −sf(γ )U ]= ( , ), (γ ) 1397 e lk s rγ (49) ( − ν(γ ))E[ −sg(γ )U ] E[ −sg(γ )Bˆ (γ )]= 1 e , k=0 e (γ ) 1435 1 − ν(γ )E[e−sg(γ )U ] 1398 where (γ ) where, 1436 (−1)k (2λvsrγ )kE[(U )k] ( , ) = . ∞ (γ ) 1399 lk s rγ  (− )k( λv )k ( −ρ )k k! −sg(γ )U (γ ) 1 2 srγ e (γ ) k E[e ]= E[(U ) ]. 1437 (γ ) k! 1400 Given that the support of service times W is [0, 2rγ /v], k=0 1401 it follows from (39) that the support of the random variable (57) 1438 MADADI et al.: SHARED RATE PROCESS FOR MOBILE USERS IN POISSON NETWORKS AND APPLICATIONS 19

−ρ(γ ) − (−1k )(2λvsrγ )k (e )k 1 (γ ) ˆ(γ ) (γ ) ( , ) = E[( )k]. We now prove the first relation. Since S ≤ S ≤ log(1+ 1476 1439 Let mk s rγ k! U The (γ ) SNR), in order to prove the first inequality, it is enough to 1477 1440 moments of the random variable U are bounded (γ ) E[( )k] <( /v)k show that 1478 1441 U 2rγ . Thus, for all values of s,weget (γ ) 2 1 2 1442 that limrγ →∞ mk(s, rγ ) = 0, because ρ = λπrγ and the ˆ(γ ) lim − s log P(S > s) ≤ . (60) 1479 −ρ(γ ) k−1 s→∞ β 1443 exponential term (e ) dominates. Also for k = 2, 3, ... 1444 we have that We have 1480 2 k −ρ(γ ) k−1 ˆ(γ ) −β ˆ (2λvsrγ ) (e ) P(S > s) = P(log(1 + KL )>s(1 + Nγ )), 1481 1445 |m (s, rγ )|≤ . k k! hence 1482 1446 Therefore for all values of s, ˆ(γ ) −β ˆ P(S > s) = P KL > exp s(1 + Nγ ) − 1 1483 ∞ −β ˆ 1447 lim mk(s, rγ ) = 0. ≥ P KL > exp s(1 + Nγ ) 1484 rγ →∞ = k 2 β ˆ = P L < K exp −s(1 + Nγ ) 1485 1448 Thus, Eq (57) can be written as   2 2 − (γ ) (γ ) −ρ(γ ) (γ ) −ρ(γ ) 2 β ˆ E[ sf U ]= − v λE[ ]+ ( ), = P L < K exp −s (1 + Nγ ) 1486 1449 e 1 se 2 rγ U e o rγ β

2 2 2 ˆ ( ) = 2 β −s β −s β Nγ 1450 where, limrγ →∞ o rγ 0. = P L < K e e . 1487 (γ ) 1451 From Lemma 1 we have that, limrγ →∞ 2λrγ vE[U ]=1. 2 1452 Putting these results together we have that Using now the fact that L is an exponential random variable 1488 ˆ with parameter λπ, independent of Nγ ,weget 1489 E[ −sg(γ )Bˆ (γ )] 1453 e ˆ(γ ) −ρ(γ ) − ρ(γ ) (γ ) − ρ(γ ) P(S > s) 1490 − 2 v γ λE[ ]+ 2 ( γ ) e se 2 r U e so r 2 − 2 − 2 ˆ 1454 =    , ≥ − E −λπ β s β s β Nγ −ρ(γ ) − −ρ(γ ) v λE[ (γ )]− −ρ(γ ) ( ) + −ρ(γ ) 1 exp K e e 1491 e 1 e s2 rγ U e so rγ e 2 − 2 − 2 − 2 1455 (58) β s β 2 s β s β ≥ λπ K e exp −ξπrγ 1 − e 1 + o e , 1492

1456 so that −a 2 where we used again the bound 1 − e ≥ a + a /2andthe 1493 ˆ (γ ) ˆ (γ ) 2 −sg(γ )B 1 fact that N is Poisson with parameter ξπrγ . 1494 1457 lim E[e ]= . (59) rγ →∞ + 1 1 s Taking the log, multiplying both sides by − s and letting 1495

1496 1458 Now, from Eq (52) and (53) we get, s go to infinity gives (60).

(γ ) 2 ˆ (γ ) E[ −sf(γ )B ]= − ( − −λπrγ )E[ −sf(γ )B ]. 1459 e 1 s 1 e e D. Proof of Theorem 7 1497 (γ ) −sg(γ )B 1 Let us first prove the last relation. Using the Chernoff’s 1498 1460 Thus from the limit in (59), lim →∞ E[e ]= . rγ 1+s ˆ (γ ) (γ ) bound for the Poisson random variable N ,weget 1499 1461 Thus, the random variable V scaled by g(γ ) = 2 −λπrγ (γ ) −β 1462 λv γ ˆ 2 r e converges in distribution to an exponential ran- P N + 1 > sn log (1 + Krγ ) 1500 1463 dom variable with unit mean.  −β s log (1+Krγ ) 1464 eθ ≤ −θ , e −β 1501 sn log (1 + Krγ ) 1465 C. Proof of Theorem 6 2 θ = ξπr 1502 = /w with γ , which shows that 1466 Let us first prove the second relation. Let K p and let 1467 L denote the distance to the closest base station. We have 1 ˆ (γ ) −β lim − log P N + 1 > s log (1 + Kr ) 1503 →∞ n γ n sn log(sn) 1468 P((log(1 + SNR)>s) −β   ≥ log (1 + Kr ). 1504 = P −β > s − γ 1469 KL e 1 The converse inequality follows from the bound 1505 = P β < K 1470 L θ k es − 1 ˆ (γ ) −β −θ  P + > ( + ) ≥ , 1506  2 N 1 sn log 1 Krγ e IEEEβ Proof k! K = P 2 < −β 1471 L = ( + ) es − 1 with k sn log 1 Krγ and Stirling’s bound on the 1507   2 Gamma function. 1508 β K Let us now prove the first relation. We have 1509 1472 = 1 − exp −λπ ,  es − 1 ˆ(γ ) 1 P S < sn 1510 2 1473 where we used the fact that L is an exponential random  2 rγ 1474 variable with parameter λπ. The result then follows from the −λπv (γ ) − β = P ˆ + > + v 2 v. ≥ − −a ≥ + 2/ ≥ e N 1 sn log 1 K d 1511 1475 bound a 1 e a a 2, for a 0. 0 20 IEEE TRANSACTIONS ON INFORMATION THEORY

Fig. 18. Tagged user at origin, its serving base station X j at distance x and ( ) ( ) a point Q at a distance u along with discs Bx 0 and Bz Q . Fig. 19. The tagged user at the origin, its serving base station X j at distance x and a point U at a distance u on the line d(0,θ).

1512 For sn large enough,  where, L is a random variable denoting the distance to the 1538 closest BS. Thus, 1539 P ˆ(γ ) < 1  1513 S sn r ( ) ˆ(γ ) ∗ fL x E[ ]= E[ | = ] 1540    2 J V L x r dx − β rγ 0 fL (y)dy ˆ (γ ) 2 −λπv 0 1514 ≤ P N + 1 > s log 1 + Krγ e dv, n r E[ ∗| = ] λπ −λπx2 0 0 V L x 2 xe dx = 1541 1 − e−λπr2 1515 which allows one to conclude that    √   r π x cos(α)+ r2−x2 sin(α) = 1 1 ˆ(γ ) 1 −β −λπ 2 2 1542 1516 lim − log P S < s ≥ log (1 + Kr ). − r →∞ ( ) n γ 1 e 0 0 0 n sn log sn 2 (−λl(Bz(Q)−Bx (0))) −λπx e ududα2λπxe dx 1543    √ 1517 The upper bound follows from the inequality: 1 r π x cos(α)+ r2−x2 sin(α) ⎛ ⎞ = 4λπ 1544  2 − −λπr2  ∞ β 1 e 0 0 0  θl (−λ ( ( )∪ ( ))) ˆ(γ ) 1 −θ ⎝ 2 K ⎠ l Bx 0 Bz Q α . 1518 P S < sn = e P L > e uxdud dx (62) 1545 l! l+1 − l=0 e s 1 ( ( )) = π 2 ( ( ) ∪ ⎛  ⎞ The last equality is because l Bx 0 r and l Bx 0 1546 2 ( )) = (α) + (π − α) 2 + (π − (α, ))( 2 + 2 − θ k β Bz Q ux sin x f x u x 1547 −θ ⎝ 2 K ⎠ u−x cos(α) 1519 ≥ e P L > , 2ux cos(α)) with cos( f (α, x)) = √ . 1548 ! k+1 2+ 2− (α) k e s − 1 u x 2ux cos

−β F. Integral Geometry II 1549 1520 where k = sn log (1 + Krγ ). Let us consider a point U at a distance u from the origin on 1550 the line d(0,θ) as shown in the figure. Then, the probability 1551

1521 E. Integral Geometry I that the point is in the Voronoi cell of the BS X j is given by 1552 −λl(C(u,θ)) e ,wherel(C(u,θ)) is the area of the shaded region 1553 1522 Consider a tagged user at the origin, and let the serving in the figure. The conditional expectation is given by: 1554 1523 BS X j be at a distance x from the origin. Let Q denote a E[ ( ,θ)| = ] 1524 point a distance u from the origin. We need to consider all l 0 L x √ 1555  π  (θ)+ 2− 2 (θ) 1525 points Q that are within a distance rγ from the serving BS x cos r x sin 1 (−λl(C(u,θ))) = = 2 + 2 − (α) < = e dudθ, (63) 1556 1526 X j i.e., z X j Q u x 2uxcos rγ ,where π  0 0 1527 α = X j OQ as shown in Fig.8. where, L is the random variable denoting the distance to the 1557 1528 Let Bx (0), Bz(Q) be two discs with centers at the origin closest BS. 1558 1529 and Q, and radii x and X j Q, respectively. Let Bx (0), Bz(U) be two discs with centers at the origin 1559 1530 The conditional probability that a point at distance u from and U, and radii x and X j U, respectively. Thus, 1560 1531 origin is within the Voronoi cellIEEE of the BS serving the user Proof 1532 at the origin given it is at a distance x is the probability that E[ ( ,θ)] l 0  1561 1533 there is no other BS in the area of disc B (Q) excluding the r ( ) z fR x ( ) = E[l(0,θ)|L = x] dx 1562 1534 area of Bx 0 . Then the conditional expectation is: r ( ) 0 0 fL y dy 2 E[ ˆ(γ )| = ] r E[l(0,θ)|L = x]2λπxe−λπx dx 1535 J L x √ = 0   −λπ 2 1563 π xcos(α)+ r2−x2sin(α) 1 − e r √ (−λl(Bz(Q)−Bx (0)))    1536 = 2 e ududα, r π x cos(θ)+ r2−x2 sin(θ) 0 0 = 1 −λπ 2 1564 1537 (61) π(1 − e r ) 0 0 0 MADADI et al.: SHARED RATE PROCESS FOR MOBILE USERS IN POISSON NETWORKS AND APPLICATIONS 21

(−λl(C(u,θ))) −λπx2 1565 × e dudθ2λπxe dx [14] F. Baccelli and P. Brémaud, Palm Probabilities and Stationary Queues, 1620    √ r π x cos(θ)+ r2−x2 sin(θ) vol. 41. Springer, 2012. 1621 1 [15] J. Møller, “Random Johnson–Mehl tessellations,” Adv. Appl. Probab., 1622 1566 = 2λπ −λπr2 vol. 24, no. 4, pp. 814–844, 1992. 1623 π(1 − e ) 0 0 0 [16] E. W. Weisstein. Kolmogorov–Smirnov test. MathWorld—A Wol- 1624 (−λl(Bx(0)∪Bz(U))) 1567 × e xdudθdx. (64) fram Web Resource. [Online]. Available: http://mathworld.wolfram.com/ 1625 Kolmogorov-SmirnovTest.html 1626 AQ:3 2 1568 The last equality is because l(Bx (0)) = πr and l(Bx (0) ∪ [17] D. B. Johnson and D. A. Maltz, “Dynamic source routing in ad hoc 1627 2 2 2 wireless networks,” in Mobile Computing. 1996, pp. 153–181.1628 AQ:4 1569 Bz(U)) = ux sin(θ) + (π − θ)x + (π − f (θ, x))(u + x − (θ)) ( (θ, )) = √ u−x cos(θ) [18] V. Gupta and P. G. Harrison, “Fluid level in a reservoir with an on-off 1629 1570 2ux cos with cos f x . source,” SIGMETRICS Perform. Eval. Rev., vol. 36, no. 2, pp. 128–130, 1630 u2+x2−2ux cos(θ) 2008. 1631 [19] J. F. C. Kingman, Poisson Processes. Hoboken, NJ, USA: Wiley, 1993. 1632 1571 G. Proof of Theorem 9 [20] F. Morlot, “A population model based on a Poisson line tessellation,” in 1633 Proc. 10th Int. Symp. Modeling Optim. Mobile, Ad Hoc Wireless Netw. 1634 1572 Proof: Let Z be a geometric random variable given by: (WiOpt), May 2012, pp. 337–342. 1635 l [21] F. Baccelli, M. Klein, M. Lebourges, and S. Zuyev, “Stochastic geometry 1636 1573 P(Z = l) = (1 − α)α . (65) and architecture of communication networks,” Telecommun. Syst.,vol.7, 1637 nos. 1–3, pp. 209–227, 1997. 1638 1574 Then, the random variable T representing the time taken to 1575 download the file is the geometric sum of independent random 1576 variables given by Z (γ ) 1577 T = (Xi + I ) + Y. (66) i Pranav Madadi is a Ph.D. student at the Electrical and Computer Engineering 1639 i=0 department of The University of Texas at Austin. He is working under 1640 the supervision of Prof. Francois Baccelli and Prof. Gustavo de Veciana. 1641 1578 Thus, the Laplace transform of T is   He received his bachelor degree from Indian Institute of Technology, Hyder- 1642 abad in 2014. His primary research interest is in stochastic modeling of 1643 1579 L (s) = (1 − α) + (1 − α)αL (s)L (γ ) (s) + .. L (s) T X I Y wireless networks. He is currently working on analyzing the performance 1644 ( − α)L ( ) ( − α)L ( ) = 1 Y s = 1 Y s . of a mobile user moving in a Poisson cellular network. 1645 1580 λv (67) 1 − αL (s)L (γ ) (s) − αL ( ) 2 r X I 1 X s 2λvr+s (γ ) 1581 The last equality follows from the fact that I is exponen- 1582 tial with parameter 2λvr.

François Baccelli is Simons Math+X Chair in Mathematics and ECE at 1646 1583 REFERENCES UT Austin. His research directions are at the interface between Applied 1647 Mathematics (probability theory, stochastic geometry, dynamical systems) and 1648 1584 [1] Enterprise Mobile Data Report-Q3, Wandera, London, U.K., Oct. 2013. Communications (network science, information theory, wireless networks). 1649 1585 [2] F. Baccelli and B. Blaszczyszyn, “Spatial modeling of wireless commu- He is co-author of research monographs on point processes and queues (with 1650 1586 nicationsa stochastic geometry approach,” Foundations and Trends in P. Brémaud); max plus algebras and network dynamics (with G. Cohen, 1651 1587 Networking. Breda, The Netherlands: NOW Publishers, 2009. G. Olsder and J.P. Quadrat); stationary queuing networks (with P. Brémaud); 1652 1588 [3] T. Karagiannis, J.-Y. Le Boudec, and M. Vojnovic, “Power law and stochastic geometry and wireless networks (with B. Blaszczyszyn). Before 1653 1589 exponential decay of intercontact times between mobile devices,” IEEE joining UT Austin, he held positions in France, at INRIA, Ecole Normale 1654 1590 Trans. Mobile Comput., vol. 9, no. 10, pp. 1377–1390, Oct. 2010. Supérieure and Ecole Polytechnique. He received the France Télécom Prize 1655 1591 [4] V. Conan, J. Leguay, and T. Friedman, “Characterizing pairwise inter- of the French Academy of Sciences in 2002 and the ACM Sigmetrics 1656 1592 contact patterns in delay tolerant networks,” in Proc. 1st Int. Conf. Auton. Achievement Award in 2014. He is a co-recipient of the 2014 Stephen 1657 1593 Comput. Commun. Syst. (ICST), 2007, Art. no. 19. O. Rice Prize and of the Leonard G. Abraham Prize Awards of the IEEE 1658 1594 [5] H. Cai and D. Y. Eun, “Toward stochastic anatomy of inter-meeting Communications Theory Society. He is a member of the French Academy of 1659 1595 time distribution under general mobility models,” in Proc. 9th ACM Int. Sciences and part time researcher at INRIA. 1660 1596 Symp. Mobile Ad Hoc Netw. Comput., 2008, pp. 273–282. 1597 [6] S. Mitra et al., “Trajectory aware macro-cell planning for mobile users,” 1598 in Proc. IEEE Conf. Comput. Commun. (INFOCOM), Apr./May 2015, 1599 pp. 792–800. 1600 [7] H. Deng and I.-H. Hou, “Online scheduling for delayed mobile 1601 offloading,” in Proc. IEEE Conf. Comput. Commun. (INFOCOM), 1602 Apr./May 2015, pp. 1867–1875. Gustavo de Veciana (S’88–M’94–SM’01–F’09) received his B.S., M.S, and 1661 1603 [8] Z. Lu and G. De Veciana, “Optimizing stored video delivery for mobile Ph.D. in electrical engineering from the University of California at Berkeley 1662 1604 networks: The value of knowing the future,” in Proc. IEEE INFOCOM, in 1987, 1990, and 1993 respectively, and joined the Department of Electrical 1663 1605 Apr. 2013, pp. 2706–2714. and Computer Engineering where he is currently a Cullen Trust Professor 1664 1606 [9] J. Keilson, Markov Chain Modelsrarity and Exponentiality, vol. 28. of Engineering. He served as the Director and Associate Director of the 1665 AQ:1 1607 Springer, 2012. Wireless Networking and Communications Group (WNCG) at the University 1666 1608 Probability Approximations via the Poisson Clumping [10] D. Aldous, IEEE of TexasProof at Austin, from 2003-2007. His research focuses on the analysis 1667 1609 Heuristic, vol. 77. Springer, 2013. and design of communication and computing networks; data-driven decision- 1668 1610 [11] A. M. Makowski, “On a random sum formula for the busy period making in man-machine systems, and applied probability and queuing theory. 1669 1611 of the M/G/Infinity queue with applications,” DTIC Document, Dr. de Veciana served as editor and is currently serving as editor-at-large 1670 1612 Tech. Rep., 2001. [Online]. Available: http://www.researchgate.net/ for the IEEE/ACM TRANSACTIONS ON NETWORKING. He was the recipient 1671 1613 publication/235086381_On_a_Random_Sum_Formu%la_for_the_ of a National Science Foundation CAREER Award 1996 and a co-recipient 1672 AQ:2 1614 Busy_Period_of_the_MGInfinity_Queue_With_Applications of five best paper awards including: IEEE William McCalla Best ICCAD 1673 1615 [12] S. N. Chiu, D. Stoyan, W. S. Kendall, and J. Mecke, Stochastic Geometry Paper Award for 2000, Best Paper in ACM TODAES Jan 2002–2004, Best 1674 1616 and its Applications. Hoboken, NJ, USA: Wiley, 2013. Paper in ITC 2010, Best Paper in ACM MSWIM 2010, and Best Paper IEEE 1675 1617 [13] A. Okabe, B. Boots, K. Sugihara, and S. N. Chiu, Spatial Tessellations: INFOCOM 2014. In 2009 he was designated IEEE Fellow for his contributions 1676 1618 Concepts and Applications of Voronoi Diagrams. Hoboken, NJ, USA: to the analysis and design of communication networks. He currently serves 1677 1619 Wiley, 2009. on the board of trustees of IMDEA Networks Madrid. 1678 AUTHOR QUERIES AUTHOR PLEASE ANSWER ALL QUERIES

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IEEE Proof IEEE TRANSACTIONS ON INFORMATION THEORY 1 Shared Rate Process for Mobile Users in Poisson Networks and Applications

Pranav Madadi , François Baccelli, and Gustavo de Veciana, Fellow, IEEE

1 Abstract— This paper focuses on the modeling and analysis users moving through space. These are driven by geometric 35 2 of the temporal performance variation experienced by a mobile variations in the spatial and environmental relationships (asso- 36 3 user in a wireless network and its impact on system-level design. ciations) to the infrastructure, the presence of other users, 37 4 We consider a simple stochastic geometry model: the infrastruc- as well as random fluctuations intrinsic to wireless channels, 38 5 ture nodes are Poisson distributed while the user’s motion is the 6 simplest possible, i.e., constant velocity on a straight line. We first e.g., shadowing or fast fading. Our focus on temporal variabil- 39 7 characterize variations in the signal-to-noise ratio (SNR) process ity should be contrasted with the extensive work characterizing 40 8 and associated downlink Shannon rate, resulting from variations spatial variability as seen by randomly located users, e.g., 41 9 in the infrastructure geometry seen by the mobile. Specifically, through metrics such as coverage probability, spatial density of 42 10 by making a connection between stochastic geometry and queuing throughput, 90% quantile rate, “edge” capacity, spectral effi- 43 11 theory, the level crossings of the SNR process are shown to form 12 an alternating renewal process whose distribution is completely ciency, etc. Although space and time may be related through 44 13 characterized. For large/small SNR levels, and associated rare averages (when ergodicity holds), the temporal characteristics 45 14 events, we further derive simple distributional (exponential) of the stochastic processes modeling the capacity variations 46 15 models. We then characterize the second major contributor to of a typical mobile user are relatively unexplored beyond 47 16 such variations, namely, changes in the number of other users correlation analysis, and are of great practical interest. 48 17 sharing the infrastructure. Combining these two phenomena, 18 we study what are the dominant factors (infrastructure geom- Indeed, an increasing volume of data traffic is generated by 49 19 etry or sharing number) when a mobile experiences a very wireless devices on the move, e.g., 20-30% of cellular data 50 20 high/low shared rate. These results are then used to evaluate is generated during commute hours, [1]. In the future, with 51 21 and optimize the system-level quality of experience of the mobile increasing use of public transportation and/or the emergence 52 22 users sharing such a wireless infrastructure, including mobile of driverless cars, this volume could grow substantially. The 53 23 devices streaming video which proactively buffer content to 24 prevent rebuffering and mobiles which are downloading large Quality of Experience (QoE) seen by such users can be highly 54 25 files. Finally, we use simulation to assess the fidelity of this model dependent on the capacity variations they see as they move 55 26 and its robustness to factors which are presently not taken into through space. This is particularly the case for applications that 56 27 account. operate over longer time scales, e.g., video/audio streaming, 57 28 Index Terms— Poisson networks, wireless, signal-to-noise ratio, navigation/augmented reality, and real-time services, which 58 29 Shannon rate, mobile user, temporal variations, quality of may not be able to smooth substantial capacity variations 59 30 experience, stochastic geometry, queuing theory, level-crossings, through buffering, but are also for the opportunistic transfers 60 31 alternating renewal process, rarity and exponentiality. of large files over heterogeneous networks, e.g., WiFi offload- 61 ing. More broadly, increases in wireless network capacity 62 32 I. INTRODUCTION have been achieved through heterogeneous network densifi- 63

33 HE primary aim of this paper is to model and study cation leveraging technologies providing different coverage- 64 throughput trade offs, e.g., cellular (macro, pico, femto cells), 65 34 Tthe temporal capacity variations experienced by wireless WiFi and perhaps, in the future, mmWave access points. 66 Manuscript received September 12, 2016; revised July 5, 2017; accepted Characterizing and managing the capacity variations mobile 67 November 9, 2017. The work of P. Madadi was supported in part by the National Science Foundation, The University of Texas at Austin, under Grant users would see across such networks is a challenging but 68 NSF-CCF-1218338, in part by the Award from the Simons Foundation, The important problem towards understanding the efficiency and 69 University of Texas at Austin, under Grant 197982, and in part by NSF under performance of offloading/onloading based services. 70 Grant CNS-1343383 and Grant CNS-1731658. The work of F. Baccelli was supported in part by the National Science Foundation, The University of Texas at Austin, under Grant NSF-CCF-1218338, and in part by the Award from the A. Related Work 71 Simons Foundation, The University of Texas at Austin, under Grant 197982. There is a rich literature on modeling spatial capacity 72 The work of G. Veciana was supported in part by NSF under Grant CNS- 1343383 and Grant CNS-1731658. ThisIEEE paper was presented in part at the variabilityProof in wireless services experienced by a randomly 73 2016 Wi-Opt. located user. Of particular relevance is that based on stochastic 74 The authors are with the Department of Electrical and Computer Engi- geometry, which captures the effect of the variability in base 75 neering, The University of Texas at Austin, Austin, TX 78712 USA (e-mail: [email protected]; [email protected]; gus- station locations, as well as the variability in the environment 76 [email protected]). through shadowing, and in the channel through fading, see 77 Communicated by R. La, Associate Editor for Xxxx . e.g., [2] for a survey. In contrast, work studying the temporal 78 Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. capacity variations seen by a user on the move in a cellular 79 Digital Object Identifier 10.1109/TIT.2017.2781909 network is much more limited. 80 0018-9448 © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. 2 IEEE TRANSACTIONS ON INFORMATION THEORY

81 There is also significant related work on Delay-Tolerant Net- 4) Characterization of rare events. Conditional on a rare 137 82 works (DTN). This literature considers mobile nodes, where event, i.e., very poor or very good user rate, what is the 138 83 the contact duration and the inter-contact times are defined relative contribution of the user’s location vs network 139 84 and empirically measured from real traces [3], [4] as well as congestion? Can one characterize the time scales for rare 140 85 through underlying mobility models [5]. However, this work event occurrences? 141 86 is geared at studying opportunistic ad-hoc communication 5) Applications to QoE. What are the implications of 142 87 networks where the main question is the end to end delay of the temporal capacity variations mobiles see on their 143 88 a tagged packet, under the assumption that nodes are moving application-level QoE and system-level performance? 144 89 independently. Our focus is on studying the continuous-time For instance, what acceptable density of users streaming 145 90 stochastic process experienced by a tagged mobile user tra- video or downloading large files can be supported? 146

91 versing a static pattern of nodes e.g., base stations, modeling To the best of our knowledge the analysis of the SNR 147 92 the wireless infrastructure. and shared rate processes developed in this paper for Poisson 148 93 There certainly is a lot of interest in studying how to design wireless infrastructure is new and provides a first order answer 149 94 networks to better address the needs of mobile users or to to these basic questions. While our models are simplified, 150 95 leverage user mobility for the offloading/onloading traffic. For they should be viewed as a first and necessary step towards 151 96 example [6] studies how user mobility patterns and user per- characterizing temporal variability in more complex random 152 97 ceived QoE might drive the selection of macro-cell upgrades. structures, e.g., SINR variations which, in this paper, are 153 98 The work in [7] examines the effectiveness of algorithms for only explored via simulation for comparison to SNR process 154 99 optimizing offloading to a set of spatially distributed WiFi characteristics. Generalizations to SINR processes would be 155 100 APs. The work in [8] evaluates how proactive knowledge of desirable, particularly for systems operating in the interference 156 101 capacity variations could be used in designing new models limited regime. Such extensions may be tractable based on 157 102 for video delivery. These works exemplify applications and results known for shot noise fields (see [2, Volume 1]), yet 158 103 engineering problems which depend critically on the temporal are beyond the scope of this paper. 159 104 capacity variability that mobile users would experience, but C. Contributions and Organization 160 105 do not directly address the characteristics of such processes. Section II introduces the basic model studied in this paper. 161 We consider a tagged user moving at a fixed velocity along 162 106 B. Key Questions a straight line through a shared wireless network such that: 163 107 Two primary sources of temporal variation in a mobile’s (a) the access points/base stations form a realization of a 164 108 capacity are changes in the SNR which are associated with Poisson point process; (b) other users sharing the network 165 109 changes in the users geometric relationship to the infrastruc- also form a Poisson (or possibly Cox) point process; (c) all 166 110 ture, and the sharing number, i.e., the number of other mobiles users associate with the closest base station; (d) resources 167 111 sharing the resource. Our goal in this paper is to study the are shared equally with other users sharing the base station; 168 112 relative impacts these have on mobile users’ QoE. In this (e) a standard distance based path loss is used with neither 169 113 setting several basic questions arise: shadowing nor fading. Our goal is to characterize the shared 170 114 1) Characterization of the SNR process. As a mobile rate process seen by the tagged user by studying the SNR 171 115 user moves through a wireless network infrastructure and sharing number processes. 172 116 associating with the closest node, its SNR process and In Section III we characterize the SNR process, including 173 117 thus associated maximal rate (i.e., without accounting the intensity of peaks and valleys and, given a SNR threshold, 174 118 for sharing) will experience peaks and valleys. What is we provide a complete characterization of the on/off level 175 119 the intensity of peaks and valleys? Does it match the rate crossing process as an alternating-renewal process. Equiv- 176 120 of cell boundary crossings? Given an SNR threshold, alently, this characterizes the duration for coverage/outage 177 121 can one characterize the temporal characteristics of the events seen by the mobile. We then provide asymptotics for 178 122 on/off level crossing process associated with being above the likelihood of high and low SNR, and show that, after 179 123 and below the threshold, i.e., the coverage and outage appropriate rescaling, the time intervals between up crossings 180 124 durations? are exponential, which provides a new illustration of the “rarity 181 125 2) Characterization of the sharing number process. hence exponentiality” principle [9], [10]. 182 126 Assuming a population of other users sharing the net- Section IV provides a detailed discussion of the sharing 183 127 work, what are the characteristics of the sharing number number process, i.e., the number of other users which are 184

128 process seen by a mobile?IEEE If the network is shared by Proof 185 covered, i.e., meet an SNR threshold, and share infrastructure 129 heterogeneous users, i.e., static, pedestrian, and users on nodes with the tagged user. The process is somewhat complex 186 130 public transport and/or a road system, how will this bias and not independent of the SNR process; so we introduce 187 131 what a mobile user sees? a simpler bounding process which is used in Section V to 188 132 3) Smoothness of the effective rate process. How smooth characterize the shared rate process seen by our mobile. 189 133 is the bit rate obtained by the mobile seen as a function In particular, we highlight the contrast between the occurrence 190 134 of time? How often does this rate incur discontinuities, of rare events associated with high and low shared rates. 191 135 trend changes, large jumps or other phenomena nega- We also provide an asymptotic characterization for the 192 136 tively impacting real time applications? re-scaled inter-arrival times associated with up-crossings. 193 MADADI et al.: SHARED RATE PROCESS FOR MOBILE USERS IN POISSON NETWORKS AND APPLICATIONS 3

194 In Section VI we present a discussion of discrete event 195 simulation results which are used to assess the robustness 196 of this model to perturbations that were not yet taken into 197 account in the analysis. For example the relative impact of 198 variability associated with the changing geometry (proximity 199 of base stations) seen by a mobile versus that associated with 200 channel variability due to channel fades. We also consider the 201 robustness of our model to the changes in the direction of the 202 motion of the mobile user. 203 In Section VII we leverage our results to evaluate the QoE 204 experienced by mobile users in two concrete scenarios, which 205 give a system-level view on the performance that mobile users 206 would see. 207 Section VIII briefly discusses some important extensions of 208 our results to the case of heterogeneous wireless infrastructures 209 and the case where the locations of mobile users follow a Cox 210 process associated with a road network. The results show how Fig. 1. Mobile user motion in a sample of the Poisson cellular network. 211 heterogeneous technologies impact the temporal variations in 212 the mobile users SNR process. The results also show how a 213 mobile user on a roadway is likely to see poorer performance (L(t), t ≥ 0) be the process denoting the distance from the 250 214 than a static or pedestrian user. Section IX concludes the paper. mobile user to its closest node. 251 215 Here is the summary of our key contributions. We develop We consider downlink transmissions and assume that all 252 216 the framework to evaluate the temporal variations in the nodes transmit at a fixed power p. Based on the classical 253 217 shared rate seen by mobile users. This framework allows us power law path loss model and in the absence of fading 254 218 to derive the following new results on mobile user Quality and shadowing, the Signal-to-Noise Ratio (SNR) process, 255 219 of Experience (QoE): (1) a characterization of the outage (SNR(t), t ≥ 0), of the mobile user is 256 220 process seen by a mobile user requiring minimum rate, as an ( )−β 221 alternating renewal process, (2) an asymptotic characterization pL t SNR(t) = , for β>2, (1) 257 222 of the outage process for high rate requirements, and (3) the w 223 relative contributions of the two underlying factors, SNR and where w denotes the noise power. 258 224 sharing number process, towards the variability of the shared We assume that users are only served if their SNR exceeds 259 225 rate. We then illustrate the use of these results to study a given threshold γ . It follows from (1) that a user is served 260 226 the impact of such temporal variations on user’s QoE. This p 1/β if it is within distance rγ = ( wγ ) from the closest node. 261 227 includes a study of video streaming and file downloading for We refer to rγ as the radius of coverage. Note that a user may 262 228 mobile users. thus associate with a node but not be served. 263 (γ ) The Shannon rate process, (R (t), t ≥ 0), seen by the 264 229 II. SYSTEM MODEL mobile user is directly determined by the SNR process through 265 the relation: 266 230 Consider an infrastructure based wireless network consisting 231 of nodes e.g., base stations/WiFi hotspots, denoted through (γ ) a log (1 + SNR(t)) if L(t) ≤ rγ , 232 their locations on the Euclidean plane. The configuration of the R (t) = (2) 267 0otherwise, 233 nodes is assumed to be a realization of Poisson point process ={ , ,..} R2 λ 234 X1 X2 in with intensity . We consider a tagged where a is a constant depending on the available bandwidth. 268 235 v user moving at a fixed velocity along a straight line starting We assume that each infrastructure node shares time equally 269 236 = from the origin at time t 0. The mobile user shares the among the users it serves. We define the sharing number 270 (γ ) (γ ) 237 network with other (static) users which are spatially distributed process (N (t), t ≥ 0),whereN (t) is the number of 271 238 according to another independent Poisson point process of static users sharing the infrastructure with the mobile user if 272 ξ 239 intensity . it is served, and 0 if it is not served. In other words, if the 273 240 All users associate with the closest infrastructure node. For mobile user is served by its closest node X (t) at time t, 274 (γ ) 241 each X ∈ one can define a set of locations which are ( ) i IEEE it sharesProof the resource with N t static users. The sharing 275 \ (γ ) 242 closer to Xi than any other point in Xi .Thisisaconvex factor (F (t), t ≥ 0), is then defined by: 276 243 polygon known as the Voronoi cell associated with Xi [2]. (γ ) 1 244 The collection of such cells forms a tessellation of the plane F (t) = . (3) 277 + (γ )( ) 245 called the Voronoi tessellation see e.g., Fig.1. Thus, all the 1 N t

246 users located within the Voronoi cell of node Xi associate Finally, the shared rate process seen by the mobile user, 278 (γ ) 247 with Xi . (S (t), t ≥ 0),isgivenby 279 248 Let (X (t), t ≥ 0) denote the random process, where X (t) ∈ (γ )( ) = (γ )( ) × (γ )( ). 249 is the closest node to the mobile user at time t.Let S t R t F t (4) 280 4 IEEE TRANSACTIONS ON INFORMATION THEORY

281 Our aim is to characterize this process. In the next two 282 sections, we first study the two underlying processes namely: 283 (1) the SNR process (SNR(t), t ≥ 0) and its level sets, and (γ ) 284 (2) the sharing number process (N (t), t ≥ 0). 285 In the sequel, for any stationary random process, e.g., (γ ) (γ ) 286 (S (t), t ≥ 0),weletS represent a random variable whose 287 distribution is the stationary distribution of the associated 288 process. 289 A summary of the key notation is provided in Table I.

290 III. CHARACTERIZATION OF THE SNR PROCESS

291 This section is structured as follows: we start by considering 292 the on-off coverage structure of the SNR process and then 293 study in more detail the characteristics of its fluctuations. 294 We then analyze the scale of the inter-occurrence times of 295 certain rare events.

296 A. Analysis of the SNR Level Crossing Process

297 A first order question is whether the mobile is cov- 298 ered or not. To that end we define the SNR level crossing 299 process as follows: 300 Definition 1: Given an SNR threshold γ , the SNR level (γ ) (γ ) 301 crossing process (C (t), t ≥ 0) is defined as C (t) = 302 1(SNR(t) ≥ γ) as shown in Fig 2. 303 Clearly this is an on-off process, where the on and off Fig. 2. Level crossings of the SNR process as an on-off process (bottom) 304 periods correspond to coverage and outage periods respec- and the point process of its local maxima χ, denoted by “o” on the x axis, 305 tively. The process alternates between “on” intervals of length together with edge crossings, denoted by “×” (top). (γ ) (γ ) 306 (Bn , n ≥ 1) and “off” intervals of length (In , n ≥ 1) as We first show that the arrival process to the disc and thus the 332 307 depicted in Fig. 2. We also define the sequence of SNR up- (γ ) queue, is Poisson. We begin by proving that the arrival process 333 308 crossing times (Tn , n ≥ 1) for a given threshold γ .Let (γ ) (γ ) (γ ) has independent and stationary increments. The probability 334 309 ∼ − V Tn Tn−1 be a random variable whose distribution that there is an arrival in the next  seconds is the probability 335 (γ ) 310 is that associated with up-crossing inter arrivals as illustrated that there is a node in the set with area A = 2rγ v depicted 336 311 in Fig. 2. in Fig. 3. Since nodes are distributed according to a homo- 337 312 In order to characterize the SNR level crossing process, we geneous Poisson point process of intensity λ, the number of 338 313 establish a connection between the time-varying geometry seen nodes in any closed set of area b follows a Poisson distribution 339 314 by the mobile user and an associated queuing process. λ > (γ ) with parameter b. Thus, for any 0, the increments in the 340 315 Let D (t) denote the closed disc of radius rγ centered on arrival process have the same distribution. Also, the number 341 316 the mobile user’s location at time t. This closed disc follows (γ ) of nodes in any two disjoint closed sets are independent. 342 317 the mobile user’s motion along the straight line. Let K (t) = (γ ) Thus, the arrival process has independent stationary Poisson 343 318 |∩ D (t)| denote the number of infrastructure nodes in the increments. For a small value of , the probability that there 344 (γ ) 319 disc. is a single arrival is λA + o(). Thus, the arrival process is 345 320 The following theorem provides a simple characterization vλ (γ ) Poisson with rate 2rγ . 346 321 of (K (t), t ≥ 0) which in turn will enable the study of the Every node that enters the moving closed ball stays in it for 347 322 SNR level crossing process. (γ ) a time that depends on its entry locations and is proportional 348 323 Theorem 1: The process (K (t), t ≥ 0) is same as that to the chord length as shown in Fig. 3. This corresponds to 349 324 modeling the number of customers in an M/GI/∞ queue with (γ ) its service time in the queue. 350 325 arrival rate λ = 2rγ vλ and i.i.d. service times with density Since the mobile moves at constant velocity, the distribution 351 326 IEEE Proof ⎧ of the service times can be derived from the distribution of the 352 v2s ⎨  for s ∈[0, 2rγ /v], chord lengths, see e.g., 3. Without loss of generality suppose 353 2 2 2 2rγ 4rγ −v s 327 f (γ ) (s) = (5) the mobile moves along the x-axis. The y coordinate of a 354 W ⎩ (γ ) 0otherwise. typical node entering the disc, Y , is uniform on [−rγ , rγ ]. 355 (γ ) The random variable W representing the typical service 356 (γ )( ) 328 Proof: The entry of a node into the closed disc D t time can be represented as: 357 329 can be viewed as an arrival to the queue. The amount of time  (γ ) 2 − ( (γ ))2 330 spent by the node in D (t) corresponds to its service time 2 rγ Y (γ ) (γ ) ( ) W = . 358 331 and thus the exit from D t to its departure from the queue. v MADADI et al.: SHARED RATE PROCESS FOR MOBILE USERS IN POISSON NETWORKS AND APPLICATIONS 5

TABLE I TABLE OF NOTATION

Proof: Let us consider the M/GI/∞ queue defined in 384 Theorem 1. If the queue is idle there are no nodes within 385 a distance rγ of the tagged user. Thus, an off period of the 386 associated SNR level crossing process is equivalent to an idle 387 period of the queue. Similarly, the queue’s busy period is 388 equivalent to an on period. 389 Since the arrival process is Poisson, the inter arrival times 390 (γ ) are exponential with parameter λ = 2λvrγ . Because of 391 the memoryless property, all idle periods have the same 392 distribution. 393 ˆ (γ ) (γ ) Let B denote a random variable representing the forward 394 Fig. 3. The disc around the mobile user with area A and the chord along which the node moves. recurrence time associated with the on time defined as 395  ∞ P( ˆ (γ ) > ) = 1 P( (γ ) > ) . B x (γ ) B z dz (6) 396 E[B ] x (γ ) 359 The density of W is then given by (5), and has a mean Using the correspondence of the SNR level crossing process’s 397 E[ (γ )]= πrγ 360 W 2v . on times with the busy period of an M/GI/∞ queue, its 398 ( (γ )( ), ≥ ) 361 Thus, the process K t t 0 capturing the number distribution is that given in Theorem 12 in the Appendix A 399 (γ ) 362 of nodes in the moving disc follows the dynamics of the with service time S ∼ W and 400 363 number of customers in an M/GI/∞ queue with arrival rate (γ ) (γ ) (γ ) 2 (γ ) −ρ (γ ) ρ = rγ vλE[W ]=λπr ν = − e . 401 364 λ = 2rγ vλ and i.i.d service times following the distribution 2 γ , 1 (7) (γ ) (γ ) 365 of W . It follows that the stationary distribution for K (t) (γ ) (γ ) 2 The busy period B and idle period I are independent. 402 366 is Poisson with mean πrγ λ. Hence, the SNR level crossing process is an alternating- 403 367 Given this connection to an M/GI/∞ queuing model, renewal process. The mean of the busy period of the queue is 404 368 we can now characterize the SNR level crossing process as given by [11]: 405 369 an alternating renewal process defined as follows: ν(γ ) 370 Definition 2: A process alternating between successive E[ (γ )]= . B (γ ) (8) 406 371 on and off intervals is an alternating renewal process if λv γ ( − ν ) (γ ) 2 r 1 372 the sequences of on period (Bn , n ≥ 1) and off period (γ ) Thus, the probability that the SNR level crossing process is 407 ( , ≥ ) 2 373 In n 1 are independent sequences of i.i.d. non-negative −λπrγ “on” is 1 − e . 408 374 random variables. It is easy to see that all path loss functions which are 409 375 Theorem 2: For any γ>0, theIEEE SNR level crossing process, Proof (γ ) monotonic lead to an analogue of Theorem 2. 410 376 (C (t), t ≥ 0), is an alternating-renewal process. Further, its (γ ) (γ ) 377 typical on period, B , and off period, I , are distributed as

378 the busy and idle periods of an M/GI/∞ queue with arrival B. Fluctuations of the SNR Process 411 λ(γ ) = vλ 379 rate 2rγ and i.i.d. service times with distribution The SNR process is pathwise continuous and has continuous 412 (γ ) ∼ ( λv ) 380 given in (5). Thus, I exp 2 rγ and the busy period derivatives except at a countable set of times corresponding to 413 381 distribution can be explicitly characterized as in [11]. Also, Voronoi cell edge crossings where the mobile sees a hand- 414

382 in the stationary regime, the probability that the SNR level 415 2 off and where the first derivative of the SNR process is − −λπrγ 383 crossing process is “on” is 1 e . discontinuous. 416 6 IEEE TRANSACTIONS ON INFORMATION THEORY

The following theorem gives a characterization of the 454 asymptotic behavior of the distribution of the random variable 455 (γ ) V corresponding to the up-crossings as γ →∞and as 456 γ → 0, which can be seen as “good” and “bad” events 457 respectively. 458 Theorem 4: For all γ>0, the up-crossings of the SNR 459 (γ ) level crossing process, (Tn , n ≥ 1), constitute a renewal 460 (γ ) process. Let V be the typical inter arrival for this process, 461 and let f (γ ) = 2λvrγ ,thenasγ →∞, we have 462

(γ ) d f (γ )V −→ exp(1), 463

2 −λπrγ ⊥ ⊥ and for g(γ ) = λvrγ e ,asγ → , we have 464 Fig. 4. The disc B (Z ) around the projection point Z in two scenarios: 2 0 ⊥ h (a) where Z does not belong to the Voronoi cell of nucleus Z and (b) where (γ ) d it does. g(γ )V −→ exp(1), 465 d where −→ implies convergence in distribution. 466 v 417 Since the mobile is moving at a fixed velocity√ , the inten- Proof: The proof is given in the Appendix B. 467 418 sity of the cell-edge crossings is given by 4 v λ/π [12], [13]. The difference in the scales of the inter-event times of “good 468 419 As the mobile traverses a particular cell, there is also a time events” and “bad events” is due to the fact that the probability 469 420 where it is the closest to the associated node. The SNR process of occurrence of a bad event goes to zero faster than the 470 421 has a local maximum at these times. Perhaps surprisingly such probability of occurrence of good event. Since for bad events, 471 422 maxima may happen at the edge of the cell. Let us denote the as rγ tends to ∞, a disc of radius rγ should be empty, an event 472 −λπ 2 423 point process of times corresponding to interior maxima by rγ whose probability is e . Conversely, for good events, as rγ 473 424 χ. The intensity of χ is given in the following theorem. tends to 0, there should be at least one node in the disc of 474 2 425 Theorem 3: The intensity of the point process χ of the SNR −λπrγ √ radius rγ , an event whose probability is given by 1 − e . 475 426 maxima occurring in the interior of cells is v λ. 427 Proof: Let Z ∈ , be a typical associated node to the IV. SHARING NUMBER PROCESS 476 428 mobile user at any given time. Let us consider the projection ⊥ ⊥ 429 Z of Z onto the straight line (mobile user’s path). Then, Z , In contrast to the SNR or the Shannon rate process, which 477 ⊥ 430 is a point of χ if Z lies within the Voronoi cell of Z. Thus, take their values in the continuum, the sharing number process 478 431 the point process can be characterized by the fact that given is discrete valued and piece-wise constant. We start with 479 432 a node at a height h from the straight line, the closed ball evaluating the frequency of discontinuities and then introduce 480 ⊥ 433 Bh(Z ) of radius h centered at the projection point is empty an upper-bound process to be used in the sequel. We conclude 481 434 of nodes. To that extent, consider a rectangle of unit length this section with a study of rare events. 482 435 and height 2x, with the mobile user’s straight line path passing 436 through the center. From Mecke’s formula [14], the intensity A. Discontinuities 483 437 of the point process χ is   In the sequel, we will use the following simplified version 484 x −λπh2 1 of the Johnson-Mehl cell, see [12]. 485 438 μχ = v λ x e dh lim→∞ 2 ={ } x −x 2x Definition 3: Consider Xi the Poisson point process 486 √ √ √ 2 of nodes on R . For any threshold γ , the JohnsonÐMehl cell 487 439 = v lim λerf( λπx) = v λ. (9) →∞ (γ ) x J (Xi ) associated with Xi is the intersection of the Voronoi 488 440 cell of Xi w.r.t. with a disc of radius rγ centered at Xi ,see 489

441 Thus, the fraction of SNR maxima that happen in the Fig.5. 490 γ 442 interior of the cell is given by π/4. In other words, a proportion Thus for a fixed threshold , the mobile user is cov- 491 π − ered/served at time t if and only if it is within the Johnson– 492 443 1 4 of the SNR maxima seen by the mobile occur at the ( ) γ 493 444 cell edges. Mehl cell of its associated node X t for the radius r . We recall the following definition. 494 (γ ) Definition 4: The process (N (t), t ≥ 0) is defined as 495 445 C. Rare Events IEEE theProof number of static users present in the Johnson-Mehl cell 496 (γ )( ( )) (γ )( ( )) 446 In this subsection we consider certain rare events such as J X t if the mobile is in J X t and 0 otherwise. 497 ( (γ )( ), ≥ ) N 447 the occurrence of large SNR values. We show that the “rarity The process N t t 0 is an -valued piece-wise 498 448 hence exponentiality” principle [9], [10] applies here. We also constant process, with jumps at certain Johnson–Mehl cell 499 449 give the scale at which the inter-arrival times of high SNR edge crossings, see e.g., Fig 6. The value it assumes upon 500 450 are close to being exponentially distributed. As we shall see entering a cell is a conditionally Poisson random variable with 501 451 in Section VII, these asymptotic results can also be used a parameter that depends on the area of the cell in question. 502 452 for moderate values of SNR for settings typically found in The following theorem provides an upper bound for the jump 503 453 wireless networks. intensity: 504 MADADI et al.: SHARED RATE PROCESS FOR MOBILE USERS IN POISSON NETWORKS AND APPLICATIONS 7

(SNR(t), t ≥ 0), are not independent as is easily seen in 527 the case where γ = 0. For example, assume that the mobile 528 user experiences a low SNR, i.e., it is far from its associated 529 node. Then the area of the Voronoi cell is likely to be large. 530 This in turn implies that the mobile is more likely to share 531 its infrastructure node with a large number of users. 532 Fortunately, the sharing number process admits a simple 533 upper bound which is independent of the SNR process. This 534 upper bound is defined as follows: 535 Definition 5: We define an upper bound 536 ˆ (γ ) process,(N (t), t ≥ 0), on the sharing number process. Let 537 D(x, rγ ) denote a disc of radius rγ centered at location x. 538 ˆ (γ ) Let N (t), t ≥ 0, be the random process capturing the 539 number of static users which are in the disc D(X (t), rγ ) if 540 (γ ) the mobile is in J (X (t)),and0otherwise. 541 Fig. 5. Johnson-Mehl cells. ˆ (γ ) The stochastic process (N (t), t ≥ 0) enjoys most of the 542 (γ ) structural properties of (N (t), t ≥ 0); in particular, it is 543 piece-wise constant and one can derive natural bounds on the 544 intensity of its jumps. In addition and more importantly: 545 ˆ (γ ) (γ ) ˆ (γ ) • For all t ≥ 0, P(N (t) ≥ N (t)) = 1, i.e., N (t) 546 is an upper bound for the sharing number; this bound is 547 tight in the regime where the area of the typical Voronoi 548 1 cell ( λ ) is large compared to the area of the disc of radius 549 rγ ; 550 • When the mobile user is covered, the sharing number 551 ˆ (γ ) 2 N (t) is Poisson with parameter ξπrγ ; 552 ˆ (γ ) • The stochastic processes (N (t), t ≥ 0) and 553 (SNR(t), t ≥ 0) are independent. 554

V. S HARED RATE PROCESS 555 Fig. 6. Trace of the sharing number and shared rate processes. Recall that the shared rate is given by 556

(γ ) (γ ) (γ ) (γ ) 1 505 Theorem 5: An upper bound for the intensity of disconti- S (t) = R (t)F (t) = R (t) , (10) 557 N(γ )(t) + 506 nuities of the sharing number process is the intensity of the 1 (γ ) (γ ) 507 Johnson-Mehl cell edge crossings, which is given by: where R (t) is the Shannon rate defined in (2) and F (t) 558 508 √ the sharing factor at time t. A realization of this process is 559 v λ √ √ 2 illustrated in Fig.6. In the sequel, we use our upper bound 560 4 −λπrγ 509 (erf( λπrγ − 2 λrγ e ), (γ ) π process on N (t) to obtain a lower bound on the shared 561 rate: 562 ( ) = √2 z −t2 510 where, erf z π 0 e dt. ˆ(γ )( ) = (γ )( ) ˆ (γ )( ) = (γ )( ) 1 , 511 Proof: The intensity of cell edge crossings in a Johnson- S t R t F t R t (11) 563 Nˆ (γ )(t) + 1 512 Mehl tessellation is given in [15]. Not all cell edge crossings 513 lead to jumps as the number of static users can be the which is now a product of two independent random variables. 564 514 same across two adjacent Johnson-Mehl cells, thus the given 515 intensity is an upper bound. (γ ) A. Shared Rate Variability 565 516 Given that the value of N (t) is a non-zero constant, its The shared rate process is continuous and differentiable 566 517 time of constancy is defined as the typical amount of time it except for a countable number of points and a countable set 567 518 remains at that same constant. This time of constancy is lower of points of discontinuity for the first derivative. It is equal 568 519 bounded by the distribution of intersection of the motion’s line IEEE Proof 569 to zero when the mobile is not covered. The point process 520 with a typical Johnson–Mehl cell [15]. It is a lower bound only of discontinuities is upper bounded by the Johnson-Mehl cell 570 521 since the number of static users can be the same across two edge crossings (see Theorem 5). The point process of its local 571 522 adjacent Johnson-Mehl cells. maxima where the derivative is zero is the same as that of the 572 SNR process given in Theorem 3. 573 523 B. Upper Bound for the Sharing Number Process As can be seen from (11), the variability in the mobile user’s 574 524 Note that the two underlying processes that are used shared rate is driven by two processes: the Shannon rate and 575 525 to define the shared rate process, namely the sharing the sharing factor process. It is of interest to understand their 576 (γ ) 526 number process (N (t), t ≥ 0) and the SNR process relative contributions. To answer this question, let us consider 577 8 IEEE TRANSACTIONS ON INFORMATION THEORY

TABLE II Theorem 6: The likelihood of the rare events associated 609 ξ VALUES OF VARIANCE WITH INCREASING USER DENSITY, , with high shared rate is the same (up to a logarithmic 610 CONSTANT DENSITY OF NODES, λ = 25/π AND CONSTANT = equivalence) as that for the high SNR in the sense that 611 RADIUS OF COVERAGE rγ 500m 1 (γ ) lim − s log P(S > s) 612 s→∞ 1 = lim − s log (P(log(1 + SNR)>s)) 613 s→∞ = 2 . β (13) 614

Notice that as a direct corollary of the last theorem, 615

1 (γ ) TABLE III lim − s log P(S > s) 616 s→∞ VALUES OF VARIANCE WITH INCREASING NODE DENSITY, λ,CONSTANT DENSITY OF USERS, ξ = 50 AND CONSTANT 1 (γ ) (γ ) = lim − s log P(S > s|N = 0) 617 RADIUS OF COVERAGE rγ = 200m →∞ s 1 (γ ) (γ ) = lim − s log P(S > s, N = 0) 618 s→∞ = 2 . β (14) 619

For the rare events associated with a low shared rate, we will 620 ˆ(γ ) consider the lower bound process (S (t), t ≥ 0): 621 Theorem 7: The rare events associated with achieving low 622 shared rates are predominantly the same as the rare events 623 ˆ(γ ) associated with high sharing number given that the mobile is 624 578 the variance of S which can be written using the conditional at the covering cell edge, in the sense that for some sequence 625 579 variance formula as: −β {sn} such that sn log (1 + Krγ ) ∈ Z ∀n and limn→∞ sn → 626 ˆ(γ ) (γ ) ˆ (γ ) 2 ˆ (γ ) (γ ) 2 580 var(S )=var(R )E[F ] +var(F )(E[(R ) ]). (12) ∞, we have 627   1 (γ ) 1 581 Note that if the density of users increases, the variance of the ˆ lim − log P S < sn 628 n→∞ ( ) 582 shared factor decreases whereas the variance of the Shannon sn log sn  −β 583 rate remains constant. Thus, the variance of the shared rate 1 log(1 + Krγ ) 1 = lim − log P < 629 584 varies approximately linearly with a slope equal to the second →∞ ˆ (γ ) n sn log(sn) N + 1 s n 585 moment of the Shannon rate. By contrast if the density of −β = ( + ). 630 586 nodes increases, the variance of both the sharing factor and log 1 Krγ (15) 587 the Shannon rate vary making their relative contributions more Theorem 8: Conditioned on a very good or very bad shared 631 588 complex. rate, the relative contribution of the mobile user’s location and 632 589 Let us evaluate empirically what are the contributions of the network congestion is as follows: 633 590 the user and node density to the variability of shared rate. P( (γ ) > , (γ ) = | (γ ) > ) = , 591 For a given node density λ = 25/π and radius of coverage lim R s N 0 S s 1 (16) 634 s→∞ 592 rγ = 500m, by increasing the density of static users, the (γ ) (γ ) P(N > as log(1 + γ)− 1 | 0 < S < 1/s) = 1. (17) 635 593 variance of the shared rate decreases linearly with the variance

594 of the sharing factor. Table II gives numerical values evaluated Conditioned on a very high shared rate, the mobile user has 636 ξ = 595 via simulation. For a given static user density 50 to be close to the associated node with no other static users 637 596 and radius of coverage of 200m, by increasing the density sharing its resources. By contrast, conditioned on the mobile 638 597 of nodes the variance of the sharing factor monotonically being served and experiencing a very low shared rate, it has 639 598 increases, whereas the changes in the variance of SNR process to share its associated node with a number of users that is 640 599 is complicated due to the way it is defined. Thus, the changes inversely proportional to the desired shared rate. 641 600 in the variance of the sharing number process is difficult to From the previous theorem, we know that high shared rates 642 601 characterize with densification of nodes due to the complex IEEE are predominantlyProof the same as the event where the SNR is high 643 602 relation given in (12). Table III gives numerical values evalu- and the sharing number is zero. Thus, for a given threshold 644 603 ated by simulations. δ for the shared rate process, we can study the asymptotic 645 behavior of the distribution of the inter arrival time of shared 646 rate up-crossings as δ →∞. 647 604 B. Rare Events ˆ Corollary 1: Let Zδ denote the inter arrival time for the 648 605 Since the mobile user’s shared rate depends on two factors it ˆ(γ ) good events of the lower bound process (S (t), t ≥ 0) and 649 606 is of interest to understand their relative contributions towards let f (rδ) = 2λvrδ , then as δ → 0, we have 650 607 the events of high/low shared rate. The proofs of the following 2 −ξπrγ ˆ d 608 theorems are given in Appendix C and D respectively. e Zδ −→ exp(1). 651 MADADI et al.: SHARED RATE PROCESS FOR MOBILE USERS IN POISSON NETWORKS AND APPLICATIONS 9

ˆ = 652  Proof: The result follows from the fact that Zδ X rδ 653 i=1 V ,whereX is a geometric random variable with 2 −ξπrγ (rδ ) 654 parameter e and V the typical interval time of the 655 renewal process of up-crossings associated with the SNR 656 process of threshold δ.

657 VI. SIMULATION AND VALIDATION

658 In this section we evaluate the robustness of our mathemat- 659 ical model and associated asymptotic results to more realistic 660 settings. We use simulation to study the temporal variations of 661 the SNR process experienced by a mobile user under various 662 scenarios which are not captured by our analytical framework. 663 Our model is challenged in various complementary ways: e.g., 664 by adding fading, accounting for interference from other base 665 stations and relaxing the straight line mobility model. In each 666 case the objective is to determine to what degree our simplified Fig. 7. Q-Q plot for various thresholds. 667 mathematical model is still approximately valid, providing 668 robust engineering rules of thumb to predict what mobile users

669 see in practice. In the sequel we evaluate how quickly the convergence to 705 670 In particular, we will answer the following questions: exponential stated in Theorem 4 arises. To that end we com- 706

671 • How quickly do the SNR up crossings converge to pare the re-normalized distributions obtained via simulation to 707 672 exponential asymptotics as a function of the associated a reference exponential distribution with parameter 1 using the 708 673 thresholds? Kolmogorov-Smirnov (K-S) test. The K-S test finds the great- 709 674 • Are the results obtained robust to the presence of fast est discrepancy between the observed and expected cumulative 710 675 fading? frequencies– called the “D-statistic”. This is compared against 711 676 • Are there regimes where the temporal characteristics of the critical D-statistic for that sample size with 5% significance 712 677 the SNR process are good proxies for those of the SINR level. If the calculated D-statistic is less than the critical one, 713 678 process, e.g., high path loss? we conclude that the distribution is of the expected form, see 714 679 • Does the characterization of the SNR level crossing e.g. [16]. 715 680 process valid for other mobility models?

681 We begin by introducing our simulation methodology and the B. Convergence of Level-Crossing Asymptotics 716

682 default parameters used throughout this section. Theorem 4 shows that as the SNR threshold γ in dB 717 increases, the rescaled distribution for up-crossings of the SNR 718

γ 719 683 A. Simulation Methodology process becomes exponential. The question is how large needs to be for this result to hold. To that end we simulated 720 684 We initially consider a user moving on a straight line (road) the level crossing process for various γ and subjected it to the 721 685 at a fixed velocity of 16 m/s. The base stations are randomly K-S test as mentioned above. 722 686 placed according to a Poisson point process with intensity λ The Q-Q plots for up-crossing inter-arrivals rescaled by 723 687 such that the mean coverage area per base station is equal to f (rγ ) as introduced in the theorem are depicted in Fig. 7. 724 688 that of a disc with radius 200m. Unless otherwise specified, we As expected, as the threshold increases, the points become 725 689 consider the path loss function given by Eq. (1) with exponent linear, and for a threshold value of γ = 50 or more, it is 726 690 β = 4 and assume that all base stations transmit with equal accepted as an exponential with unit mean by the K-S test. 727 691 power p = 2W. The signal strength received by the mobile −2 In practice a SNR of 50 dB is not realistic for wireless users. 728 692 user is recomputed every 10 seconds. The total distance However, as seen from the Q-Q plots in Fig.7, for moderate 729 693 traveled by the mobile is 750 km along a straight line. The values of γ such as 0.1, the up-crossing inter-arrivals can be 730 694 number of simulation runs considered to evaluate the means well approximated by an exponential with parameter 1/f (rγ ). 731 695 is 1000. The number of handovers experienced by the mobile Further, the absolute error in the mean of the inter-arrival of 732 696 while traveling this distance is on average 4269. up-crossings for moderate value of γ = 1 is approximately 733 697 IEEE Proof We calibrate the thermal noise power to the cell-edge only 9.3 s. 734 698 user. Let L be a random variable denoting the distance of a 699 typical user to the closest base station. Define dedge by the C. Robustness of Level-Crossing Asymptotics to Fading 735 700 relation P(L ≤ dedge) = 0.9. Since in our simulation setting 2 − ln(0.1) Next we study the effect that channel fading might have 736 701 P(L ≤ l) = 1 − exp(λπl ),wehavededge = πλ .Ifwe on the level-crossing asymptotics. We consider channels with 737 702 fix the desired SNR at the cell edge to be SNRedge, this then −β pd Rayleigh fading with unit mean, so that the SNR experienced 738 w = edge 703 determines the noise power to be .Wesetthe by the tagged mobile user at a distance d from the base station 739 SNRedge −β 704 SNRedge to be 1 dB. is given by Hpd /w,whereH is a fading random variable 740 10 IEEE TRANSACTIONS ON INFORMATION THEORY

Fig. 8. SNR process in presence of fading with mean 1. Fig. 9. Threshold above which the inter-arrival of up-crossing converges to an exponential with parameter 1, for different variance of fading.

741 which is exponential with unit mean. The coherence time is 742 set to tc = 0.423/fd,where fd is the Doppler shift given by = v v 743 fd c fo where is the vehicle velocity, c is the speed of 744 light and fo = 900MHz is the operating frequency. This gives 745 a coherence time tc = 0.007 s. Thus, fading (power) changes 746 every 0.007 seconds. A realization of the SNR process with 747 fading is exhibited in Fig. 8. 748 If we assume no fast fading, the realizations of the SNR 749 process seen by a mobile are continuous and differentiable 750 and level crossings are well defined. However when the 751 fluctuations associated with fast fading are added to the model, 752 level crossings will exhibit bursts of up-crossings associated 753 with fast fading when the mean channel gains reach the 754 thresholds, see Fig. 8. Thus, in our simulations, we avoid these 755 bursts of up-crossings by considering the first up-crossing 756 and by suppressing all subsequent up crossings (if any) for 757 an appropriate time period. We take a time period for the Fig. 10. Threshold above which the inter-arrival of up-crossing converges 758 suppression of up-crossings equal to twice the expected on to an exponential with parameter 1, for different path loss exponent. (rγ ) 759 time of 2E[B ] [11]. 760 In order to vary the variance while keeping the mean of the To that end we simulated the SINR process for a setting 779 761 fading equal to 1, we now consider fading which is a mixture with a high path loss exponent of β = 4 and found once again 780 762 of exponentials. For this process, we would expect that for that the rescaled distribution for the up-crossing inter-arrivals 781 763 fading with mean one, if variance is small, the appropriately converges to an exponential with parameter 1. The test requires 782 764 rescaled inter-arrival distribution for up-crossings which are a threshold γ = 31.7dB. We then evaluated, for different 783 765 suppressed would once again be asymptotically exponen- path loss β, what threshold values are needed to obtain a 784 766 tial with parameter 1. In other words we expect geometric similar convergence. As shown in Fig. 10, the threshold in 785 767 variations associated with base station locations to dominate question increases as β decreases. Further we found that for 786 768 channel variations. Whereas, if the fading variance is high, one β<3.5, we no longer have the desired convergence property. 787 769 might expect the SNR threshold required to see convergence In summary, for high path-loss exponents β ∈[3.5, 4], the up- 788 770 to an exponential to increase. Fig. 9 exhibits such thresholds as crossing asymptotics for the SNR and SINR processes are 789 771 a function of the fading variance.IEEE As can be seen for fading Proof similar. 790 772 variances exceeding 8, the channel variations dominate the 773 geometric variations and thus up-crossing asymptotics differ

774 from those of Theorem 4. E. Robustness to Generalized Mobility Models 791

So far we have focused on a mobile moving strictly along 792 775 D. Robustness of Level-Crossing Asymptotics to Interference a straight line. Next we evaluate to what extent one can relax 793 776 So far we have focused on the SNR process. One might this requirement. To that end, we consider a variant of the 794 777 ask to what degree the Signal-to-Interference-plus-Noise random way point model, [17], where a user moves at a 795 778 Ratio (SINR) process, share similar characteristics. constant velocity, v meters per sec, but every t0 secs, with 796 MADADI et al.: SHARED RATE PROCESS FOR MOBILE USERS IN POISSON NETWORKS AND APPLICATIONS 11

cellular or WiFi network and (2) large file downloads using 835 WiFi. In both applications, we consider a service model where 836 the users are served only if the SNR experienced by them 837 is above a certain threshold. Thus, the users are served only 838 during the on period of the SNR process. We illustrate how the 839 characterization of the level-crossing process can be leveraged 840 to improve the QoE of the tagged user. 841

A. Stored Video Streaming to Mobile Users 842

Let us consider a scenario where mobile users are viewing 843 stored videos which are streamed over a sequence of wireless 844 downlinks. The users are distributed according to a Poisson 845 point process of density ξ and are moving independently of 846 each other. Consider a policy where a mobile user is served 847 Fig. 11. Three specific mobile user motion paths. by a node only if the SNR experienced is greater than a 848 threshold γ . Thus, the nodes serve the mobile users within 849 1/β the radius of coverage rγ = (p/(γ w)) . A lower threshold 850 797 probability p0, selects a new independent direction (angle) γ corresponds to a lower transmission rate (when served), 851 (−θ ,θ ) 798 uniformly distributed in 0 0 . a higher probability of coverage and sharing with a large 852 799 We conducted simulations where for a given SNR threshold, number of other mobile users. Conversely a higher threshold 853 γ θ v 800 , various values for the parameters, p0, 0, and t0 were implies higher transmission rate and sharing with fewer other 854 801 varied and the following quantities were measured: mobile users. For simplicity we consider rebuffering as the 855 802 • The means of the on-off periods; primary metric for user’s video QoE [8]. 856 803 • The independence of the on-off periods; The playback buffer state of the tagged mobile user moving 857 804 • The distribution of the up-crossing inter-arrivals. at a fixed velocity v on a straight line, can be modeled as a 858 fluid queue. The arrival rate to the queue alternates between 859 805 We considered three particular sets of values for the parameters (γ ) an average ergodic rate, h(γ ) = E[R |SNR >γ] and zero 860 806 (v, t0, p0,θ0), each reflecting a specific kind of traffic: (1) Car 861 807 on a highway: (20, 10, 0.001,π/45), (2) Car in an urban area: depending on whether the mobile is being served or not. Let η 862 808 (10, 1, 0.1,π/4) and (3) Wanderer: a pedestrian moving in a denote the video playback rate in bits. Hence, as long as the η 863 809 city: (1, 0.1, 0.6,π). Fig.11 illustrates a snapshot of the paths buffer is non-empty, the fluid depletion rate of the queue is . 864 810 of the three cases. Re-buffering of the video is directly linked to the proportion − 865 811 The mean of the on-off periods of the SNR process is of time the playback buffer is empty, which is given by 1 ρ(γ,ξ) ρ(γ,ξ) 866 812 within a 5 percent error margin in the case (1) but they do ,wheretheload factor, , [14] of the queue is:   813 not match for the other two cases for moderate values of the (γ ) ν h(γ ) 1 814 SNR threshold. However, for certain high threshold values, ρ(γ,ξ) = E |SNR >γ . (18) 867 η ˆ (γ ) + 815 γ ∈[20dB, 60dB] the mean coverage time for the case (2), N 1 816 is within an acceptable error margin of 10 percent. Further, (γ ) where ν is the probability that the alternating renewal 868 817 the mean on time for both the second and third cases is larger process associated with the arrival rate to the fluid queue is 869 818 than the mean on time in the case where the mobile is moving “on”. 870 819 along a straight line. The first natural question one can ask is whether there is a 871 820 The on-off periods are also highly correlated for the choice of γ such that the fluid queue is unstable, thus ensuring 872 821 later two cases, while independence approximately holds in no rebuffering in the long term. In other words, does there exist 873 822 case (1). The distribution of the up-crossing inter-arrivals for a a γ>0 such that ρ(γ,ξ) > 1? Given our policy and the QoE 874 823 specific high threshold value of γ = 50 dB is now considered. metric, the network provider may choose a lower value of 875 824 The convergence to an exponential of unit mean is only threshold,γ , as long as the typical mobile user in the long run 876 825 observed for the first case. In summary, the robustness of our experiences no rebuffering. 877 826 model is only guaranteed for slight changes in the direction. Now, for simplicity let us consider a constant mean trans- 878 827 Major changes in the direction have a strong effect on the 1 IEEE missionProof rate κ = a log (1 + γ ) E instead of the 879 (γ ) 828 characterization of on-off periods as an alternating renewal N +1 829 process with the proposed distribution. average ergodic rate. This constant bit rate is when the network 880 does not rely on adaptive coding/decoding. For additional 881 motivation for this scenario, see [8]. The load factor ρ(γ,ξ) 882 830 VII. APPLICATIONS is then given by 883 831 In this section we use our framework to evaluate  −b 832 application-level performance of mobiles in a shared wire- a log(1 + γ) 1 − e γ 2/β   833 less network. In particular we consider two very different ρ(γ,ξ) = E 1 , (γ ) (19) 884 834 wireless scenarios: (1) video streaming to mobiles sharing a η N + 1 12 IEEE TRANSACTIONS ON INFORMATION THEORY

and κ>η, the fluid queue alternates between busy and 901

idle period representing the periods when the video is un- 902 interrupted or frozen, respectively. The distribution of the on 903 (γ ) (γ ) periods B and that of the off periods I of the M/GI/∞ 904

queue discussed in Section determine the distribution of the 905 busy period B f and that of the idle period I f of the fluid 906 queue. When denoting by κ the constant input rate during on 907

period and by η the constant output rate when the queue is 908 non-empty, the Laplace transform of B f is given by [18] 909

L ( ) = L (γ ) ( σ + λ(σ − )( − L ( ))), B f s B s 1 1 B f s (20) 910 (γ ) where σ = κ/η > 1, and the idle period I f has an exp(λ ) 911 distribution. 912

B. WiFi Offloading 913

WiFi offloading helps to improve spectrum efficiency and 914 Fig. 12. The load factor of the fluid queue for a single user (top curve) and reduce cellular network congestion. One version of this 915 for a positive density of users as a function of γ (curves below the top curve). scheme is to have mobile users opportunistically obtain 916 For the latter curves, the number of users per base station are ξ/λ = 1, 4and 10 from top to bottom. Here b = 1. All functions can be multiplied by an data through WiFi rather than through the cellular network. 917 arbitrary positive constant when playing with a and η. Offloading traffic through WiFi has been shown to be an 918 effective way to reduce the traffic on the cellular network, 919 when available WiFi is faster and uses less energy to transmit 920 the data. 921 Let us consider a scenario where the mobile users download 922 a large file from the service provider, relying on WiFi hotspots, 923 distributed according to a Poisson point process of intensity 924 λ, rather than from the cellular base stations. The users are 925 distributed according to a Poisson point process of density ξ 926 and are moving independently of each other. Assume that the 927 WiFi hotspots have a fixed coverage area, i.e., the mobile user 928 connects to WiFi only if it is within a certain distance r from 929 the hotspot. Thus, higher the density of hotspots, λ,deployed 930 by the provider the better the performance experienced by 931 mobile users which rely only on them. We consider the time 932 it takes to complete the download as the primary metric for 933 user’s quality of experience. 934 Consider again the case without adaptive coding/decoding 935 ∗ Fig. 13. Level set curve of ρ(γ ,ξ)= 1 for an arbitrary positive constants and the tagged mobile user moving on a straight line at 936 a and η. constant velocity v. Then, the shared rate experienced by the 937 1 mobile user is the constant κ = a log (1 + γ ) E as 938 N(γ )+1   2 p β (γ ) defined above. In addition, the mobile user experiences an 939 885 where b = λπ w and E[1/(N + 1)] can be calculated alternating on and off process, as characterized in Theorem 2. 940 886 by numerical integration as described in the section VIII A. Below, for the sake of mathematical simplicity, we assume 941 887 It is easy to check that the function ρ(γ,ξ) has a unique ∗ ∗ that the file size F is exponential with parameter δ and that the 942 888 maximum γ on (0, ∞). A plot of (19) and the value of γ mobile user starts to download the file at the beginning of an 943 889 are exhibited in Fig. 12. Notice that for these parameters, the ∗ on period. Let T be a random variable denoting the time taken 944 890 value of γ increases with ξ. (λ) to download the file. Consider the event J ={F >κB }, 945 891 For a given base station density λ and density of users ξ, (λ) ∗ where B is a random variable distributed like a typical on 946 892 IEEEγ Proof one can evaluate the SNR threshold value for which the period seen by the mobile (see Theorem 2) independent of F 947 893 load factor ρ is maximum. Fig.13 illustrates the level set curve ∗ and let 948 894 of ρ(γ ,ξ) = 1 for various values of λ and ξ. In this setup, η α = P( ) = P( >κ (λ)) = L (δκ). 895 given the video consumption rate , it is possible to answer J F B B(λ) (21) 949 896 questions like what is the minimum density of base stations Now, define the non-negative random variable X as the busy 950 897 required to serve a certain density of users such that in the long (λ) period, B , conditioned on event J, whose c.d.f. is given by: 951 898 term the video streaming is uninterrupted for all the users.  γ x 899 Remark 1: In the case where there exists no threshold for P( < ) = 1 −δκz ( ) X x e fB(λ) z dz (22) 952 900 which the condition for no long term rebuffering is satisfied α 0 MADADI et al.: SHARED RATE PROCESS FOR MOBILE USERS IN POISSON NETWORKS AND APPLICATIONS 13

(λ) 953 Conditioned on the event, {F ≤ κ B },letY denote the straight line. Let us define the mobile sharing number process 1002 F (γ ) 954 time taken to download the file, i.e., κ . The c.d.f. of the non- (Np (t), t ≥ 0) as the number of users sharing the node 1003 955 negative random variable Y is then given by: associated with the tagged user when it is served and zero 1004  yκ otherwise. Thus, at any given time t, the tagged user shares 1005 P( < ) = 1 δ −δzP( (λ) > z ) . (γ ) 956 Y y e B dz (23) its resources with a random number of users Np (t) which 1006 1 − α 0 κ is Poisson with a parameter depending on the area of the 1007 957 Notice that Johnson-Mehl cell. 1008 1 L ( ) = L ( + δκ), For simplicity, we define the shared rate process 1009 958 X s B(λ) s (24) (γ ) α (S (t), t ≥ ) 1010  ∞ c 0 as 1 −y (δκ+s) (λ) ⎧   959 L (s) = δκe P(B > y)dy. (25) Y − α ⎨ 1 1 0 (γ ) a log (1 + γ ) E (γ ) if L(t) ≤ rγ , + Sc (t) = Np 1 (27) 1011 960 The following representation of the Laplace transform of T is ⎩ 0otherwise. 961 an immediate corollary of the on-off structure:

962 Theorem 9: Consider a network of WiFi hotspots distrib- Since the distribution of the area of the Voronoi cell is 1012 λ (γ ) 963 uted according to a Poisson point process of intensity and unknown, we approximate Np to be Poisson with parameter 1013 ˆ(γ ) ˆ(γ ) 964 radius of coverage r shared by mobile users distributed inde- ξE[J ],whereJ denotes the area of the Johnson-Mehl 1014 ξ 965 pendently according to a Poisson point process of intensity . cell of radius rγ , conditioned on the fact that the tagged 1015 966 Assuming that the tagged mobile user starts to download the user is within a distance rγ from the associated node, which 1016 967 file at the beginning of an on period, the Laplace transform of introduces an additional bias. We can compute the expectation 1017 ∼ (δ) 968 the time taken to download a file of size F exp is given with the help of integral geometry as shown in Appendix E. 1018 969 by We found the value of the expectation using numerical inte- 1019 (γ ) ( − α)L ( ) gration and compared the mean number of users E[N ]= 1020 L ( ) = 1 Y s . p 970 T s (26) ξE[ ˆ(γ )] − αL ( ) 2λvr J with the sample mean obtained from simulation for 1021 1 X s 2λvr+s various parameters λ,rγ . To validate the approximation of 1022 (γ ) 971 Proof: Proof is given in the Appendix G. ˆ(γ ) Np by a Poisson random variable with parameter ξE[J ], 1023 972 It is remarkable that the Laplace transform of T admits a (γ ) (λ) we compared the value of E[1/(Np + 1)] calculated using 1024 973 quite simple expression in terms of that of B . Other and numerical integration with that from simulations. We found 1025 974 more general file distributions can be handled as well when that the calculated value of the expectation is within the 95% 1026 975 using classical tools of Laplace transform theory. Note that this confidence interval of the simulated mean. 1027 976 setting also leads to interesting optimization questions such as 2) Cox Process: Let us consider a population model where 1028 977 the optimal density of WiFi hotspots needed to be deployed roads are distributed according to a Poisson line process of 1029 978 for lower expected download times. 2 intensity λr on R [19]. Then independently on each road, 1030 we consider users distributed according to a stationary Poisson 1031 979 VIII. VARIANTS point process of intensity λt . This is known as a Cox process 1032 980 In this section, we consider two generalizations of our and we denote it by u [19]. This model can be used to 1033 981 framework: (1) we move from sharing with static users to represent a car motion on a road network. 1034 982 sharing with mobile users, and (2) we move from homoge- For a line L of the line process, let us denote the orthogonal 1035 983 neous to heterogeneous wireless infrastructures. projection of the origin O on L by (θ, r) in polar coordinates. 1036 For θ ∈[0,π) and r ∈ R, (θ, r) is unique. Thus, a Poisson 1037 984 A. Sharing the Network With Mobile Users line process with intensity λr is the image of a Poisson point 1038 985 Until here we have considered a tagged mobile user sharing process with the same intensity on half-cylinder [0,π)× R. 1039 986 the network with static users. We now consider two scenarios: Suppose all users on the roads are moving arbitrarily 1040 987 (1) that where the other users are mobile and are initially but independently from each other. Thus, at any instant the 1041 988 distributed according to a homogeneous Poisson point process, distribution of users on a given road remains Poisson. Consider 1042 989 and (2) that where they are mobile but restricted to a random a tagged user moving along a given road, then we have the 1043 990 road network, i.e., form a Cox process. following theorem by [20]. 1044 991 1) Homogeneous Poisson Case: Consider the case where Theorem 10: u is stationary, isotropic, with intensity 1045 992 other users sharing the network are initially located according πλr λt . From the point of view of the tagged user i.e., under 1046

993 IEEEξ Proof 1047 to a homogeneous Poisson point process of intensity ,and its Palm distribution, the point process is the union of three 994 subsequently exhibit arbitrary independent motion. This is a counting measures: (1) the atom at the origin, O, (2) an 1048 995 possible model for pedestrian motion. It follows from the independent λt -Poisson point process on a line through O with 1049 996 displacement theorem for Poisson point processes [2] that the a uniform independent angle and (3) the stationary counting 1050 997 other users at any time instant will remain a Poisson point measure u. 1051 998 process of intensity ξ. Following our previous framework, let us define an another 1052 ( (γ )( ), ≥ ) 999 As considered before, the users are served only if they are sharing number process Nd t t 0 as the number of 1053 1000 at a distance less than rγ from the closest node. Consider users sharing the node associated with the tagged user when 1054 1001 a tagged mobile user moving at a fixed velocity along a it is served and zero otherwise. 1055 14 IEEE TRANSACTIONS ON INFORMATION THEORY

(γ ) 1056 Evaluation of the mean of Nd . Suppose the tagged user Note that the SNR level crossing process as previously 1109 1057 is at the origin O. Let the associated node X (t) be at a defined is again an alternating renewal process which now 1110 1058 distance x from the origin. Let d(0,θ) be a line through the depends on the heterogeneous resource deployment. 1111 1059 origin with θ uniform from [0,π). From the aforementioned Theorem 11: For heterogeneous networks with preferential 1112 (γ ) 1060 theorem, the number of sharing users Nd can be split into association to micro base stations, the probability that the 1113 1061 two terms: Ns denoting the number of sharing users from stationary SNR level crossing process seen by a tagged user 1114 2 ˆ 2 −π(λrγ +λrˆγ ) 1062 stationary u and Nl denoting the number of sharing users is “on” is 1 − e . Also, the mean time for which 1115 ( ,θ) 1063 on the line d 0 . the process is “on” is given by 1116 1064 Given any convex body Z, N (Z) denotes the number of s 2 ˆ 2 π(λrγ +λrˆγ ) − 1065 sharing users from stationary u present in Z,andE[Ns (Z)] e 1 ˆ(γ ) . (29) 1117 1066 πλ λ ( ) ˆ is given by r t area Z [21]. Let J denote the area as 2v(λrγ + λrˆγ ) 1067 defined before. Thus, ˆ(γ ) Proof: In order to characterize the SNR level crossing 1118 1068 E[N ]=πλ λ E[J ], s r t process, we establish a connection to a Boolean model. 1119 ˆ(γ ) 1069 where, E[J ] is evaluated using integral geometry (see Assume that the mobile user is moving with unit velocity. 1120 ( , ) 1070 Appendix E). Let B Xi rγ denote the closed ball of radius rγ centered at 1121 ( ˆ , ˆ ) ˆ ˆ 1071 Now, let l(0,θ) denote the length of the line d(0,θ) in the Xi and B Xi rγ a closed ball of radius rγ centered at Xi . 1122 ˆ(γ ) 1072 area J . Then, N , the number of sharing users on the line The union of all these closed balls forms a Boolean model 1123 l   1073 d(0,θ), is Poisson with parameter λ l(0,θ). Thus, E[N ]= t l ˆ λ E[ ( ,θ)] E = ∪ ∈ B(X , rγ ) ∪ ∪ ˆ ˆ B(X , rˆγ ) . (30) 1124 1074 t l 0 . Xi i Xi ∈ i 1075 One can evaluate E[l(0,θ)] using integral geometry (see  1076 Appendix F). We have, Now, assume that E is intersected by the directed line l.Note 1125 (γ ) E[ ]=E[ ]+E[ ]=πλ λ E[ ˆ(γ )]+λ E[ ( ,θ)]. that the Boolean model under consideration has independent 1126 1077 Nd Ns Nl r t J t l 0  convex grains and thus the intersection E ∩ l yields an 1127 1078 (28) alternating sequence of “on” and “off” periods which are 1128 (γ ) (γ ) 1079 Thus, the mean number of users sharing the tagged user’s independent. Let Bh and Ih be random variables denoting 1129 1080 association node is larger when the users are distributed the length of a typical on and off periods respectively. 1130 (γ ) 1081 according to a Cox process than when the users are distributed The distribution of the length of off period Ih is easy 1131 1082 according to a Poisson point process, assuming that both have to establish using the contact distribution functions and is 1132 ∗ 1083 the same mean spatial intensity. exponential with parameter λ [12]: 1133

∗ −λ∗l f (γ ) (l) = λ e , 1134 1084 B. Mixture of Pedestrian and Road Network Ih 1085 Suppose now we have two types of users : drivers who stay ∗ ˆ (γ ) (γ ) where, λ is 2(λ+λ)E[R ].HereR is the random variable 1135 1086 on roads, and pedestrians which are unconstrained. If pedestri- h h denoting the radius of the closed ball and is given by 1136 1087 ans are supposed to follow a Poisson point process, from their ⎧ 1088 point of view, the number of sharing users corresponds to the λ ⎨ γ , (γ ) r w.p. λ+λˆ 1089 sum of a stationary Poisson point process and a stationary = Rh λˆ (31) 1137 ⎩ˆγ . 1090 Poisson line process. On the other hand, from a driver’s point r w.p. λ+λˆ 1091 of view, the number of sharing users corresponds to the same Thus, the mean off period under the assumption of unit 1138 1092 sum, but in addition with a Poisson point process on a road velocity is 1139 1093 the driver is taking. Thus, the mean number of sharing users 1094 is always greater for drivers. In other words, pedestrians are (γ ) 1 1 E[I ]= = . 1140 1095 likely to share their infrastructure node with fewer users than h ∗ ˆ λ 2(λrγ + λrˆγ ) 1096 drivers.

Now, in the stationary regime, the probability that the 1141

1097 C. Heterogeneous Networks mobile user is either within a distance rˆγ from a micro-BS or a 1142 ˆ distance rγ from a macro-BS i.e., it’s SNR level crossing 1143 1098 Let us consider a deployment of micro-base stations = ˆ ˆ process is “on”, is given by the volume fraction [12]: 1144 1099 {X1, X2,..} distributed according to some homogeneous Pois- ˆ 1100 λ IEEE Proofˆ ¯ son process of intensity independent of the existing macro- (−(λ+λ)V ) c = 1 − e , (32) 1145 1101 base stations ={X1, X2,..}. Assume that all micro-BS 1102 transmit at a fixed power pˆ. where 1146 1103 Let us consider a mobile user moving at a fixed velocity v γ π(λr 2 + λˆrˆ2) 1104 along a straight line. For a given SNR threshold , the mobile ¯ = πE[( (γ ))2]= γ γ . V Rh (33) 1147 1105 is served by a micro-base station if its distance from its closest λ + λˆ 1/β 1106 micro-BS is less than rˆγ = (pˆ/wγ ) . Otherwise it is served 2 ˆ 2 −π(λrγ +λrˆγ ) 1107 by a macro-base station provided its distance from the closest Thus, c = 1 − e . The probability evaluated above 1148 1108 macro-BS is less than rγ . does not depend on the velocity of the mobile user and thus 1149 MADADI et al.: SHARED RATE PROCESS FOR MOBILE USERS IN POISSON NETWORKS AND APPLICATIONS 15

Fig. 14. Comparing mean-on time for heterogeneous and homogeneous Fig. 15. Comparing volume fraction for heterogeneous and homogeneous networks. networks.

v E[ (γ )] 1150 holds for any constant velocity . The mean on period Bh no rebuffering in the long term. Let the event Q denote that 1182 1151 can be evaluated using the following relation the tagged user is served, then 1183  (γ )   E[B ] a log(1 + γ)P(Q) 1 = h , ρ(γ,ξ) = E |Q . (34) 1184 1152 c (γ ) (γ ) η (γ ) E[ ]+E[ ] Np + 1 Bh Ih Assuming our approximation is valid, we need to find 1185 1153 which results in 1 2 ˆ 2 1186 π(λrγ +λrˆγ ) E (γ ) in the case of heterogeneous network. Let G and (γ ) e − 1 Np +1 1154 E[B ]= . h ˆ H be the events denoting that the tagged user is served by 1187 2(λrγ + λrˆγ ) micro BS and macro BS respectively, then 1188   1155 Since, the mobile user is moving with a fixed velocity v, E 1 | 1156 the mean time for which the mobile user is “on” is given by (γ ) Q 1189 E[ (γ )] N + 1 Bh p      1157 v . 1 1 1158 E (γ ) |G P(G) + E (γ ) |H P(H ) Np +1 Np +1 1159 These results provide an analytical characterization of the = . P( ) (35) 1190 1160 impact of heterogeneous densification on the mobile user’s Q

1161 temporal performance. Let us now compare the performance Since, the micro and macro base stations are distributed 1191 1162 improvement seen by the mobile user in a heterogeneous independently, the mobile user experiences two independent 1192 1163 network as compared to that of a homogeneous network. alternating renewal processes. Thus, in the stationary regime, 1193 1164 The graphs in Fig. 14 and Fig. 15 illustrate the difference the probability that mobile is served by macro BS is the 1194 1165 in the expected on-times and volume fraction of the networks product of the probabilities that the mobile is in “on” period of 1195 1166 respectively. alternating renewal process of macro BS and in “off” period 1196 1167 Notice that as micro-base stations are added, the expected of that of micro BS which gives: 1197 1168 on time decreases and later increases. The initial decrease ˆ 2 ˆ 2 2 −λπrˆγ −λπrˆγ −λπrγ 1169 is due to the inclusion of many relatively small length P(G) = 1 − e , P(H ) = e (1 − e ). (36) 1198 1170 on-times resulting from the micro-base stations in the voids ˆ(γ ) ˆ(γ ) Let J and J denote the area similar to that of what 1199 1171 of the homogeneous network. However, the volume fraction 1 2 we considered before. Given the mobile is served by macro 1200 1172 increases monotonically with the addition of micro-base BS, we need to consider the users which are within the area 1201 1173 stations. ˆ(γ ) J1 excluding the area covered by micro BS in this area. The 1202 ˆ(γ ) IEEE areaProof covered by the micro BS in J is approximated to be 1203 1 (ˆ ) γ (ˆ ) 1174 D. Stored Video Streaming in Heterogeneous Networks ν rγ × E[ ˆ ] ν rγ J1 ,where is the fraction of space associated 1204 1175 Consider a scenario as introduced earlier where mobile users with micro-BS as given by (7). Then 1205   (γ ) 1176 are viewing a video being streamed over a sequence of wireless ξE[Jˆ ] 1 1 − e 2 1177 downlinks, but now served by a heterogeneous network with E |G = , 1206 (γ ) + ξE[ ˆ(γ )] 1178 micro and macro BS. In this setting we once again consider Np 1 J2   (γ ) 1179 a fluid queue representing the tagged mobile user’s playback ξE[Jˆ ](λπˆ rˆ2 ) 1 1 − e 1 γ 1180 buffer state similar to Section VII A, and ask whether there is a E | = . (γ ) H (γ ) 1207 γ + ξE[ ˆ ](λπˆ ˆ2) 1181 choice of such that the fluid queue is unstable, thus ensuring Np 1 J1 rγ 16 IEEE TRANSACTIONS ON INFORMATION THEORY

deployed by the operator to serve a certain density of users, 1227 given the density of macro-BS λ such that the video streaming 1228 is uninterrupted for all the users. Thus, operators evaluate the 1229 cost incurred to serve a higher density of users by deploying 1230 micro-BS and in case the cost incurred is higher, an operator 1231 might consider other technologies. 1232 Remark 2: In heterogeneous networks, the interference 1233

from macro-BS to micro-BS is of major concern. One can 1234 introduce blanking, to reduce the effect of such interference 1235

i.e., assume that with certain probability f micro-BS transmit 1236 and with probability 1 − f macro BS transmit. 1237 Also, by considering heterogeneous networks with cellular 1238

BS and WiFi hotspots, there is no such problem of interference 1239 since both cellular BS and WiFi hotspots operate at different 1240

frequencies. 1241

∗ Fig. 16. γ with increase in micro- BS density for a certain fixed density of IX. CONCLUSION 1242 macro-BS for an arbitrary positive constants a and η in heterogeneous case. As explained in the introduction, the analysis of temporal 1243 variations of the shared rate experienced by a mobile user 1244 requires the characterization of both the functional distribution 1245 of a continuous time, continuous space stochastic process (rate 1246 process) constructed on a random spatial structure (e.g., 1247 the Poisson Voronoi tessellation) and of another stochastic 1248 process (sharing number process) constructed on a random 1249 distribution of static users. This paper addressed the simplest 1250 question of this type by focusing on the underlying SNR 1251 process and the sharing number process in the absence of 1252 fading. This allowed us to derive an exact representation of 1253 the level crossings of the stochastic process of interest as 1254 an alternating renewal process with a full characterization of 1255 the involved distributions and of the asymptotic behavior of 1256 rare events. The simplicity and the closed form nature of 1257 this mathematical picture are probably the most important 1258 observations of the paper. We also showed by simulation 1259 ∗ that this very simple model provides a good representation 1260 Fig. 17. Level set curve of ρ(γ ,ξ)= 1 for an arbitrary positive constants a and η in heterogeneous case. of the salient characteristics which happen for more complex 1261 systems, such as those with fading when the fading variance is 1262 small enough, or those based on SINR rather than SNR when 1263 the path loss exponent is large enough or those with changes 1264 1208 For a given density of macro-BS λ, the density of micro- ˆ in the direction of motion, when the change in direction is 1265 1209 BS λ and the density of users ξ, we can evaluate the SNR ∗ small. This model is hence of potential practical use as is, 1266 1210 value γ for which the load factor ρ(γ,ξ) given in (34) is ∗ in addition to being a first glimpse at a set of new research 1267 1211 maximum. Fig.16 illustrates the optimal SNR value (γ ) with ˆ questions. 1268 1212 varying density of micro BS λ. Notice that the optimal gamma 1213 value initially decreases with the density of micro-BS. X. FUTURE WORK 1269 1214 Since in our setting the micro BS have smaller transmission 1215 power, for a given threshold γ , the radius of coverage for The most challenging questions on the mathematical side 1270 1216 micro BS is smaller than that of the macro BS i.e., rˆγ < are as follows: (1) The understanding of the tension between 1271 1217 rγ . Since the micro-BS are given higher preference, with the randomness coming from geometry (studied in the present 1272

1218 IEEE Proof 1273 an increase in their density, the optimal threshold decreases paper), from sharing the network with other users (also studied 1219 in order to increase their coverage. Also, the cost incurred in the present paper) and that coming from propagation (only 1274 1220 by increasing coverage of macro-BS is compensated by the studied by simulation here): it would be nice to analytically 1275 1221 increased density and coverage of micro-BS till a certain quantify when one dominates the other. (2) The extension 1276 1222 density. of the analysis to SINR processes, which is our long term 1277 ∗ 1223 Fig.17 illustrates the level set curve of ρ(γ ,ξ) = 1for aim and will require significantly more sophisticated mathe- 1278 ˆ 1224 various values of λ and ξ for a given density of macro-BS λ. matical tools (e.g. based on functional distributions of shot 1279 1225 Given this setup, it is possible to answer questions like what is noise fields), than those used in the present paper. (3) The 1280 1226 the minimum density of micro-base stations that needs to be characterization of the distribution of the shared rate process. 1281 MADADI et al.: SHARED RATE PROCESS FOR MOBILE USERS IN POISSON NETWORKS AND APPLICATIONS 17

1282 (4) The extension the case where all other users are mobile. and 1328  1283 To begin with, we might consider a stationary tagged user, with 2 rγ /v E[ (γ )]= P( (γ ) > ) 1284 an independent homogeneous Poisson point process of mobile U U u du 1329 0 1285 users moving on independent straight lines. Considering the (γ )  −ρ 2 rγ /v e (γ ) ˆ (γ ) 1286 Voronoi cell of the associated base station of the tagged user, ρ P(W >u) = (e − 1)du. (42) 1330 ν(γ ) 1287 we might try to characterize the number of mobile users 0 / /∞ 1288 entering and leaving the cell as an M GI queue. Thus, Thus, 1331 1289 the sharing number process associated with the stationary user   −ρ(γ ) 2r/v e (γ ) ˆ (γ ) 2rγ 1290 will be equivalent to the number of customers in a queue. (γ ) ρ P(W >u) E[U ]= e du − . (43) 1332 ν(γ ) v 1291 Later, one could extend this to the case where the tagged user 0 1292 is mobile. ˆ (γ ) Since W is the forward recurrence time associated with 1333 1293 On the practical side, the main future challenges are linked (γ ) the service time W defined in (6), 1334 1294 to the initial motivations of this work, namely in the prediction  ∞ 1295 and optimization of the user quality of experience. Many sce- ˆ (γ ) 1 (γ ) P(W > u) = P(W > t)dt, 1335 E[ (γ )] 1296 narios refining those studied can be considered. For instance, W u 1297 the stationary analysis of the fluid queue representing video (γ ) ρ(γ ) (γ ) where, E[W ]= vλ from (7) and P(W > t) = 1336 1298 streaming should be completed by a transient analysis and by  2rγ 2 2 2 1299 a discrete time analysis. This alone opens an interesting and 1/2rγ 4rγ − v t from (5). Thus, 1337 1300 apparently unexplored connection between stochastic geome- (γ ) 1301 try and queuing theory with direct implications to video QoE. E[U ] 1338 −ρ(γ )   2rγ 2rγ  e v λv v r2 −v2t2dt 2rγ = u 4 γ − 1302 APPENDIX (γ ) e du 1339 ν 0 v 1303 ∞  A. Busy Period of the M/GI/ Queue −ρ(γ )   2rγ e v λ 2rγ 2 − 2 2rγ / /∞ u 4rγ x dx 1304 Theorem 12 (Makowski [11]): Consider an M GI = e du − 1340 ν(γ ) v 1305 queue with arrival rate λ and generic service time W. Let M 0     γ  2r −λ 1 2 −v2 2+ 2 ( √ uv ) 1306 denote an N-valued random variable which is geometrically v 2 u 4rγ u 2rγ arctan 1 4rγ2 −v2u2 = e du 1341 1307 distributed according to ν(γ ) 0 − P( = ) = ( − ν)(ν)l 1, = , ,... 2 γ 1308 M l 1 l 1 2 (37) −λπrγ 2r − e 1342 v 1309 with   2 2 −λπrγ γ 2 γ −ρ 1 r −λrγ q(z) 2e r 1310 ν = 1 − e and ρ = λE[W]. (38) = e dz − . (44) 1343 −λπr2 − γ v 0 v + 1 e 1311 Consider the R -valued random variable U distributed The second equality is by the change of variables vt = x 1344 1312 according to and the last equality√ is by the change of variables z = 1345 ˆ 1 2 √ z 1 −ρP(W≤u) uv/rγ and q(z) = z 4 − z + 2arctan( ).Now,from 1346 1313 P(U ≤ u) = ( − e ), u ≥ , 2 − 2 ν 1 0 (39) 4 z Equation (44) we have, 1347 ˆ  1314 2 where, W is the forward recurrence time associated with the γ 2 (γ ) r −λrγ q(z) { , ≥ } lim E[U ]∼ e dz. 1348 1315 generic service time W. Let Un n 1 be an i.i.d. sequence →∞ rγ v 0 1316 independent of the random variable M. Let B denote a typical ˆ rγ 2 −λr2 q(z) rγ 2 −λr2 2z 1317 busy period. Then the forward recurrence time B associated γ γ Now, the integral v 0 e and v 0 e dz are 1349 1318 with B admits the following random sum representation: asymptotically equal as rγ goes to ∞. 1350 M The asymptotic equality of the two functions as rγ goes to 1351 ˆ ∞ 1319 B =d Ui , (40) means, that the relative error of the approximate equality 1352 i=1 goes to 0 as rγ goes to ∞ i.e., 1353 = 2 2 1320 where d denotes equality in distribution. 2 −λrγ q(z) − −λrγ 2z 0 e e dz = . 1354 lim −λ 2 0 rγ →∞ 2 e rγ 2z 1321 B. Proof of Theorem 4 0 + (γ ) IEEE Proof 1355 1322 Lemma 1: For a R -valued random variable U defined With the help of the Taylor series expansion, we get that 2 1323 by (39) and service time distribution by (5). We have the C/rγ −λ 2 ( ) −λ 2 e rγ q z − e rγ 2zdz 1324 following limit: 0 lim = 0, 1356 →∞ 2 −λ 2 (γ ) rγ e rγ 2z 1325 lim 2λrγ vE[U ]=1. 0 rγ →∞ for some constant C. Thus, we need to prove that 1357 1326 Proof: From Equations (7) and (39): 2 2 2 −λrγ q(z) −λrγ 2z 2 e − e dz −ρ(γ ) C/rγ (γ ) (γ ) (γ ) e ρ P( ˆ > ) lim = 0. 1358 P( > ) = ( W u − ), ≥ , →∞ 2 −λ 2 1327 U u (γ ) e 1 u 0 (41) rγ rγ 2z ν 0 e 18 IEEE TRANSACTIONS ON INFORMATION THEORY

2 2 −λrγ q(z) −λrγ 2z (γ ) 1359 Now let us observe the function h(z) = e − e . U is also [0, 2rγ /v] and all its moments are bounded 1402 (γ ) k k 1360 The derivative of the function h(z) is E[(U ) ] <(2rγ /v) . Thus, for all values of s, we get that 1403   2 2  lim → l (s, rγ ) = 0. 1404 ( ) = − − 2 −λrγ q(z) + −λrγ 2z 2 rγ 0 k 1361 h z 4 z e 2e rγ = , ..    For all k 2 3 we also have that 1405 2 2 −λrγ 2z 2 2 −λr [q(z)−2z] 2 k 1362 = e rγ 2 − 4 − z e . (45) (2λvsrγ ) |l (s, rγ )|≤ . 1406 k k! = / 2  1363 Thus by substituting z C √rγ and using the Taylor series ∞ Therefore, for all values of s, limrγ →0 = lk(s, rγ ) = 0. 1407 2 k 2 1364 expansion of [q(z) − 2z] and 4 − z ,weget Thus, Eq (49) can be written as 1408  2 −2Cλz (γ ) 1365 lim h (C/rγ ) = e [o(rγ )]=0. −sf(γ )U (γ ) rγ →∞ E[e ]=1 − s2λrγ vE[U ]+o(rγ ). 1409 2 1366 The derivative of function h(z) is 0 for z = C/rγ as rγ goes Now, substituting into Eq (48) we have 1410 ∞ ( ) / 2 (γ ) 1367 to i.e., the function h z has a local maximum at C rγ for −sf(γ )Bˆ E[e ] 1411 1368 some constant C and it can be observed that it is decreasing (γ ) (γ ) (γ ) 1 − ν + 2sλvrγ E[U ](1 − ν ) + so(rγ ) 1369 = for C 3. = , (50) 1412 − ν(γ ) + ν(γ ) λv E[ (γ )]+ ( ) 1370 Therefore, the function h(z) is a decreasing function from 1 2 s rγ U so rγ / 2 1371 3 rγ to 2 and the area under its curve has an upper bound from which it follows that 1413 1372 that goes to 0. (γ ) −sf(γ )Bˆ E[ ]= . 1414 1373 Thus, the two integrals are asymptotically equal as rγ goes lim e 1 (51) rγ →0 (γ ) 1 1374 ∞ E[ ]∼ γ ∞ to , which implies that U 2vrγ λ as r goes to . 1415 1375 From the results in [11] we have that (γ ) (γ ) (γ ) ˆ (γ ) 1376 Proof: V is the sum of a busy period B and E[ −sf(γ )B ]= − (γ )E[ (γ )]E[ −sf(γ )B ], (γ ) e 1 sf B e (52) 1416 1377 idle period I of an M/GI/∞ queue. We know that the 1378 distribution of the idle period is exponential with parameter and also that 1417 λv (γ ) (γ ) (γ ) 1379 2 rγ and that the busy period B and idle period I are (γ ) ν E[B ]= . (53) 1418 1380 independent. λv ( − ν(γ )) (γ ) 2 rγ 1 1381 The Laplace transform of the random variable V scaled Thus, we get 1419 1382 by 2λvrγ is given by 2 −λπrγ (γ ) ( (γ ) ) = (γ ) ( (γ ) ) (γ ) ( (γ ) ), (γ ) 1 − e ˆ (γ ) 1383 V f s B f s I f s (46) E[ −sf(γ )B ]= − E[ −sf(γ )B ]. e 1 s −λπ 2 e 1420 e rγ 1384 where, 1 Using the limit in (51), we get 1421 1385 (γ ) ( f (γ )s) = . (47) (γ ) I + −sf(γ )B 1 s lim E[e ]=1. (54) 1422 ( (γ ) ) rγ →0 1386 So, asymptotically, as rγ goes to 0, I (γ ) f s converges (γ ) 1387 in distribution to an exponential random variable with unit Thus, the distribution of the random variable V scaled 1423 1388 mean. by f (γ ) = 2λvrγ converges in distribution to an exponential 1424 1389 Next, we need to prove that asymptotically, the busy period random variable with unit mean. 1425 (γ ) (γ ) 1390 B scaled by f goes to one. Let us consider the forward Now, consider a continuous function g(γ ) such that 1426 ˆ (γ ) 1391 recurrence time of the busy period, B whichisdefinedin limrγ →∞ g(γ ) = 0. The Laplace transform of the random 1427 (γ ) 1392 (6). From Theorem 2 we get, variable V scaled by g(γ ) is given by 1428 (γ ) − (γ ) (γ )  (γ ) (γ ) ( − ν )E[ sU ] − ˆ − M 1 e (γ ) = (γ )(g(γ )s) (γ ) (g(γ )s), (55) 1429 E[ s B ]=E[ s i=1 Ui ]= , V (s) B I 1393 e e (γ ) 1 − ν(γ )E[e−sU ] where, 1430 1394 and it follows that 1 (γ ) (γ ) − (γ ) (γ ) (g(γ )s) = . (56) 1431 ˆ (γ ) ( − ν )E[ sf U ] I + (γ )( / v λ) E[ −sf(γ )B ]= 1 e . 1 sg 1 2 rγ 1395 e (γ ) (48) 1 − ν(γ )E[e−sf(γ )U ] ∞ ( (γ ) ) Asymptotically, as rγ goes to , I (γ ) g s goes to 1. 1432 1396 Further, we have that By the same arguments as those for the up-crossing case 1433

IEEE∞ Proof 1434 we have that (γ )  E[ −sf(γ )U ]= ( , ), (γ ) 1397 e lk s rγ (49) ( − ν(γ ))E[ −sg(γ )U ] E[ −sg(γ )Bˆ (γ )]= 1 e , k=0 e (γ ) 1435 1 − ν(γ )E[e−sg(γ )U ] 1398 where (γ ) where, 1436 (−1)k (2λvsrγ )kE[(U )k] ( , ) = . ∞ (γ ) 1399 lk s rγ  (− )k( λv )k ( −ρ )k k! −sg(γ )U (γ ) 1 2 srγ e (γ ) k E[e ]= E[(U ) ]. 1437 (γ ) k! 1400 Given that the support of service times W is [0, 2rγ /v], k=0 1401 it follows from (39) that the support of the random variable (57) 1438 MADADI et al.: SHARED RATE PROCESS FOR MOBILE USERS IN POISSON NETWORKS AND APPLICATIONS 19

−ρ(γ ) − (−1k )(2λvsrγ )k (e )k 1 (γ ) ˆ(γ ) (γ ) ( , ) = E[( )k]. We now prove the first relation. Since S ≤ S ≤ log(1+ 1476 1439 Let mk s rγ k! U The (γ ) SNR), in order to prove the first inequality, it is enough to 1477 1440 moments of the random variable U are bounded (γ ) E[( )k] <( /v)k show that 1478 1441 U 2rγ . Thus, for all values of s,weget (γ ) 2 1 2 1442 that limrγ →∞ mk(s, rγ ) = 0, because ρ = λπrγ and the ˆ(γ ) lim − s log P(S > s) ≤ . (60) 1479 −ρ(γ ) k−1 s→∞ β 1443 exponential term (e ) dominates. Also for k = 2, 3, ... 1444 we have that We have 1480 2 k −ρ(γ ) k−1 ˆ(γ ) −β ˆ (2λvsrγ ) (e ) P(S > s) = P(log(1 + KL )>s(1 + Nγ )), 1481 1445 |m (s, rγ )|≤ . k k! hence 1482 1446 Therefore for all values of s, ˆ(γ ) −β ˆ P(S > s) = P KL > exp s(1 + Nγ ) − 1 1483 ∞ −β ˆ 1447 lim mk(s, rγ ) = 0. ≥ P KL > exp s(1 + Nγ ) 1484 rγ →∞ = k 2 β ˆ = P L < K exp −s(1 + Nγ ) 1485 1448 Thus, Eq (57) can be written as   2 2 − (γ ) (γ ) −ρ(γ ) (γ ) −ρ(γ ) 2 β ˆ E[ sf U ]= − v λE[ ]+ ( ), = P L < K exp −s (1 + Nγ ) 1486 1449 e 1 se 2 rγ U e o rγ β

2 2 2 ˆ ( ) = 2 β −s β −s β Nγ 1450 where, limrγ →∞ o rγ 0. = P L < K e e . 1487 (γ ) 1451 From Lemma 1 we have that, limrγ →∞ 2λrγ vE[U ]=1. 2 1452 Putting these results together we have that Using now the fact that L is an exponential random variable 1488 ˆ with parameter λπ, independent of Nγ ,weget 1489 E[ −sg(γ )Bˆ (γ )] 1453 e ˆ(γ ) −ρ(γ ) − ρ(γ ) (γ ) − ρ(γ ) P(S > s) 1490 − 2 v γ λE[ ]+ 2 ( γ ) e se 2 r U e so r 2 − 2 − 2 ˆ 1454 =    , ≥ − E −λπ β s β s β Nγ −ρ(γ ) − −ρ(γ ) v λE[ (γ )]− −ρ(γ ) ( ) + −ρ(γ ) 1 exp K e e 1491 e 1 e s2 rγ U e so rγ e 2 − 2 − 2 − 2 1455 (58) β s β 2 s β s β ≥ λπ K e exp −ξπrγ 1 − e 1 + o e , 1492

1456 so that −a 2 where we used again the bound 1 − e ≥ a + a /2andthe 1493 ˆ (γ ) ˆ (γ ) 2 −sg(γ )B 1 fact that N is Poisson with parameter ξπrγ . 1494 1457 lim E[e ]= . (59) rγ →∞ + 1 1 s Taking the log, multiplying both sides by − s and letting 1495

1496 1458 Now, from Eq (52) and (53) we get, s go to infinity gives (60).

(γ ) 2 ˆ (γ ) E[ −sf(γ )B ]= − ( − −λπrγ )E[ −sf(γ )B ]. 1459 e 1 s 1 e e D. Proof of Theorem 7 1497 (γ ) −sg(γ )B 1 Let us first prove the last relation. Using the Chernoff’s 1498 1460 Thus from the limit in (59), lim →∞ E[e ]= . rγ 1+s ˆ (γ ) (γ ) bound for the Poisson random variable N ,weget 1499 1461 Thus, the random variable V scaled by g(γ ) = 2 −λπrγ (γ ) −β 1462 λv γ ˆ 2 r e converges in distribution to an exponential ran- P N + 1 > sn log (1 + Krγ ) 1500 1463 dom variable with unit mean.  −β s log (1+Krγ ) 1464 eθ ≤ −θ , e −β 1501 sn log (1 + Krγ ) 1465 C. Proof of Theorem 6 2 θ = ξπr 1502 = /w with γ , which shows that 1466 Let us first prove the second relation. Let K p and let 1467 L denote the distance to the closest base station. We have 1 ˆ (γ ) −β lim − log P N + 1 > s log (1 + Kr ) 1503 →∞ n γ n sn log(sn) 1468 P((log(1 + SNR)>s) −β   ≥ log (1 + Kr ). 1504 = P −β > s − γ 1469 KL e 1 The converse inequality follows from the bound 1505 = P β < K 1470 L θ k es − 1 ˆ (γ ) −β −θ  P + > ( + ) ≥ , 1506  2 N 1 sn log 1 Krγ e IEEEβ Proof k! K = P 2 < −β 1471 L = ( + ) es − 1 with k sn log 1 Krγ and Stirling’s bound on the 1507   2 Gamma function. 1508 β K Let us now prove the first relation. We have 1509 1472 = 1 − exp −λπ ,  es − 1 ˆ(γ ) 1 P S < sn 1510 2 1473 where we used the fact that L is an exponential random  2 rγ 1474 variable with parameter λπ. The result then follows from the −λπv (γ ) − β = P ˆ + > + v 2 v. ≥ − −a ≥ + 2/ ≥ e N 1 sn log 1 K d 1511 1475 bound a 1 e a a 2, for a 0. 0 20 IEEE TRANSACTIONS ON INFORMATION THEORY

Fig. 18. Tagged user at origin, its serving base station X j at distance x and ( ) ( ) a point Q at a distance u along with discs Bx 0 and Bz Q . Fig. 19. The tagged user at the origin, its serving base station X j at distance x and a point U at a distance u on the line d(0,θ).

1512 For sn large enough,  where, L is a random variable denoting the distance to the 1538 closest BS. Thus, 1539 P ˆ(γ ) < 1  1513 S sn r ( ) ˆ(γ ) ∗ fL x E[ ]= E[ | = ] 1540    2 J V L x r dx − β rγ 0 fL (y)dy ˆ (γ ) 2 −λπv 0 1514 ≤ P N + 1 > s log 1 + Krγ e dv, n r E[ ∗| = ] λπ −λπx2 0 0 V L x 2 xe dx = 1541 1 − e−λπr2 1515 which allows one to conclude that    √   r π x cos(α)+ r2−x2 sin(α) = 1 1 ˆ(γ ) 1 −β −λπ 2 2 1542 1516 lim − log P S < s ≥ log (1 + Kr ). − r →∞ ( ) n γ 1 e 0 0 0 n sn log sn 2 (−λl(Bz(Q)−Bx (0))) −λπx e ududα2λπxe dx 1543    √ 1517 The upper bound follows from the inequality: 1 r π x cos(α)+ r2−x2 sin(α) ⎛ ⎞ = 4λπ 1544  2 − −λπr2  ∞ β 1 e 0 0 0  θl (−λ ( ( )∪ ( ))) ˆ(γ ) 1 −θ ⎝ 2 K ⎠ l Bx 0 Bz Q α . 1518 P S < sn = e P L > e uxdud dx (62) 1545 l! l+1 − l=0 e s 1 ( ( )) = π 2 ( ( ) ∪ ⎛  ⎞ The last equality is because l Bx 0 r and l Bx 0 1546 2 ( )) = (α) + (π − α) 2 + (π − (α, ))( 2 + 2 − θ k β Bz Q ux sin x f x u x 1547 −θ ⎝ 2 K ⎠ u−x cos(α) 1519 ≥ e P L > , 2ux cos(α)) with cos( f (α, x)) = √ . 1548 ! k+1 2+ 2− (α) k e s − 1 u x 2ux cos

−β F. Integral Geometry II 1549 1520 where k = sn log (1 + Krγ ). Let us consider a point U at a distance u from the origin on 1550 the line d(0,θ) as shown in the figure. Then, the probability 1551

1521 E. Integral Geometry I that the point is in the Voronoi cell of the BS X j is given by 1552 −λl(C(u,θ)) e ,wherel(C(u,θ)) is the area of the shaded region 1553 1522 Consider a tagged user at the origin, and let the serving in the figure. The conditional expectation is given by: 1554 1523 BS X j be at a distance x from the origin. Let Q denote a E[ ( ,θ)| = ] 1524 point a distance u from the origin. We need to consider all l 0 L x √ 1555  π  (θ)+ 2− 2 (θ) 1525 points Q that are within a distance rγ from the serving BS x cos r x sin 1 (−λl(C(u,θ))) = = 2 + 2 − (α) < = e dudθ, (63) 1556 1526 X j i.e., z X j Q u x 2uxcos rγ ,where π  0 0 1527 α = X j OQ as shown in Fig.8. where, L is the random variable denoting the distance to the 1557 1528 Let Bx (0), Bz(Q) be two discs with centers at the origin closest BS. 1558 1529 and Q, and radii x and X j Q, respectively. Let Bx (0), Bz(U) be two discs with centers at the origin 1559 1530 The conditional probability that a point at distance u from and U, and radii x and X j U, respectively. Thus, 1560 1531 origin is within the Voronoi cellIEEE of the BS serving the user Proof 1532 at the origin given it is at a distance x is the probability that E[ ( ,θ)] l 0  1561 1533 there is no other BS in the area of disc B (Q) excluding the r ( ) z fR x ( ) = E[l(0,θ)|L = x] dx 1562 1534 area of Bx 0 . Then the conditional expectation is: r ( ) 0 0 fL y dy 2 E[ ˆ(γ )| = ] r E[l(0,θ)|L = x]2λπxe−λπx dx 1535 J L x √ = 0   −λπ 2 1563 π xcos(α)+ r2−x2sin(α) 1 − e r √ (−λl(Bz(Q)−Bx (0)))    1536 = 2 e ududα, r π x cos(θ)+ r2−x2 sin(θ) 0 0 = 1 −λπ 2 1564 1537 (61) π(1 − e r ) 0 0 0 MADADI et al.: SHARED RATE PROCESS FOR MOBILE USERS IN POISSON NETWORKS AND APPLICATIONS 21

(−λl(C(u,θ))) −λπx2 1565 × e dudθ2λπxe dx [14] F. Baccelli and P. Brémaud, Palm Probabilities and Stationary Queues, 1620    √ r π x cos(θ)+ r2−x2 sin(θ) vol. 41. Springer, 2012. 1621 1 [15] J. Møller, “Random Johnson–Mehl tessellations,” Adv. Appl. Probab., 1622 1566 = 2λπ −λπr2 vol. 24, no. 4, pp. 814–844, 1992. 1623 π(1 − e ) 0 0 0 [16] E. W. Weisstein. Kolmogorov–Smirnov test. MathWorld—A Wol- 1624 (−λl(Bx(0)∪Bz(U))) 1567 × e xdudθdx. (64) fram Web Resource. [Online]. Available: http://mathworld.wolfram.com/ 1625 Kolmogorov-SmirnovTest.html 1626 AQ:3 2 1568 The last equality is because l(Bx (0)) = πr and l(Bx (0) ∪ [17] D. B. Johnson and D. A. Maltz, “Dynamic source routing in ad hoc 1627 2 2 2 wireless networks,” in Mobile Computing. 1996, pp. 153–181.1628 AQ:4 1569 Bz(U)) = ux sin(θ) + (π − θ)x + (π − f (θ, x))(u + x − (θ)) ( (θ, )) = √ u−x cos(θ) [18] V. Gupta and P. G. Harrison, “Fluid level in a reservoir with an on-off 1629 1570 2ux cos with cos f x . source,” SIGMETRICS Perform. Eval. Rev., vol. 36, no. 2, pp. 128–130, 1630 u2+x2−2ux cos(θ) 2008. 1631 [19] J. F. C. Kingman, Poisson Processes. Hoboken, NJ, USA: Wiley, 1993. 1632 1571 G. Proof of Theorem 9 [20] F. Morlot, “A population model based on a Poisson line tessellation,” in 1633 Proc. 10th Int. Symp. Modeling Optim. Mobile, Ad Hoc Wireless Netw. 1634 1572 Proof: Let Z be a geometric random variable given by: (WiOpt), May 2012, pp. 337–342. 1635 l [21] F. Baccelli, M. Klein, M. Lebourges, and S. Zuyev, “Stochastic geometry 1636 1573 P(Z = l) = (1 − α)α . (65) and architecture of communication networks,” Telecommun. Syst.,vol.7, 1637 nos. 1–3, pp. 209–227, 1997. 1638 1574 Then, the random variable T representing the time taken to 1575 download the file is the geometric sum of independent random 1576 variables given by Z (γ ) 1577 T = (Xi + I ) + Y. (66) i Pranav Madadi is a Ph.D. student at the Electrical and Computer Engineering 1639 i=0 department of The University of Texas at Austin. He is working under 1640 the supervision of Prof. Francois Baccelli and Prof. Gustavo de Veciana. 1641 1578 Thus, the Laplace transform of T is   He received his bachelor degree from Indian Institute of Technology, Hyder- 1642 abad in 2014. His primary research interest is in stochastic modeling of 1643 1579 L (s) = (1 − α) + (1 − α)αL (s)L (γ ) (s) + .. L (s) T X I Y wireless networks. He is currently working on analyzing the performance 1644 ( − α)L ( ) ( − α)L ( ) = 1 Y s = 1 Y s . of a mobile user moving in a Poisson cellular network. 1645 1580 λv (67) 1 − αL (s)L (γ ) (s) − αL ( ) 2 r X I 1 X s 2λvr+s (γ ) 1581 The last equality follows from the fact that I is exponen- 1582 tial with parameter 2λvr.

François Baccelli is Simons Math+X Chair in Mathematics and ECE at 1646 1583 REFERENCES UT Austin. His research directions are at the interface between Applied 1647 Mathematics (probability theory, stochastic geometry, dynamical systems) and 1648 1584 [1] Enterprise Mobile Data Report-Q3, Wandera, London, U.K., Oct. 2013. Communications (network science, information theory, wireless networks). 1649 1585 [2] F. Baccelli and B. Blaszczyszyn, “Spatial modeling of wireless commu- He is co-author of research monographs on point processes and queues (with 1650 1586 nicationsa stochastic geometry approach,” Foundations and Trends in P. Brémaud); max plus algebras and network dynamics (with G. Cohen, 1651 1587 Networking. Breda, The Netherlands: NOW Publishers, 2009. G. Olsder and J.P. Quadrat); stationary queuing networks (with P. Brémaud); 1652 1588 [3] T. Karagiannis, J.-Y. Le Boudec, and M. Vojnovic, “Power law and stochastic geometry and wireless networks (with B. Blaszczyszyn). Before 1653 1589 exponential decay of intercontact times between mobile devices,” IEEE joining UT Austin, he held positions in France, at INRIA, Ecole Normale 1654 1590 Trans. Mobile Comput., vol. 9, no. 10, pp. 1377–1390, Oct. 2010. Supérieure and Ecole Polytechnique. He received the France Télécom Prize 1655 1591 [4] V. Conan, J. Leguay, and T. Friedman, “Characterizing pairwise inter- of the French Academy of Sciences in 2002 and the ACM Sigmetrics 1656 1592 contact patterns in delay tolerant networks,” in Proc. 1st Int. Conf. Auton. Achievement Award in 2014. He is a co-recipient of the 2014 Stephen 1657 1593 Comput. Commun. Syst. (ICST), 2007, Art. no. 19. O. Rice Prize and of the Leonard G. Abraham Prize Awards of the IEEE 1658 1594 [5] H. Cai and D. Y. Eun, “Toward stochastic anatomy of inter-meeting Communications Theory Society. He is a member of the French Academy of 1659 1595 time distribution under general mobility models,” in Proc. 9th ACM Int. Sciences and part time researcher at INRIA. 1660 1596 Symp. Mobile Ad Hoc Netw. Comput., 2008, pp. 273–282. 1597 [6] S. Mitra et al., “Trajectory aware macro-cell planning for mobile users,” 1598 in Proc. IEEE Conf. Comput. Commun. (INFOCOM), Apr./May 2015, 1599 pp. 792–800. 1600 [7] H. Deng and I.-H. Hou, “Online scheduling for delayed mobile 1601 offloading,” in Proc. IEEE Conf. Comput. Commun. (INFOCOM), 1602 Apr./May 2015, pp. 1867–1875. Gustavo de Veciana (S’88–M’94–SM’01–F’09) received his B.S., M.S, and 1661 1603 [8] Z. Lu and G. De Veciana, “Optimizing stored video delivery for mobile Ph.D. in electrical engineering from the University of California at Berkeley 1662 1604 networks: The value of knowing the future,” in Proc. IEEE INFOCOM, in 1987, 1990, and 1993 respectively, and joined the Department of Electrical 1663 1605 Apr. 2013, pp. 2706–2714. and Computer Engineering where he is currently a Cullen Trust Professor 1664 1606 [9] J. Keilson, Markov Chain Modelsrarity and Exponentiality, vol. 28. of Engineering. He served as the Director and Associate Director of the 1665 AQ:1 1607 Springer, 2012. Wireless Networking and Communications Group (WNCG) at the University 1666 1608 Probability Approximations via the Poisson Clumping [10] D. Aldous, IEEE of TexasProof at Austin, from 2003-2007. His research focuses on the analysis 1667 1609 Heuristic, vol. 77. Springer, 2013. and design of communication and computing networks; data-driven decision- 1668 1610 [11] A. M. Makowski, “On a random sum formula for the busy period making in man-machine systems, and applied probability and queuing theory. 1669 1611 of the M/G/Infinity queue with applications,” DTIC Document, Dr. de Veciana served as editor and is currently serving as editor-at-large 1670 1612 Tech. Rep., 2001. [Online]. Available: http://www.researchgate.net/ for the IEEE/ACM TRANSACTIONS ON NETWORKING. He was the recipient 1671 1613 publication/235086381_On_a_Random_Sum_Formu%la_for_the_ of a National Science Foundation CAREER Award 1996 and a co-recipient 1672 AQ:2 1614 Busy_Period_of_the_MGInfinity_Queue_With_Applications of five best paper awards including: IEEE William McCalla Best ICCAD 1673 1615 [12] S. N. Chiu, D. Stoyan, W. S. Kendall, and J. Mecke, Stochastic Geometry Paper Award for 2000, Best Paper in ACM TODAES Jan 2002–2004, Best 1674 1616 and its Applications. Hoboken, NJ, USA: Wiley, 2013. Paper in ITC 2010, Best Paper in ACM MSWIM 2010, and Best Paper IEEE 1675 1617 [13] A. Okabe, B. Boots, K. Sugihara, and S. N. Chiu, Spatial Tessellations: INFOCOM 2014. In 2009 he was designated IEEE Fellow for his contributions 1676 1618 Concepts and Applications of Voronoi Diagrams. Hoboken, NJ, USA: to the analysis and design of communication networks. He currently serves 1677 1619 Wiley, 2009. on the board of trustees of IMDEA Networks Madrid. 1678 AUTHOR QUERIES AUTHOR PLEASE ANSWER ALL QUERIES

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