Fundamentals of Mobility in Cellular Networks: Modeling and Analysis

Xingqin Lin, Radha Krishna Ganti, Philip J. Fleming and Jeffrey G. Andrews

Abstract—Mobility modeling and analysis is a key issue in velocity. This process repeats at each waypoint. Optionally, the networks, but there are surprisingly few fundamental mobile user can have a random pause time at each waypoint results. In this paper we propose a new and quite general random before moving to the next waypoint. A drawback of the RWP waypoint (RWP) mobility model which is valid over the entire plane. We derive key properties of the proposed mobility model mobility model is that the stationary spatial distribution including transition length, transition time and spatial node tends to concentrate near the centre of the finite domain [4]. distribution. Then the RWP mobility model is applied to study Thus, we propose a new RWP mobility model defined on the the handover rate in cellular networks under both deterministic entire plane and then derive its associated properties. (hexagonal) and random (Poisson) base station (BS) models. Closed form expressions for handover rate can be obtained, most We then apply this new RWP model to study handover of which to our knowledge are the first of their kind. For example, rate in cellular networks whose BSs are modeled in two these results show the expected property that the handover rate is proportional to the square root of base station density. Also, we opposite ways. First is the traditional hexagonal grid, which find that Poisson-Voronoi model for BS coverage areas is about as represents an extreme in terms of regularity and is completely accurate in terms of mobility (particularly handover) evaluation deterministic. The second is to model the BS locations as as the ubiquitous hexagonal model. drawn from a Poisson point process (PPP), which creates a set of BSs with completely independent locations [5], [6]. I.INTRODUCTION Thus, most any real-world deployment would seem to fall The support of mobility is a fundamental aspect of wireless in between these two extremes, since real BS deployments networks [1], [2]. is taking on new are neither perfectly regular nor independent. Under the PPP importance and complexity in emerging cellular networks, BS model, the can be viewed as a Poisson- which have ever-smaller and more irregular cells. Mobility has Voronoi tessellation if the mobile users are assumed to connect generally been handled heuristically, and indeed there are very to the nearest BSs. Closed form expressions are obtained few analytic results or even widely accepted models on the for handover rate under both models. To the best of our fundamentals of mobility, e.g. the handover rate as a function knowledge, most of these expressions are new. of speed in a cellular network. One approach is to construct mobility models based on em- The analysis in this paper requires stochastic geometric pirical mobility patterns, known as trace models [1]. Though tools, which are becoming increasingly sophisticated and pop- considered accurate, they require the knowledge of many ular [7]–[10]. As far as mobility is concerned, [11] proposed a detailed network characteristics and rule out general analytical framework to study the impact of mobility in cellular networks models and results. Therefore, so-called synthetic models are modeled as Poisson-Voronoi tessellation. In particular, the often utilized to analyze the impact of mobility on network authors proposed Poisson line process to model the road performance and to evaluate protocols in simulation. Popular system, along which the mobile users move. Thus, the mobility synthetic models include random walk, random waypoint pattern in [11] is of large scale while the RWP mobility model (RWP), and Gauss-Markov [1]. in this paper is of small scale. Our study can be viewed as a complementary one to [11] and gives valuable insights. For In this paper, we focus on RWP mobility model orig- example, our result indicates that if cells decrease in size such inally proposed in [3]. In this model, mobile users move that the BS density per unit area is increased by 4 times, then in a finite domain A. At each turning point, the mobile the handover rate would be doubled. Also, recent work showed user selects the destination point (referred to as waypoint) that the PPP model for BSs was about as accurate in terms of uniformly distributed in A and chooses the velocity from some SINR distribution as the hexagonal grid for a representative distribution. Then the user moves along the line connecting its urban cellular network [12]. Interestingly, we find that this starting waypoint to the newly selected waypoint at the chosen observation is also true for mobility evaluation, though the Xingqin Lin and Jeffrey G. Andrews are with Department of Electrical & Poisson-Voronoi model yields slightly higher handover rate Computer Engineering, The University of Texas at Austin, USA. (E-mail: than hexagonal model. [email protected], [email protected]). Radha Krishna Ganti is with Department of Electrical Engineering, Indian Institute of Technology, Madras, Before ending this section, we remark that most of the India. (E-mail: [email protected]). Philip J. Fleming is with Nokia Siemens Networks (E-mail: phil.fl[email protected]). This research was supported by proofs in this paper are omitted for brevity. For interested Nokia Siemens Networks. readers, we refer to [13]. λ = 0.1 λ = 0.01 II.PROPOSED RWP MOBILITYMODEL 1 12

0 10 −1 We describe the proposed RWP mobility model in this 8 −2 6 section. As in the description of traditional RWP model −3 4 (see, e.g., [4]), the movement trace of a node can be −4 −5 2 formally described by an infinite sequence of quadruples: −6 0 −1 −0.5 0 0.5 1 1.5 2 2.5 −15 −10 −5 0 {(Xn−1, Xn,Vn,Sn)}n∈N , where n denotes the n-th move- λ = 0.001 λ = 0.0001 ment period. During the n-th movement period, Xn−1 denotes 60 70 50 60 40 the starting waypoint, Xn denotes the target waypoint, Vn 50 30 40 denotes the velocity, and Sn denotes the pause time at the 20 30 10 X 20 waypoint n. 0 2 10 We next describe the way of selecting waypoints in . −10 R −20 0 −20 −10 0 10 20 30 −40 −20 0 20 40 60 80 Given the current waypoint Xn−1, the next waypoint Xn is chosen such that the included angle between the vector Xn − Xn−1 and the abscissa is uniformly distributed on [0, 2π] and Fig. 1. Sample traces of the proposed RWP mobility model with λ equals −1 −2 −3 −4 the transition length Ln =k Xn − Xn−1 k is a nonnegative 10 , 10 , 10 and 10 respectively random variable. This selection of waypoints is independent and identical for each movement period. {L } where L are independent and identically Rayleigh Though there is a degree of freedom in modeling the n n∈N n distributed with random transition lengths, we focus on a particular choice in 1 this paper. Specifically, the transition lengths {L1,L2, ...} are E[L] = √ . (2) chosen to be independent and identically distributed (i.i.d.) 2 λ with cumulative distribution function (cdf) Note that the transition lengths are not i.i.d. in the traditional RWP mobility model. For example, a node currently located P (L ≤ l) = 1 − exp(−λπl2), l ≥ 0 (1) near the border of the finite domain tends to have a longer i.e., the transition lengths are Rayleigh distributed. This se- transition length while a node located around the centre of the lection bears an interesting interpretation: Given the waypoint finite domain statistically has a shorter transition length for Xn−1, a homogeneous PPP Φ(n) with intensity λ is generated the next movement period. and then the nearest point in Φ(n) is selected as the next Nevertheless, the random waypoints Xn are i.i.d. in the waypoint, i.e., Xn = arg minx∈Φ(n) k x − Xn−1 k . The traditional model, which is obvious since these waypoints are selection of velocity and pause time follow the traditional selected uniformly from a finite domain and independently RWP mobility model. In particular, velocities Vn are i.i.d. over movement periods. This property forms the basis of the with distribution PV (·). Pause times Sn are also i.i.d. with analysis of the traditional RWP mobility model (see, e.g., [4], distribution PS(·). [14]). In contrast, the waypoints in our proposed model are Under the proposed model, different mobility patterns can not i.i.d. but form a Markov process. be captured by choosing different λ’s. Larger λ statistically B. Transition time implies that the transition lengths L are shorter. These λ’s may be appropriate for mobile users walking and shopping We define transition time as the time a node spends during in the city. In contrast, smaller λ statistically implies that the the movement between two successive waypoints. We denote transition lengths L are longer. These λ’s may be appropriate by Tn the transition time for the movement period n. Then for driving users, particularly those on the highways. This T = L/V where we omit the period index n since Tn are i.i.d.. intuitive result can also be observed in Fig. 1, which shows 4 Denote V ∈ R as the range of the random velocity V . Given sample traces of the proposed RWP mobility model. any velocity distribution PV (·), the probability distribution of T is given as follows. III.STOCHASTICPROPERTIESOFTHE PROPOSED RWP Proposition 1. The cdf of the random transition time T is given MOBILITYMODEL by Z 2 2 In this section, we study the various stochastic properties P (T ≤ t) = 1 − exp(−λπv t )dPV (v), t ≥ 0. (3) of the proposed RWP mobility model. Stochastic properties of V interests include transition length, transition time and spatial As a specific application of Prop. 1, the following corollary node distribution. gives closed form expressions for transition times under two types of velocity distributions. A. Transition length Corollary 1. 1) If V ≡ ν where ν is a positive constant, the We define transition length as the Euclidean distance be- pdf of transition time T is tween two successive waypoints. In the proposed model, the 2 −λπν2t2 transition lengths can be described by a stochastic process fT (t) = 2πλν te , t ≥ 0. (4) 2) If V is uniformly distributed on [vmin, vmax], the pdf of resources such as transmit power and spectrum. Thus, analytic transition time T is results on handover rate is useful for network dimensioning.

g(vmin) − g(vmax) For brevity, we only consider the scenario with constant fT (t) = , t ≥ 0, (5) velocity, i.e., V ≡ ν. (vmax − vmin)t

2 2 1 √ where g(x) = xe−λπt x + Q( 2πλtx) is non- A. Hexagonal model λ1/2t Z ∞ 2 1 − u In this subsection, we focus on hexagonal cellular networks. increasing, and Q(x) = √ e 2 du. 2π x We assume the typical mobile user is located at the origin. We now derive the mean of transition time which will be Then the expected number of handovers during one movement used later. Instead of applying the cdf of T , we notice that period can be computed as Z ∞ L 1 1 1 X Z E[T ] = E[ ] = E[L]E[ ] = √ dPV (v) (6) E[N] = n P (dA(r, θ)) (9) V V 2 λ V v n=1 Cn 1 From (6), we can easily obtain that E[T ] = √ if V ≡ ν. where P (dA(r, θ)) is the probability distribution of the way- 2ν λ point density fX1 (r, θ) given in Lemma 1 and Cn denotes the C. Spatial node distribution area covered by the n-th layer neighbouring cells. However, In this subsection, we study spatial node distribution. To exact computation of N by (9) is tedious. Thus, we propose this end, let X0 and X1 be two successive waypoints. Given the following approximation formula X , we are interested in the probability that the moving ∞ 2π (2n+1)R 0 X Z Z node resides in some measurable set A during the movement E[N]app = n fX1 (r, θ)rdrdθ (10) 0 (2n−1)R from X0 to X1. For brevity, we only give the spatial node n=1 distribution with the assumption that the mobile node does not where R = pC/π with C being the hexagonal cell size, have pause time. i.e., we approximate the n-th neighbouring layer by a ring Theorem 1. Assume that Sn ≡ 0, ∀n, and that X0 is at the with inner radius (2n − 1)R and outer radius (2n + 1)R. This origin. Then the spatial node distribution between X0 and X1 approximation captures the essence of (9) and yields Prop. 2 is characterized by the pdf f(r, θ) given by Proposition 2. Let d be the side length of the hexagonal √ λ cell and λ the mobility parameter. The expected number of f(r, θ) = exp(−λπr2) (7) πr handovers during one movement period E[N]app is given by ∞ √ ! X 3 3 Proof: See Appendix A. E[N] = exp − (2n + 1)2λd2 (11) app 2 The physical interpretation of f(r, θ) goes as follows. Let n=0 dA(r, θ) be a small area around the point (r, θ) under the polar L U coordinate. Then the probability P (dA(r, θ)) that the moving and is bounded as E[N]app ≤ E[N]app ≤ E[N]app where node resides in some measurable set A during the movement r q √  L π 2 E[N]app , √ Q 3 3λd (12) from X0 to X1 is approximately given by 6 3λd2 r  q √  P (dA(r, θ)) ≈ f(r, θ) · |dA(r, θ)| (8) U π 2 E[N]app , √ 1 − Q 3 3λd (13) 6 3λd2 where |dA(r, θ)| denotes the area of the set dA(r, θ). Also, as 2 noted in the proof in Appendix A, f(r, θ) can be regarded as Moreover, the difference 4Napp(λd ) between the upper bound the ratio of the expected proportion of transition time in the and lower bound is a strictly increasing function of λd2 2 set dA(r, θ) to the area |dA(r, θ)|. and is within the range (0, 1). In particular, 4Napp(λd ) → 2 2 2 0 as λd → ∞, and 4Napp(λd ) → 1 as λd → 0. IV. APPLICATION TO CELLULAR NETWORKS: AHANDOVER STUDY Using Prop. 2, the handover rate and the corresponding bounds readily follow by further dividing the expected tran- In this section, we study the impact of mobility on handover sition time E[T ] which has been derived in Section III-B. In rate using the proposed model. Handover rate is formally particular, when d → 0, defined as follows. r Definition 1. The handover rate is defined as the expected num- π ν Happ ∼ √ + o(1). (14) ber E[N] of handovers during one movement period divided by 6 3 d the expected transition time. Mathematically, handover rate is Though derivation by (9) is not tractable, the exact handover given by H = E[N]/E[T ]. rate in hexagonal model can be obtained using generalized Handover rate is directly related to network signalling solution of Buffon’s needle problem [15] as follows. overhead. Clearly, handover rate is low in large cells. How- Proposition 3. Let d be the side length of the hexagonal ever, large cells impose higher requirement on other network cell and λ the mobility parameter. The expected number of handovers E[N] during one movement period is 25 Simulation ( λ = 10 −2) √ −3 r Simulation ( λ = 10 ) 2 3 1 Simulation ( λ = 10 −4) Analytic Result ( λ = 10 −2) E[N] = (15) 20 Analytic Result ( λ = 10 −3) 3πd λ Analytic Result ( λ = 10 −4) The handover rate H = E[N]/E[T ] is then given by √ s 15 4 3 ν H = (16) 10

3π d Number of Handover We remark that the insights obtained by either approximate 5 or exact approach are√ the same. Note that the hexagonal cell 2 size sH is given by 3 3d /2 in hexagonal tiling. So (14) can 0 p π √ν 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 be written as H ∼ . Similarly, the exact method µ app 4 sH Intensity of Base Station ( ) q √ 4 3 √ν yields H = π 2 s . Both results imply that handover rate H Fig. 2. Handover in Poisson-Voronoi model with velocity ν ≡ 1. is inversely proportional to the square root of cell size sH .

B. Poisson-Voronoi model 16 Poisson−Voronoi handover rate Hexagonal exact handover rate Hexagonal approximate handover rate In this subsection, we focus on cellular networks modeled 14 by Poisson-Voronoi tessellation [10]. We first give a brief 12 description of Poisson-Voronoi tessellation. Consider a locally 2 10 finite set φ = {xi} of points xi ∈ R , referred to as nuclei. e 8 The Voronoi cell Cxi (φ) of point xi is defined as Handover Rat 2 6 Cxi (φ) = {y ∈ R :k y − xi k2≤k y − xj k2, ∀xj ∈ φ}. 4 2 Let x be the Dirac measure at x, i.e., for A ∈ R , x(A) = 1 if x ∈ A, and 0 otherwise. Then spatial point process Φ can 2

P 0 be written as Φ = xi , and Poisson-Voronoi tessellation is 0 10 20 30 40 50 60 70 80 90 100 i Intensity of Base Station ( µ ) defined as follows [16]. Definition 2. For a spatial Poisson point process Φ = P  i xi Fig. 3. Impact of BS density on handover rate: velocity ν ≡ 1, BS intensity 2 on R , the union of the associated Voronoi cells, i.e., V(Φ) = µ = 1 and mobility parameter λ = 1.

∪xi∈ΦCxi (Φ), is called Poisson-Voronoi tessellation. In cellular networks modeled by Poisson-Voronoi tessella- 4 √ written as H = ν/ s , which implies that handover rate tion, the BSs are the nuclei located according to some PPP π H 2 Φ in R . Besides, each BS xi serves mobile users which are is inversely proportional to the square root of cell size. This is located within its Voronoi cell Cxi (Φ). The latter assumption is consistent with the results in hexagonal model. Also, we note 4 √ equivalent to the hypothesis of nearest BS association strategy. that the handover rate H = π ν µ has been derived in [17] In the sequel, we also assume that PPP Φ modeling the using heuristic arguments rather than a formal proof. deployment of the BSs is homogeneous and its intensity is Fig. 2 illustrates that the analytical result (17) matches the denoted by µ. Then we can prove Prop. 4. simulation result quite well. Also, we compare the handover Proposition 4. Let µ be the intensity of the homogeneously rate of Poisson-Voronoi model, exact and approximate han- PPP distributed BSs and λ the mobility parameter. The expected dover rate of hexagonal model in Fig. 3 as a function BS intensity. They all indicate that handover rate grows linearly number of handovers E[N] during one movement period is √ with the square root of the BS intensity µ. r 2 µ We finally evaluate the three types of analytic results on E[N] = (17) π λ handover by simulating the proposed RWP mobility model The handover rate H = E[N]/E[T ] is then given by using the real-world data of macro-BS deployment in a cellular 4 √ network. The results are shown in Fig. 4. It can be seen H = ν µ (18) that Poisson-Voronoi model is about as accurate as hexagonal π model in predicting the number of handovers. Meanwhile, The insight obtained here is consistent with those of regular the approximate analytic result underestimates the number of hexagonal tessellation. In Poisson-Voronoi tessellation with handovers but more accurate in high λ region. the nuclei being homogeneous PPP Φ of intensity µ, the V. CONCLUSIONS expected value of the size sH of typical Voronoi cell is given by sH = E[|Co(Φ)|2] = 1/µ [7], where Co(Φ) is the typical In this paper, we study mobility in cellular networks. To this cell and |Co(Φ)|2 denotes the area of Co(Φ). So H can be end, we first propose a tractable RWP mobility model defined ~ 4.5 where ∆l denotes the length of the small intersection if L1 Analytic result by hexagonal model (approximate) Analytic result by hexagonal model (exact) Analytic result by Poisson−Voronoi model intersects dA, and we apply Lemma 1 in the second equality Simulation based on data from real network 4 in (21). The first equality in (21) follows since 3.5 Z ∞ ~ 3 E[|L1 ∩ dA|] = ∆l · fX1 (x, θ) xdxdθ (22) r 2.5 and |dA| = r · dθ · ∆l. So

Number of Handovers 2 √ exp(−λπr2) λ exp(−λπr2) 1.5 f(r, θ) = = (23) 2πrE[L1] πr 1 √ where we use the result that E[L1] = 1/2 λ in the last 0.5 20 40 60 80 100 120 140 160 180 200 Mobility Parameter ( λ ) equality. This completes the proof.

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