Uncoordinated Massive Wireless Networks: Spatiotemporal Models and Multiaccess Strategies

918 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 27, NO. 3, JUNE 2019 Uncoordinated Massive Wireless Networks: Spatiotemporal Models and Multiaccess Strategies

Giovanni Chisci , Member, IEEE,HeshamElSawy , Senior Member, IEEE, Andrea Conti , Senior Member, IEEE, Mohamed-Slim Alouini , Fellow, IEEE, and Moe Z. Win , Fellow, IEEE

Abstract— The massive wireless networks (MWNs) enable require ultra-reliable-low-latency communications while mon- surging applications for the Internet of Things and cyber physical itoring and system automation may require massive number of systems. In these applications, nodes typically exhibit stringent connections with more flexible constraints in terms of latency power constraints, limited computing capabilities, and sporadic and reliability. Therefore, developing a scalable wireless net- traffic patterns. This paper develops a spatiotemporal model to work that is able to accommodate the traffic generated from characterize and design uncoordinated multiple access (UMA) massively many nodes with different QoS requirements is strategies for MWNs. By combining stochastic geometry and queueing theory, the paper quantifies the scalability of UMA via mandatory to unleash the potential and benefits of the smart the maximum spatiotemporal traffic density that can be accom- world era. modated in the network, while satisfying the target operational The access and reuse of wireless resources are fundamen- constraints (e.g., stability) for a given percentile of the nodes. tal challenges for MWNs. In the context of IoT, there are The developed framework is then used to design UMA strategies several evolving low-power wireless technologies, such as that stabilize the node data buffers and achieve desirable latency, LoRa, SigFox, On-Ramp Wireless, and Weightless, to address buffer size, and data rate. such challenges [3]. The 3GPP LTE standard also pro- Index Terms— Wireless networks, Internet of Things, uncoor- vides solutions for accommodating IoT traffic within the dinated multiple access, meta distribution of the SINR. cellular networks such as device-to-device communications, machine-to-machine communications, and narrowband IoT (NB-IoT) [4]. Each of these technologies provides innovative I. INTRODUCTION solutions in terms of energy efficiency, spectrum sharing, ASSIVE WIRELESS NETWORKS (MWNs) enable and traffic prioritization to fulfill the diverse requirements Mthe communication among many nodes including of IoT applications. Due to its distributed nature and simple sensors, actuators, machines, vehicles, drones, and many operation,uncoordinated multiple access (UMA) is crucial for other smart objects (things), and will significantly con- all these technologies [3]Ð[6]. In particular, ALOHA-based tribute to the supply of big data and automation of the protocols are largely used to either transmit data or request foreseen smart world. Realizing such a massive wire- scheduling. Synchronous ALOHA is of particular interest due less connectivity is an important foundation for the Inter- to its higher performance when compared to its asynchronous 1 net of things (IoT) and cyber physical systems (CPSs) counterpart. Nevertheless, the performance of UMA degrades [1], [2]. Such massively many connected nodes are required as the number of nodes grows [3]Ð[5], which raises the to communicate with each other and connect to the Internet following fundamental questions: i) how to determine the with an application specific quality of service (QoS). For scalability of UMA in MWNs in terms of per-node traffic instance, public safety and autonomous driving applications intensity, node density, and required transmission rate; and ii) how to design a MWN with guaranteed stability and target Manuscript received December 18, 2017; revised October 28, 2018; performance level for a given network percentile. The answers accepted December 7, 2018; approved by IEEE/ACM TRANSACTIONS ON to these questions will shed lights over the scalability of UMA NETWORKING Editor E. Yeh. Date of publication April 11, 2019; date of in MWNs, highlight its operational bottlenecks, and enable current version June 14, 2019. This work was supported in part by FAR and applications for massive IoT and CPSs. the “5x1000” Young Researcher Mobility Project, University of Ferrara, Italy, in part by the KAUST Sensor Research Initiative under Award OSR-2015- SENSORS-2700, and in part by the National Science Foundation under Grant A. Related work CCF-1525705. This paper was presented in part at the 2017 IEEE International Conventionally, multiple access and scheduling schemes Symposium on Wireless Communication Systems. (Corresponding author: are characterized and designed via mathematical models that Giovanni Chisci.) involve interacting queues [10]Ð[15]. However, pessimistic G. Chisci and A. Conti are with the Department of Engineering and CNIT, University of Ferrara, 44121 Ferrara, Italy (e-mail: [email protected]; collision models are often used and only networks with [email protected]). small size are studied. More realistic signal-to-interference- H. ElSawy is with the Electrical Engineering Department, King Fahd plus-noise ratio (SINR)-based two-queue interactions are University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia (e-mail: investigated in [16] and [17]; however, a generalization to [email protected]). MWNs appears to be a daunting task. Integrated graph and M.-S. Alouini is with the Computer, Electrical and Mathematical Science and Engineering Division, King Abdullah University of Science and Technol- queueing theory models for interacting queues are presented ogy, Thuwal 23955, Saudi Arabia (e-mail: [email protected]). in [18] and [19]. However, such approaches do not capture the M. Z. Win is with the Laboratory of Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, MA 02139 USA (e-mail: 1Note that network synchronization is viable in cellular and GPS-supported [email protected]). MWNs such as NB-IoT [7], or through distributed network synchronization Digital Object Identifier 10.1109/TNET.2019.2892709 [8], [9]. 1063-6692 © 2019 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. CHISCI et al.: UNCOORDINATED MWNs: SPATIOTEMPORAL MODELS AND MULTIACCESS STRATEGIES 919

Fig. 1. Macroscopic and microscopic scales of the spatiotemporal model: in blue the intended link, in red other active links, and in green the inactive links. aggregate effect from non-neighboring nodes, which may be mainly depends on their position. On the microscopic scale, significant in MWNs. In conclusion, previous models ignore queueing theory is used to account for the per-node buffer the spatial and physical layer effects on queue interactions, state and transmission protocol state, which are determined which are fundamental in MWNs. by the traffic generation, UMA strategy, and link quality (see Spatial and physical layer attributes of wireless networks are Fig. 1(c)). We also consider the intrinsic interdependency usually captured via stochastic geometry models [20]Ð[25], between the macroscopic- and microscopic-scale analysis, where node positions are described according to point which enables an accurate characterization of link qualities processes. Stochastic geometry based models account for radio by considering the aggregate interference from active nodes propagation in conjunction with the theory of point processes (i.e., accounting for the buffer and protocol states). The meta to model concurrent transmissions that impose mutual interfer- distribution of the SINR [37]Ð[40] is used to account for the ence among spatially coexisting nodes. However, the majority diverse qualities of links coexisting in the network and to of stochastic geometry based models ignore temporal attributes classify them into location-dependent QoS-classes. To this end, such as traffic generation and queue occupation, as well as we quantify the network scalability via the γ-stability and γ- queueing key performance indicators (KPIs) such as latency operativity Pareto frontiers, in which γ is a design parameter, and buffer occupation, all fundamental to characterize MWNs. in terms of per-node traffic intensity, node density, target To jointly account for the spatial and temporal attributes transmission rate, and UMA persistence. Operating beyond the of MWNs, recent studies combine stochastic geometry and γ-stability frontier implies that the (1 − γ)-percentile of the queueing theory [26]Ð[32]. The spatiotemporal interactions network operates with unstable queues, thus overflowing their between nodes with non-saturated data buffers are studied in data buffer. Analogously, beyond the γ-operativity frontier at [26]Ð[30]. However, the analysis in [26]Ð[28] assumes high least the (1 − γ)-percentile of the network does not achieve mobility scenarios such that an independent realization of the target KPI value (e.g., end-to-end latency or buffer size). the network occur at every time slot. The spatiotemporal This paper develops a framework to design networks with dynamics in static large scale networks is characterized in a guaranteed stability and a desired KPI value for a target [29]Ð[31]. However, [29] and [31] utilizes a dominant system percentile of the nodes in the MWN. The main contributions model where nodes in the network are always active even of this paper can be summarized as follows: if they have empty queues in order to simplify the analysis, • develop a holistic mathematical model that accounts for which can be quite pessimistic for MWNs. In [30], the uplink the intricate spatiotemporal interactions in MWNs; traffic for IoT-enabled cellular networks is analyzed, where • present the γ-stability (γ-operativity) Pareto frontiers that every queue in the network behaves like the typical queue, guarantee stable queues (KPI value) for the γ-percentile which is not generally true for MWNs. The performance of of the MWN; and UMA is characterized in [32] for static MWNs, which are • propose UMA strategies to balance the tradeoffs between the most challenging to analyze due to the spatiotemporal transmission deferral, power-ramping, and persistence. correlation of the interference at different locations. However, only UMA with single transmission power and best-effort C. Notation and Organization traffic is considered. An alternative approach to characterize 1) Notation: Along the paper we use the upright serif font, latency via bounds in static MWNs is proposed in [33], which e.g., v, for random variables (RVs) and the italic roman font, is based on an integrated stochastic geometry and network e.g., v for their instantiation. Vectors are bolded, e.g., v and calculus model. v; matrices are bolded and uppercase, e.g., V and V ;andsets B. Contribution are uppercase, e.g., S and S, as point processes, e.g., Π and This paper presents a spatiotemporal model for UMA in · 2 Π. The notation [ ] is used to denote the transpose operator MWNs, as depicted in Fig. 1(a). The approach relies on the and V [i,j] is used to denote the ith-row jth-column element. joint utilization of stochastic geometry and queueing theory. V [:,j] and V [i,:] are used to denote all elements in the jth On the macroscopic scale, stochastic geometry is used to column and all elements in the ith row, respectively. With a account for the mutual interference among active nodes (i.e., slight abuse of notation we use v[i] to denote the ith element those with non-empty data buffer, as in Fig. 1(b)), which [i] v of a vector. The functions fv(·), Fv(·),andF˜v(·) denote 2This work focus on synchronous systems, however can be generalized to the probability density function (PDF), cumulative distribution asynchronous systems by modifying the stochastic geometry part according function (CDF), and complementary cumulative distribution to [34]Ð[36] to account for partially overlapping transmissions in time. function (CCDF) of the RV v, respectively. We denote by 920 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 27, NO. 3, JUNE 2019

P{·}, E{·}, P◦{·},andE◦{·} the probability, expectation, Palm probability, and Palm expectation with respect to (w.r.t.) the typical point, respectively. With the over-bar we denote the complement operator, i.e., v˜ =1− v; · denotes the ceiling

function; and ½S (n) is the indicator function, which assumes value one if n ∈S and zero otherwise. 2) Organization: Section II presents the system model and the methodology; Section III shows the macroscopic analysis corresponding to the stochastic geometry part of the framework; Section IV delves into the microscopic analysis which corresponds to the queueing theory part of the framework; Section V develops the analysis framework for the spatiotemporal model and introduces the KPIs; Section VI presents numerical results; and, ﬁnally, Section VII provides the ﬁnal remark.

II. SYSTEM MODEL A. Spatial and Physical Layer Parameters This paper considers MWNs in which nodes are scattered in R2 according to a Poisson bipolar network (PBN) [41, Def. Fig. 2. Three QoS-classes classification for the TSP within a realization of 5.8]. That is, the potential transmitters belong to a Poisson N = N = N =1 2 + the PBN operating with ALOHA for c p t where the filled point process (PPP) Π = {yj ∈ R ,j ∈ N }, with intensity λ (empty) shapes represent transmitters (receivers). and each transmitter communicates with a dedicated receiver located at a fixed distance R in a uniformly random direction. When a transmission failure occurs, the node changes the A realization of the PBN is shown in Fig. 1(a). For simplicity, transmission power at the next channel access according to we consider the unbounded path loss model (r)=r2η,where the employed power-ramping scheme. A node uses the same r is the propagation distance and η is the amplitude loss expo- transmission power Pi with i ∈{1, 2,...,Np} (starting from nent. We consider zero-mean additive white Gaussian noise the initial power level P1) for up to Nt consecutive retrans- 2 5 (AWGN) with average power σ and a Rayleigh fading with missions, where Nt is a design parameter. If the transmission unitary mean power gain. All channel gains are independent is still unsuccessful, the next power level Pi+1 is utilized for and identically distributed (IID) in space and time. Each trans- the next Nt retransmissions. We consider two power-ramping mission is defined by a frequency band and power level pair. schemes, i.e., ramping-up and ramping-down. Ramping-up Uniformly random channel selection among a common set (i.e., Pi+1 >Pi) is a greedy choice to prioritize delayed of Nc frequency channels is adopted at each channel access, transmissions at the expense of increasing the interference which is denoted as frequency ALOHA (F-ALOHA) [42]. level and deteriorating the transmission success probability The transmission power is selected among Np different power (TSP) of other nodes’ packets. In contrast, ramping-down levels in the set P = {P1,P2,...,PNp } according to the (Pi+1

3Bernoulli traffic arrival is a simple and widely accepted traffic model [29], 5Note that consecutive retransmissions do not necessarily occur in consec- [31]. Other traffic generation models can be adopted at the cost of more utive time slots due to the ALOHA random backoff. involved queueing theory analysis as described in Section IV. 6The proposed ramping scheme is based on the acknowledgments. More 4Positive and negative acknowledgement packets are sent instantaneously sophisticated channel-aware and coordinated power control schemes are left over an error-free control channel. for future extensions. CHISCI et al.: UNCOORDINATED MWNs: SPATIOTEMPORAL MODELS AND MULTIACCESS STRATEGIES 921

As shown in Fig. 1, the macroscopic effect of the network • For each QoS-class n ∈N, use the arrival interference and the microscopic effect of the per-node behav- probability a and the class dependent TSPs ior are interrelated. Furthermore, the static network topology D[:,n] to find the marginal steady-state distribution [0] [1] [Np] leads to a location-dependent packet departure probability of πn =[πn ,πn ,...,πn ] of node physical phases, [0] each queue, as depicted in Fig. 2, where nodes at sparse loca- where πn is the probability of empty buffer and tions are more likely to have successful transmissions while [np] πn is the probability of being in a retransmission nodes at congested locations are more likely to experience phase that employs the power level np. transmission failures. For a given link, the packet departure • Aggregate the per-class states to get the overall probability depends on i) the location of the intended receiver w steady-state marginal distribution of node physi- relative to the other transmitters, ii) the power employed by the [i] 1 N [i] intended transmitter, and iii) the status of all other transmitters cal phases, where w = N n=1 πn . (i.e., node activity and transmission power). For a given (c) Iterative solution Np [i] network realization Π = Π,letnp ∈Np = {1, 2,...,Np} be • Initialize w such that i=0 w =1. • the power level index and znp,j be the SINR at the intended Iterate (a) and (b) as in Fig. 3 until convergence. receiver of the transmitter located at yj ∈ Π when operating III. MACROSCOPIC STOCHASTIC GEOMETRY ANALYSIS with the power level Pnp . Further, let pnp,j P{znp,j > θ|Π = Π} be the TSP of node j. A departure from the queue To characterize packet departures from the spatially interact- of the jth wireless link occurs upon the joint independent ing queues of Fig. 1 it is necessary to characterize the location- events of channel access (any of the Nc available) and trans- dependent TSP across the network (see Fig. 2). mission success, i.e., with probability papnp,j.Foranyfixed Consider the point process of the transmitters with activity Π ˙ + = Π,thevaluespnp,j along the network are realizations of marks, i.e., Π = {yj, aj, ρj; j ∈ N } where marks aj are p p the RV np . The distribution of the TSP np across the network IID Bernoulli RVs with probability pfa = pa/Nc, due to the is denoted as the meta distribution of the SINR with CCDF F-ALOHA channel access scheme, and ρj ∈P∪{0} are ˜ P◦{p } Fpnp (φ) np >φ which indicates the fraction of links discrete marks of the node physical phases. The marks aj, in the network successfully transmitting with a probability for j ∈ N+, are independent due to the random channel greater than φ [37]. To obtain the meta distribution it is selection and independent ALOHA transmissions. In contrast, necessary to account for node activities (i.e., physical phases), the marks ρj are spatially and temporally correlated due to + which are modeled by the IID RVs ρj ∈P∪{0} for j ∈ N the static network topology that leads to location-dependent with identical distributions w =[w[0],w[1],...,w[Np]] where SINRs. As shown in Fig. 2, nodes at sparse locations have w[0] is the probability of empty buffer (i.e., physical phase 0) frequent successful transmissions, thus emptying their queues and w[i] is the probability of having non-empty buffer while and being less likely to explore all the power levels provided being in any of the retransmissions (i.e., logical phases) of by the protocol described in Sec. II-B. On the other hand, 7 the power level Pi (i.e., physical phase i). Furthermore, nodes at congested locations are more likely to have transmis- to obtain the distribution w by an averaging step over all the sion failures, thus accumulating packets in their queues and location-dependent distributions of node physical phases, it is exploring all power levels due to several retransmissions. Such necessary to solve a finite number of queueing theory problems a spatiotemporal correlation between node transmission phases each of which require a TSP value drawn from the meta complicates the analysis and necessitates the introduction of distribution. Thus, there is a coupling between the macroscopic the following approximation, which will be further validated and microscopic scale problems which is circumvented by in Section VI. using an iterative solution. To analyze a MWN, we utilize Approximation 1 (Physical-Phase Marks IID Distribution): the following steps also depicted in Fig. 3. The marks ρj for all j are IID over different nodes and (a) Macroscopic analysis via stochastic geometry time slots according to the spatially averaged steady-state distribution w =[w[0],w[1],...,w[Np]].8 ∈N • For every power level np p, find the statistical Remark 1: Approximation 1 is commonly known as mean- p w th moments of np as a function of ,wheretheb field approximation [26], [30], which ignores the spatiotem- moment is denoted by Mb,np . poral correlation between the physical phases of interfering ∈N • For every power level np p, approximate nodes. Intuitively, the exact locations of interfering nodes p · the PDF f np ( ) via the beta distribution having and their correlated activities are of less significance to the moments M1,np and M2,np . SINR distribution at an intended receiver. Instead, the number ∈N • For every power level np p, discretize (intensity) of the nodes employing each transmission power p · the PDF f np ( ) into N equiprobable values along with their relative locations with respect to the intended d each corresponding to a QoS-class, i.e., np = receiver are the main factors influencing the SINR, which are [1] [2] [N] [dnp ,dnp ,...,dnp ]. captured via the mean-field approximation. [i] [np,i] The meta distribution of the SINR is characterized in the • Construct the TSP matrix D = padnp ,where each row of D contains the TSPs for all QoS- following by Approximation 1. classes using the same power level np ∈Np and each column contains the TSPs for all power levels A. Meta Distribution Analysis within the same QoS-class n ∈N = {1, 2,...,N}. rx Let yi ∈ Π and yi be the locations of a transmitter and of (b) Microscopic analysis via queueing theory its intended receiver, respectively. The TSP for the intended

8The spatially averaged steady-state distribution w is obtained in Section IV 7In the paper some distributions are given in the form of a stochastic vector, via queueing theory by averaging over all possible location-dependent distri- of which components sum up to one. butions of the node physical phases. 922 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 27, NO. 3, JUNE 2019

Fig. 3. Methodology of analysis and iterative algorithm within the proposed spatiotemporal model. receiver can be characterized in terms of the transmission with power and the marked PPP of interferers as b ∞ Np [i] k 1−η π b k+1 w θnp,i n ,i P{zn ,i |Π } − η p p p >θ = Π (1a) βb,np = ( 1) pfa u du −2η η k 0 u + θnp,i Pn hiR k=1 i=1 = P p >θΠ rx −2η 2 (4) ρj ajhjyj − yi + σ 2η [i] th j : yj ∈Π\{yi} where θnp = θR /Pnp , θnp,i = θnp Pi,andw is the i (1b) 1 1 1 2 w − η 2 element in .Letκ = πΓ(1 η )Γ(1 + η )θ R ,thefirst where hi is the channel gain of the intended link and σ moment of pnp can be written as is the mean power of the zero-mean AWGN. Note that the ⎧ ⎫ relative locations of the interfering transmitters are fixed. Np 1 ⎨ η ⎬ −bθ σ2 [i] Pi However, the randomness in (1) is due to the spatiotemporal np − M1,np = e exp ⎩ κλpfa w ⎭. (5) variation of fading, random channel selection, node activities, i=1 Pnp and transmission phases. In addition, due to the location- yrx dependent interference seen by each receiver i , pnp,i across In case of Np =1(single-transmission-power scheme) the p the network can be considered as realizations of the RV np integral expression in (4) reduces to the following closed forms ˜ with CCDF Fpnp (φ), namely the meta distribution of the for b =1, 2 respectively as SINR. Such a meta distribution reveals the fraction of the 2 − θ1σ [0] network having a TSP greater than φ and, thanks to the M1 = e exp{−κλpfaw˜ } (6a) ergodicity of the PPP, can be obtained via the Palm probability [0] 2 pfaw˜ 1 ˜ P◦{p } −2 θ1σ [0] as Fpn (φ)= np >φ [37]. By following [37], the PDF M2 = e exp − 2κλpfaw˜ 1 − 1 − p 2 η of pnp can be accurately approximated by the beta distribution as follows9 (6b) M (α+1)−1 1,np [0] [0] 1−M − φ 1,np (1 − φ)α−1 where w˜ =1 w . fpnp (φ)= M α (2) Proof: See Appendix. B 1,np ,α 1−M1,np Remark 2: The simplified expressions in (6) show that the spatially averaged TSP is exponentially decreasing in λ, pfa, (M1,np −M2,np )(1−M1,np ) [0] with α = 2 where B(·, ·) is the com- and w˜ . Hence, having less traffic per node (i.e., higher M2,np −M 1,np probability of empty buffer) or more channels to access can plete beta function and the moments M1,np and M2,np are given in the following Theorem 1. significantly improve the TSP. Furthermore, (6) reveals that th →∞ → Theorem 1: The b order statistical moment of the TSP given the constant product λpa,forλ and pa 0 then → 2 p pn in a PBN as described in Section II is given by M2 M1 , thus reducing the variance of np . This means p that the TSP becomes location-independent. ◦ b −bθ σ2 E {p } np {− } Mb,np np = e exp λβb,np (3) The expression in (5) shows the positive impact of ramping up Pnp for the intended link. However, it also captures the 9An exact but complex integral expression for the meta distribution can negative impact of ramping up the power of interfering nodes 1 be obtained via Gil-Pelaez inversion theorem. However, it is shown in [37] [i] η that the beta approximation via moment matching is very tight, which highly w Pi . Thus, power ramping has a dual effect, which is simplifies the subsequent analysis and computations. discussed in more detail in Section VI. CHISCI et al.: UNCOORDINATED MWNs: SPATIOTEMPORAL MODELS AND MULTIACCESS STRATEGIES 923

Algorithm 1 Meta distribution quantization

Require: N, M1,np ,andM2,np . d [n] − Ensure: A TSP vector np such that (s.t.) Fpnp (dnp ) [n+1] ∈N ∈N\{ } Fpnp (dnp )=1/N for all np p and n N . 1: for np =1, 2,...,Np do

2: Consider fpnp (φ) by (2); 3: Set φ1 =0and φN+1 =1; 4: for n =1, 2,...,N − 1 do 5: Retrieve the values of φn and φn+1 s.t.

φn+1 − 1 Fpnp (φn) Fpnp (φn+1)= fpnp (φ) dφ = φn N

Fig. 4. Quantization of the distribution of the TSP for N =10. [n] 6: Compute dnp s.t.

d[n] φ B. Network Uniform Partitioning np n+1 1 fpn (φ) dφ = fpn (φ) dφ = ; p [n] p Consider that the PBN links are divided into different φn dnp 2N QoS-classes, each according to its location-dependent TSP (or equivalently queue departure probability). If each possible p 7: end for value of the continuous RV np is considered as a TSP, then, [n ,:] [n] 8: D p dn d n an inﬁnite number of QoS-classes are required to characterize Assign = p and f np (d p )=1/N ; the location-dependent performance of the nodes. For tractabil- 9: end for ity, we partition the network into N QoS-classes by resorting 10: Return D. to the following approximation. Approximation 2 (Meta Distribution Quantization): Chose of packets ln in the transmitter buffer, the physical power- ∈ N p N and quantize the continuous RV np with distrib- level phase, and the logical retransmission phase (see Fig. 5). p dn ∈ ution f np (φ) as in (2) to the uniform discrete RV p As discussed in Section III, the location-dependent perfor- [1] [2] [N] dnp =[dnp ,dnp ,...,dnp ] where each value has probability mance of nodes leads to temporally correlated packet departure

1/N [43]. The elements of dnp are selected such that the mean probabilities for each queue (see Figs. 2 and 4). To capture error such location-dependence while avoiding queueing memory complications, we resort to the following approximation. [n] Ep {arg min d − pn } Approximation 3 (QoS-Aware But Temporally-Independent np n np p 1 Departure Rates): Recall that D is the TSP matrix obtained by [n] − discretizing the meta distribution of the SINR in Algorithm 1. = fpnp (φ)argmin dnp φ dφ 0 n At a generic time slot, the departure probability of a packet from a queue that belongs to the nth QoS-class and operates [n ,n] is minimized. The procedure is described in Algorithm 1 and with power Pn is given by paD p . A queue does not ∈P p illustrated in Fig. 4. The process is repeated for all np and change its QoS-class over time. The departures from the same the number of classes is selected to achieve sufficient accuracy queue in different time slots with transmission powers Pnp in the description of the distributions of the TSP. and Pnp are considered to be independent with probabilities As shown in Algorithm 1, the Np × N matrix D can be [n ,n] [n ,n] paD p and paD p , respectively. constructed by using the vectors dnp corresponding to each [i,j] [j] Approximation 3 suggests that the mean departure proba- power level. The element D = di defines the TSP of th bility of each queue is determined by its static location, which the queue representing a link of the j QoS-class when the does not change over a sufficiently large number of time slots. intended transmitter operates with power level Pi.Asshown However, the departure probability from each queue is still D in Fig. 3, the matrix of TSPs represents the interface random and independent from one time slot to another due between the stochastic geometry and the queueing theory to the independent random variations of the channel gains, analyses. An example of the network partitioning proposed of the interfering nodes’ activities, and of transmission powers. in this section is illustrated in Fig. 2 for a single power level, Based on Approximation 3, the queueing model for each link i.e., Np =1,andN =3QoS-classes. Since only a single is represented via a Geo/Ph/1 DTMC [44, Ch. 5.8] with the power level is considered, TSPs are given in the 1 × 3 vector [1] [2] [3] QoS-class dependent state diagram shown in Fig. 5, where d ∈{d =0.56,d =0.71,d =0.84} [:,n] , such that each the packet departure probabilities are given by paD .In value represents 1/3 of the PBN links. Geo/Ph/1 queues, the phase type distribution [44, Ch. 2.5.3] of departures is constructed via an absorbing Markov chain [44, Ch. 3.4.4] that accounts for all the transmission phases IV. MICROSCOPIC QUEUEING THEORY ANALYSIS that a queue belonging to a given QoS-class may experience To address the microscopic-scale analysis, each wireless until a packet departure.10 To build the transition matrix of link of the PBN, (e.g., that belonging to the nth QoS-class) is considered as a queueing system (see Fig. 1) that is modeled 10In the absorbing Markov chain, the absorption denotes successful trans- via a discrete-time Markov chain (DTMC) to track the number mission. Sn in (8) define the dynamic of the phases before absorption. 924 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 27, NO. 3, JUNE 2019

m =(np − 1)Nt + nt. The index np of the transmission power used by the intended transmitter can be retrieved from [m] [n] 11 m as np = m/Nt and, hence, sn = padnp . Since the transmission of each new packet starts with power np =1 and nt =1, the initialization vector for the absorbing Markov chain is β =[1, 0,...,0] ∈ RM×1. th By considering Sn, sn,andβ, the DTMC of the n QoS- class link is dedscribed by a quasi-birth-and-death (QBD) process [44] with the following block transition matrix ⎡ ln=0 ln=1 ln=2 ln=3 ln=4 ··· ⎤ ln=0 Bn Cn l =1 ⎢ E A A ⎥ n ⎢ n 1,n 0,n ⎥ ln=2 ⎢ A2,n A1,n A0,n ⎥ Pn = ⎢ ⎥ ln=3 ⎣ A2,n A1,n A0,n ⎦ ...... (10)

where ln represents the different levels (i.e., number of packets th in the queue) within a generic n class queue, Bn = M×1 1×M a˜ ∈ R, En =ãsn ∈ R , C = aβ ∈ R , M×M M×M A0,n = aSn ∈ R , A1,n = asnβ +ãSn ∈ R , M×M A2,n =ãsnβ ∈ R , and the empty blocks are filled by Fig. 5. The DTMC for a class n queue with two power levels P = {P1,P2} and Nt =3retransmissions for each power level. zeroes. The steady-state distribution of a queue with transition matrix Pn is then obtained by solving the following system of the considered Geo/Ph/1 DTMC, we first define the following equations Nt × Nt matrices, x P x ⎧ n n = n (11a) ⎨pã if i = j xn1 =1 (11b) [i,j] [np,n] Hn ,n = paD˜ if j = i +1 (7a) [0] [1] [2] [l] p ⎩ where xn =[xn , xn , xn ,...,xn ,...] is the steady-state 0 otherwise probability vector for all levels l =1, 2,... of the nth QoS- [0] [l] [np,n] [i,j] paD˜ if i = Nt and j =1 class queue; xn is the probability of empty buffer; xn = J [1] [2] [M] np,n = (7b) 0 otherwise [xl,n,xl,n,...,xl,n ] is the distribution of phases in level l, [m] and x is the probability of being at level l and phase m. − D˜ [np,n] − D[np,n] l,n where pã =1 pa and =1 .The(7a) Since the considered queueing model is ergodic and irre- contains the probabilities of ALOHA deferral and horizontal ducible [44, Ch. 3.4.4], the system of equations in (11) has a transition between the logical phases (i.e., retransmissions) unique solution if the queue is stable. To check the stability, within the same power level, while (7b) contains the prob- we first construct the matrix An = A0,n + A1,n + A2,n = abilities of horizontal transition from one power level to the snβ + Sn, and solve the system Np Nt next. Given that there are different power levels with ψ A ψ retransmissions within each power level, the transient state n n = n (12a) sub-stochastic matrix for the absorbing Markov chain is given ψn1 =1 (12b) by the following block matrix ψ [1] [2] [M] ⎡ ⎤ for n =[λn ,λn ,...,λn ], which is the marginal distrib- H1,n J1,n ution of all transmission phases when the buffer is non empty. ⎢ H J ⎥ ⎢ 2,n 2,n ⎥ The queue is stable if and only if (i.f.f.) [44] ⎢ . . ⎥ Sn = ⎢ .. .. ⎥ (8) ψnA2,n1 > ψnA0,n1. (13) ⎣ H J ⎦ Np−1,n Np−1,n Condition (13) leads to two different types of analysis, namely J H Np,n Np,n stable-QoS-class and unstable-QoS-class analysis. × where the empty blocks are filled by zeroes. The M M A. Stable-QoS-class Analysis matrix Sn,whereM = NpNt, accounts for all power If the condition in (13) is satisfied, then the QoS-class levels and retransmission phases until a packet is successfully represented by (11) is stable and has a unique solution. Then, transmitted (i.e., absorption). In case of F-ALOHA with no using the systematic matrix analytic method (MAM) [44], power ramping (i.e., M = Np = Nt =1) the transient state the system in (11) has the following solution sub-stochastic matrix is simply given by Sn = Hn + Jn. −1 The absorption probabilities from each transient state phase is [0] CT I − R −1 1 xn = 1+ ( n) (14a) defined by the M-sized vector [0] [l] xn CT for l =1 [1] [2] [M] x = (14b) sn = 1 − Sn1 =[s ,s ,...,s ] (9a) n [1] l−1 n n n xn Rn for l>1 [m] where 1 is a column vector of ones of the proper size and sn 11The superscript m is used for brevity where the specification of np and is the probability that a packet is transmitted and successfully nt is not needed. In general, a transmission phase is always associated with decoded within a link of class n for a transmitter in phase a single transmission power-retransmission index pair. CHISCI et al.: UNCOORDINATED MWNs: SPATIOTEMPORAL MODELS AND MULTIACCESS STRATEGIES 925

−1 M×M where T = I − asnβ − a˜Sn − Rna˜snβ , I ∈ R is A. Stability Conditions M×M the identity matrix and Rn ∈ R is the rate matrix given [1] [2] [M] According to Loynes theorem [45], a stable queue has the by Rn = aSnT .Letζn =[ζn ,ζn ,...,ζn ] be the vector packet departure probability greater than the packet arrival containing the probabilities of being in any transmission phase probability, which guarantees finite mean buffer size and irrespective of the buffer level, then waiting time in queue. The NC and SC for stability can be [1] −1 obtained through the mildest and most severe interference ζn = x (I − Rn) . (15) n scenarios, respectively. Note that NC and SC are computed 12 Note that ζn is a sub-stochastic vector as it does not contain directly and do not require any iterative solution. As an the probability of empty buffer. The steady-state marginal example, for a ramping-up strategy, the NC can be obtained nc distribution for the powers levels πn is given by by setting w = w =[1− a, a, 0,...,0], which captures the [0] [0] mildest interference scenario by activating the nodes with fresh πn = xn (16a) packets only. In contrast, the SC can be obtained by setting M w = wsc =[0,...,0, 1], which captures the most severe [np] [m] πn = ½M(m) ζn (16b) interference scenario where all nodes are active regardless of m=1 their buffer state as shown in the following corollary. ∈M { } Corollary 1: ½M t p Assuming an interference limited regime, where (m) equals one if m = m : m /N = n th and zero otherwise, which ensures that the sum is over the the b -order moment of the TSP in the mildest and in the logical phases (i.e., retransmissions) within the same physical most severe interference scenarios are, respectively, given by phase (i.e., transmission power) with index np. ∞ b 2η k k nc πλ b apfaθR (−1) Mb =exp 2η η−1 du B. Unstable-QoS-class Analysis η k u + θR η k=1 0 u If the condition in (13) is not satisfied, then the QoS-class (19a) represented by (11) is unstable and the node buffers will ∞ b 2η k k overflow with probability one. In this case, the probability sc πλ b pfaθR (−1) [0] M =exp du . of empty buffer is zero, i.e., xn =0, and the marginal b k 2η η−1 η k=1 u + θR u η distribution of the phases is the one obtained from solving (12). 0 Hence, the steady-state marginal distribution πn for the power (19b) levels is given by Proof: Follows directly from Theorem 1 by ignoring noise [0] πn =0 (17a) and assuming that all active transmitters operate with equal nc sc M powers, and setting w = w and w = w . nc sc nc sc [np] [m] D D πn = ½M(m) λn . (17b) By using Mb and Mb , the TSP matrices and , m=1 respectively, can be constructed via Algorithm 1. By using Dnc and Dsc, the NC and SC can be respectively verified C. Class Aggregation for each QoS-class via condition (13). It is obvious that a As shown in Fig. 3, the class aggregation is the conjunction satisfied SC condition implies a satisfied NC condition and between the queueing theory and stochastic geometry analysis, that unsatisfied NC implies unsatisfied SC condition. Hence, thus computing the spatially averaged w. From Approxi- the NC and SC verification lead to the following cases: i) mation 1, the spatial and temporal correlations among the both NC and SC are satisfied; ii) neither of NC and SC are physical-phase marks of the interfering nodes are ignored and satisfied; and iii) the NC is satisfied and the SC is unsatisfied. the aggregate interference is characterized assuming IID node When both (neither) conditions are satisfied, then it can be th physical-phase distribution w. directly concluded that the n QoS-class is stable (unstable). Remark 3: We compute the IID node physical-phase dis- However, in the third case, further investigation through the th tribution w of Approximation 1 by an average over the N iterative algorithm is required to check the stability of the n equiprobable QoS-classes obtained by Approximation 2. For QoS-class. i =0, 1,...,Np we have N B. Iterative Algorithm [i] 1 [i] w w = πn . (18) The iterative algorithm used to find the steady state is N n=1 depicted in Fig. 3 and presented in Algorithm 2. The Algo- rithm 2 solves the fixed point equation w = g(w),wherew is the steady-state averaged marginal distribution of the node V. A NALYSIS FRAMEWORK physical phases. Starting from a valid stochastic vector w, To characterize the network performance and obtain the the function g(·) involves the steps described in Section II-C, KPIs, the coupling between the stochastic geometry and the depicted in Fig. 3, and particularized in Algorithm 2. Given queueing theory parts needs to be solved. Specifically, the TSP the complicated and multi-stage nature of such fixed point matrix D requires the spatially averaged physical-phase distri- equation, addressing the solution unicity and convergence is bution w. Meanwhile, w requires D to be computed. Before seldom possible.13 For our model, the existence and unicity delving into the iterative solution, it is worth highlighting the of the solution and the convergence of the iterative procedure necessary condition (NC) and sufficient condition (SC) for stability of each QoS-class. 13The solution unicity and convergence were established for simpler models (e.g., only spatially averaged analysis and single transmission power) that 12A sub-stochastic vector has components that are probabilities but it does involve iterative solution between stochastic geometry and queueing theory not sum up to one. [26], [28]. 926 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 27, NO. 3, JUNE 2019

Algorithm 2 Iterative solution (networks of Geo/Ph/1 queues) Considering the space of the system parameters R = 2 + 2 2 Require: λ, pa, a, N, θ, η, R, σ , Nc, Np, Nt, P,andβ. {λ, θ, a, pa,Nc}∈G=[R ] × [0, 1] × N and the function G→ Ensure: D and for all n =1, 2,...,N, xn for stable classes γsu : [0, 1],wedefinetheγ-stability region as and ψn for unstable classes. Rs {R R } [1] [1] γ : γsu( ) >γ (21) 1: Inizialize k=1, 1, w s.t. w 1 =1. [k] [k−1] 2: while max w − w ≥ do i.e., as the region where a fraction of the nodes greater than n γ 14 R 3: for np ∈P do operates with stable queues. For a given , in addition to the de facto γsu obtained through (20), the percentile of the 4: Compute the moments M1,n , M2,n by (3), (4); p p PBN satisfying the NC and SC conditions can be obtained by 5: Run Algorithm 1 to populate D; nc utilizing Corollary 1 though the following steps: i) use Mb 6: end for sc nc sc and Mb in Algorithm 1 to obtain, respectively, D and D ; ∈N nc sc 7: for n do ii) use D and D to construct the queueing model for each 8: Construct Sn and sn by (7), (8), and (9); QoS-class and check (13); and iii) utilize (20) to obtain the 9: Construct An = snβ + Sn; percentile PBN satisfying the NC and SC conditions. 10: Obtain ψn by solving (12); Remark 4: Since stability implies finite latency, γsu is of 11: if ψnA2,n1 > ψnA0,n1 then particular interest for best-effort traffic, which requires even- th 12: Queues of the n class are stable; tual packet delivery and is tolerant to latency. 13: Obtain xn through (14); 14: Evaluate πn through (15) and (16) D. Key Performance Indicators 15: else th Algorithm 2 provides the steady-state probabilities (i.e., xn 16: Queues of the n class are unstable; ψ [0] and n), the moments of the SINR (i.e., M1,np and M2,np ), 17: Set πn =0and use ψn to find πn via (17); and the TSP matrix D, from which a variety of KPIs can be 18: end if computed [46], [47]. For instance, following [44, Ch. 5.8] for 19: end for Geo/Ph/1 queues, the average number of packets in the system [k+1] 20: w and the average buffer length of a node that belongs to the Calculate by (18); th 21: end while n QoS-class are given, respectively, by 22: Return xn for stable classes, ψn for non-stable classes, [1] −2 ¯ln = E {ln} = xn (I − R) 1 (22a) M1,np , M2,np ,andD. [1] −2 q¯n = E {qn} = xn R(I − R) 1. (22b)

Let tn and tqn denote the total latency and the waiting can be argued by the following facts: i) each legitimate w time in the buffer (i.e., the number of time slots that a packet leads to a unique distribution of the TSP, and hence, a unique spends in the queue before the service starts), respectively. The TSP matrix D; ii) each TSP matrix D leads to a unique average total latency and average waiting time in the buffer queueing steady-state solution πn for each class; and iii) the th for the n class are provided by the discrete time version of fixed point equation operates over a stochastic vector (i.e., [i] [i] the Little’s law [46], respectively, as 0 ≤ w ≤ 1, 0 ≤ [g(w)] ≤ 1, w1 =1,and[g(w)]1 =1) and the lefthand side w spans the entire multidimensional t¯n = E {tn} = E {ln} /a (23a) domain. It is worth highlighting that in case of satisfied NC t¯qn = E {tqn} = E {qn} /a. (23b) but unsatisfied SC for a given QoS-class, there may exist two solutions for the queueing system steady-state probability The service latency ts (i.e., the number of time slots required [0] vector πn. One solution has πn =0(i.e., no empty buffers) to complete the transmission once a packet arrives at the head due to the unsatisfied SC. The other solution would have of the buffer) accounts for ALOHA transmission deferrals and [0] πn > 0 if there exists a steady-state configuration w that transmission failures due to the event {SINR ≤ θ} within each lead to a satisfied (13). Since the stable solution is the one of transmission phase. The average service latency for a node interest, we initialize w for k =1as w[1] =[1−a, a, 0,...,0] belonging to the nth QoS-class is given by in Algorithm 2. It is worth noting that the aforementioned Np [i] arguments about convergence of Algorithm 2 are supported ¯ E {t } πn 1 via extensive Monte Carlo simulation as shown in Section VI. tsn = sn = [0] D[i,n] (24) i=1 1 − xn pa [i] [0] where πn /(1 − xn ) represents the average time spent in the C. Network Stability physical phase i. It is worth noting that the average waiting The network percentile that operates with stable queues is time in the buffer for unstable queues is infinite, while the given by average service latency is always finite according to (24). The network percentile that achieves a target KPI value, N 1 e.g., total latency constrained to t ,isgivenby

γsu = ½S (n) (20) N N n=1 1

γto = ½T (n) (25) S { ψ A 1 ψ A 1} where = n : n 2,n > n 0,n ;andwhere N n=1 ψn, A2,n,andA0,n are obtained from Algorithm 2 for every QoS-class after convergence. Note that (20) follows from the 14Due to the ergodicity of the PPP, the notion of the network γ-stability Loynes theorem along with the stability condition in (13). coincides with the typical queue -stability deﬁned in [29]. CHISCI et al.: UNCOORDINATED MWNs: SPATIOTEMPORAL MODELS AND MULTIACCESS STRATEGIES 927

Fig. 6. (a) CCDF of the TSP for P = {−30} [dBm];(A):λ =1.0, pa =0.2,andθ = −30 [dB];(B):λ =0.1, pa =0.6,andθ = −23 [dB];(C): λ =0.1, pa =0.2,andθ = −30 [dB]. (b) CCDF of the TSP for λ =0.2, Nc =1, Np =3, Nt =1, P = {−30, −32, −34} [dBm] (ramping-down), pa =0.2,andθ = −30 [dB]. (c) Mean number of packets in the system as a function of node density for Nc =4, Np =1, P = {−30} [dBm], pa =0.4, θ = −30 [dB]. For all the subﬁgures we used: N =10, η =2, R =10[m], σ2 = −90 [dBm], a =0.1[packets/slots].

where T = {n : t¯n ≤ t }. Analogously to (21), we define the γ-operativity region by the function γto : G→[0, 1] as o Rγ {R : γto(R) >γ} (26) which determines the region of the points in the space of network parameters that satisfies a target performance (e.g., constrained total latency, queue length, etc.). In contrast to γsu, that is more suited for best-effort type of traffic, γto is a metric more suited for latency-sensitive type of traffic.

VI. NUMERICAL RESULTS We start by validating the developed spatiotemporal model, particularly Approximations 1-3, via the Monte Carlo simulations described in the sequel.

A. Monte Carlo Simulation Setting Fig. 7. CCDF of the mean total latency ¯t across the network for simple λ =0.1 θ = −30 R =10 a =0.1 The simulations are conducted over an arbitrary, yet fixed, ALOHA with , [dB], [m], and . realization of the PBN with wrapped-around boundaries.15 × 5 2 the simulation kernel from which all temporal statistics are Simulations are performed over a squared area of 9 10 [m ], −3 which is sufficient to have ergodicity with the considered node computed. We use =10 and a kernel of 10,000 iterations. intensities λ = {0.1, 0.2} [nodes/m2] such that the realized The SINRs of the receivers of all active links along with the network is considered as typical. Only the fading realizations transmission powers of their intended transmitters are recorded and node activities change over the temporal domain. Each to construct the meta distribution of SINRs for all power levels. simulation run represents a time slot where independent channel gains are instantiated, packets are generated, and node B. Validation of the Framework interactions are established according to their protocol states Figs. 6(a) and 6(b) show the CCDF of the TSP for dif- and relative positions. The queue level, transmission power, ferent nodes intensities, ALOHA access probabilities, and and retransmission phase of each queue are tracked. For a transmission rates. In particular, Fig. 6(a) is plotted for a transition from one time slot to another, packets are indepen- single power level ALOHA protocol and Fig. 6(b) is plotted dently generated in all buffers with probability a. A transmitter for a power-ramping ALOHA protocol with three power with a non-empty buffer becomes active with probability pa levels. In both figures, the theoretical curves are plotted using and uniformly selects one of the Nc channels. The SINRs at (2) and the moments are obtained from Algorithm 2 after the receivers of all active links are determined based on the convergence. Both figures show a good agreement between interferers that are simultaneously active on the same channel. the theoretical (solid lines) and simulation (markers) results, Then, one packet is subtracted only from queues of the links where N =10QoS-classes are considered for the theoretical with SINR greater than θ. The simulation starts with all idle analysis. In particular, the Monte Carlo simulations with a nodes and runs for a sufficiently high number of time slots [k] fixed realization of the PBN account for the spatially cor- until the steady-state is reached. Let w be the estimated th related node activities and transmission powers (relaxed in distribution of the physical phases at the k simulation run, Approximation 1), the exact fine-grained location-dependent then the steady-state is reached when max w [k]−w [k−1] < performance (relaxed in Approximation 2), and the per-node where is a sufficiently small threshold (stop condition). The temporally correlated transmission success (relaxed in Approx- time slot at which the steady-state is reached is the time of imation 3). In addition to validating the developed mathemat- 15The wrap-around simulation model is utilized to eliminate favorable inter- ical framework, the figures also show the importance of the ference conditions induced by network edges and their effects on queueing system parameters (i.e., node intensity, access probability, and departure probabilities. transmission rates) on the distribution of the TSP. 928 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 27, NO. 3, JUNE 2019

Fig. 8. Stability regions of F-ALOHA over three different parameter planes. We show the γ-stablity regions for N =10(colorplot) and the deterministic £ ¢ 2 stability frontier for the average performance (marked dashed curve). Where it is not differently stated λ =0.1 nodes/m , Nc =4, η =2, R =10[m], 2 θ = −30 [dB], σ = −90 [dBm], P = {−30} [dBm], a =0.1[packets/slots],andpa =0.8 [attempts/slots].

C. Performance Evaluation ability vs. stability tradeoff for best-effort17 traffic in MWNs. Fig. 6(c) shows the average number of packets in the system In particular, the maximum spatial traffic density of nodes as a function of the node density for the first (n =1,red that can be accommodated subject to a γ-stability constraint is diamonds) and last (n =10, green stars) QoS-classes with shown by considering F-ALOHA with Nc =4. In particular, N =10. The case of N =1(i.e., one queueing model for Fig. 8(a) shows the (λ, a)-pairs that F-ALOHA can accommodate subject to a given γ-stability constraint with pa =0.8. all the network with departure rate paM1) is also considered. Although an infinite queue size is considered in the analysis, To accommodate more nodes may necessitate a lower traffic the figure shows that a finite queue with reasonable buffer generation to maintain the same γ-percentile of stable nodes. size is required as long as the node operates within a stable Fig. 8 also shows the importance of choosing the proper value QoS-class. If the queue is unstable, the packets will overflow for the access probability pa of the F-ALOHA to ensure a stable network operation. Fig. 8(b) shows the regions on the with probability one regardless of the buffer size. Note that the 2 curve of the highest class (i.e., N =10) experiences a change (pa, a)-plane where γ-stability holds for λ =0.1 [node/m ]. of slope at the point where a lower class becomes unstable. It can be observed that pa must be greater than a to allow Such variation occurs because all nodes within the unstable any fraction of the network to be stable, since a sufficiently class (i.e., 1/N fraction of the total nodes) become always persistent channel access is a necessary but not sufficient active, thus increasing the interference. condition for stability. Note that, for a given γ and increasing Fig. 7 shows the CCDF of the mean total latency across the a, the minimum pa ensuring γ-stability diverges increasingly network, which is obtained by the per-class characterization from pa = a. Fig. 8(c) shows the Pareto frontiers for regions through (23a) with a sufficient number (i.e. N = 100)of of the (pa, λ)-pairs. It can be observed that there is a given classes. Due to the ALOHA transmission deferrals, the chan- range of pa values that ensures stability for a given λ.This is because the tradeoff between conservative and aggressive nel access probability pa determines the minimum packet latency. Transmitted packets are susceptible to larger latencies channel access should be balanced to maintain stable queues. due to channel variations and interference, which leads to Employing a conservative access policy with low values of transmission failures. However, as the traffic becomes lighter pa misses many transmission opportunities, accumulates packets in buffers, and leads to instability. On the other hand, (i.e., low a), the interference is relieved and pa becomes the limiting factor for the latency. employing an aggressive access policy with high values of Fig. 8 shows the Pareto frontier for the system parameters pa aggravates the interference, leads to excessive transmission that ensures queuing stability for a fraction γ of the nodes. failures due to low SINR, thus leading to instability. Each colored region shows the pairs of network parameters Fig. 9 shows the Pareto frontiers, as well as the NC and that are required to ensure stability for a given fraction of SC frontiers, for different operational objectives of the con- nodes. The blue dashed line with markers, denoted as stability sidered uncoordinated MWNs. Particularly, Fig. 9 considers frontier for spatially averaged performance, shows the region the uncoordinated MWN operation with (a) γ-stability for γ =0.99 constraint, (b) γ-operativity for a latency t ≤ 5 of the network parameters leading to a spatially averaged ≤ departure rate greater than the packet generation rate,16 i.e., [slots] and γ =0.5,and(c)γ-operativity for a latency t 3 [slots] and γ =0.05. Note that the frontiers obtained by s Rav {R : paM1(R) >a} (27) considering dominant systems, i.e., NC and SC conditions, do not respectively provide tight upper and lower bounds wherewehaveconsideredthefunctionM1 : G→[0, 1]. of the actual performance. This may either lead to an over Such classical approach does not convey any information conservative transmission protocol with a consistent waste of on the fraction of nodes that have unstable queue. On the resources or to an over aggressive one that violates the KPI contrary, our approach based on the meta distribution reveals constraints. In particular, designing the network according to the percentile of stable and unstable nodes. The results shown the SC constraints may lead to serving a much lower intensity in Fig. 8 are of primary importance to characterize the scal- of nodes that the network can support. On the other hand,

16For simplicity, in the case of (27) we show the condition for single phase, 17Best-effort trafﬁc is tolerant to latency and, therefore, a ﬁnite latency is i.e., Geo/Geo/1 queues. the main requirement. CHISCI et al.: UNCOORDINATED MWNs: SPATIOTEMPORAL MODELS AND MULTIACCESS STRATEGIES 929

TABLE I UMA STRATEGY DESIGN

designing the network according to the NC may result in serving a lot more nodes than the network should support and violating the constraints on KPI values. Hence, Fig. 9 manifests the importance of the developed spatiotemporal model to determine the network scalability under the required network operational constraints.

D. Network Design To show the impact of the power-ramping strategy, we present a network design with λ =0.1, θ = −23 [dB], σ2 = −90 [dBm], a =0.1, η =2, R =10[m]and N =10. The considered setting is in accordance with LoRa dense ad-hoc scenario (i.e., λ =0.1 [nodes/m2])forlow- power consumption and short-range communications. Table I presents four different design of UMA strategies in terms of number of powers, power-ramping scheme, and persistence: Fig. 9. Pareto frontiers for the necessary (dashed), sufficient (dotted), and (i) Single power, (ii) Ramping-down 1, (iii) Ramping-down 2, actual (solid) conditions for ALOHA for (A): γ-stability, with a =0.1 and γ =0.999;(B):γ-operativity, with t = 5 [slots] and γ =0.5; and (C): (iv) Ramping-up 1, and (v) Ramping-up 2. Table I also shows with t = 3 [slots] and γ =0.05. Other variables are as in Fig. 8. the performance corresponding to each scheme in terms of the fraction γsu of stable nodes, the fraction γto of nodes with latency less than t time slots, and the spatially averaged prob- of uncoordinated MWN. Macroscopic- and microscopic-scale ability of empty buffer w[0]. The results in the table emphasize analyses are carried out through an iterative procedure that that ramping-down power control with some persistence (i.e., tracks the number of packets in the transmitter buffers, channel Ramping-down 1) outperforms all other schemes.18 This is access, transmission power, and the retransmission phase. because ramping-down the power relieves the interference The developed framework accounts for the location-dependent and the persistence on each power level ensures sacrificing performance and interference temporal correlation via the the performance of nodes belonging to low QoS-classes. meta distribution of the SINR. In particular, the network Ramping-down without persistence may mistakenly degrade is partitioned into N equiprobable QoS-classes, where the the performance of nodes belonging to high QoS-classes performance of a given QoS-class is explicitly determined and that experience instantaneous severe fading conditions. The the percentile performance of the network is characterized. table also shows that ramping-up the power has the worst To this end, the concepts of γ-stability and γ-operativity are performance among all schemes, which is because ramping- introduced, respectively, to characterize the percentile of the up the power aggravates the interference without enhancing network with stable nodes and that with desired performance. the performance of nodes at congested locations. Moreover, The tradeoff between scalability, stability, and other KPIs the single-power ALOHA scheme outperforms a ramping- is shown via the Pareto frontiers of the γ-stability and γ- down scheme when it is not properly designed. operativity regions, by accounting for the per-node traffic Remark 5: The employed power ramping scheme is simple intensity, node density, target transmission rate, and uncoor- and distributed. Each device locally determines its power level dinated access persistence. The analysis accounting only for from its buffer state and its previous transmission experience. dominant systems is inadequate and it is shown that location- Nevertheless, the number of power levels N , persistence per dependent queueing states must be accounted for designing the p network. To this end, the developed framework is also used level Nt, ramping strategy (i.e., up or down), and power level values in P are determined offline taking into account the to design UMA with different power-ramping schemes. The global network performance along with the application QoS importance of power-ramping design is manifested through a requirements (see, e.g., Fig. 8). This leads to a context-aware case study that reveals the superiority of the ramping-down design and low-complexity operation. scheme over the ramping-up scheme when considering the network performance. The results also manifest the importance VII. FINAL REMARK of power level persistence to mitigate the effect of fading on This paper combines stochastic geometry and queueing the- the power control and the dominance of the access probability ory to develop a spatiotemporal model for design and analysis on the minimum achievable latency. This work quantifies the scalability of UMA in MWNs, unleashes a traffic-aware 18Note that a similar insight was proved in [48] for uplink in cellular network design, and paves the way to new applications for networks where nodes have saturated buffers. IoT and CPSs. 930 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 27, NO. 3, JUNE 2019

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[36] A. Munari, P. Mähönen, and M. Petrova, “A stochastic geometry Andrea Conti (S’99–M’01–SM’11) received the approach to asynchronous ALOHA full-duplex networks,” IEEE/ACM Laurea (summa cum laude) in telecommunications Trans. Netw., vol. 25, no. 6, pp. 3695Ð3708, Dec. 2017. engineering and the Ph.D. in electronic engineer- [37] M. Haenggi, “The meta distribution of the SIR in Poisson bipolar ing and computer science from the University of and cellular networks,” IEEE Trans. Wireless Commun., vol. 15, no. 4, Bologna, Italy, in 1997 and 2001, respectively. pp. 2577Ð2589, Apr. 2016. He is a Professor at the University of Ferrara, Italy. [38] M. Salehi, A. Mohammadi, and M. Haenggi, “Analysis of D2D underlaid Prior to joining the University of Ferrara, he was cellular networks: SIR meta distribution and mean local delay,” IEEE with the Consorzio Nazionale Interuniversitario per Trans. Commun., vol. 65, no. 7, pp. 2904Ð2916, Jul. 2017 le Telecomunicazioni and with the IEIIT-Consiglio [39] Y. Wang, M. Haenggi, and Z. Tan, “The meta distribution of the SIR for Nazionale delle Ricerche. In Summer 2001, he was cellular networks with power control,” IEEE Trans. Commun., vol. 66, with the Wireless Systems Research Department at no. 4, pp. 1745Ð1757, Apr. 2018. AT&T Research Laboratories. Since 2003, he has been a frequent visitor to the [40] H. ElSawy and M.-S. Alouini, “On the meta distribution of coverage Wireless Information and Network Sciences Laboratory at the Massachusetts probability in uplink cellular networks,” IEEE Commun. Lett., vol. 21, Institute of Technology, where he presently holds the Research Affiliate no. 7, pp. 1625Ð1628, Jul. 2017. appointment. His research interests involve theory and experimentation of Stochastic Geometry for Wireless Networks [41] M. Haenggi, . Cambridge, wireless systems and networks including network localization, distributed U.K.: Cambridge Univ. Press, 2013. sensing, adaptive diversity communications, and network secrecy. He is [42] V. Chandrasekhar and J. G. Andrews, “Spectrum allocation in tiered cel- recipient of the HTE Puskás Tivadar Medal and co-recipient of the IEEE lular networks,” IEEE Trans. Commun., vol. 57, no. 10, pp. 3059Ð3068, Communications Society’s Stephen O. Rice Prize in the field of Communi- Oct. 2009. [43] S. Graf and H. Luschgy, Foundations of Quantization for Probability cations Theory and of the IEEE Communications Society’s Fred W. Ellersick Distributions (Lecture Notes in Mathematics), 1st ed. Berlin, Germany: Prize. Springer-Verlag, 2000. Dr. Conti has served as editor for IEEE journals, as well as chaired interna- [44] A. S. Alfa, Applied Discrete-Time Queues,2nded.NewYork,NY,USA: tional conferences. He has been elected Chair of the IEEE Communications Springer, 2016. Society’s Radio Communications Technical Committee. He is a co-founder [45] R. M. Loynes, “The stability of a queue with non-independent inter- and elected Secretary of the IEEE Quantum Communications & Information arrival and service times,” Math. Proc. Cambridge Philos. Soc., vol. 58, Technology Emerging Technical Subcommittee. He is an elected Fellow of no. 5, pp. 497Ð520, 1962. the IET and has been selected as an IEEE Distinguished Lecturer. [46] D. P. Bertsekas and R. G. Gallager, Data Networks, 2nd ed. Upper Saddle River, NJ, USA: Prentice-Hall, 1992. [47] L. Kleinrock, Queuing Systems: Theory, 1st ed. Hoboken, NJ, USA: Wiley, 1975. [48]R.Arshad,L.H.Afify,Z.Liu,H.ElSawy,T.Y.Al-Naffouri,and M.-S. Alouini, “On the effect of uplink power control on temporal Mohamed-Slim Alouini (S’94–M’98–SM’03–F’09) retransmission diversity,” IEEE Wireless Commun. Lett., vol. 8, no. 1, was born in Tunis, Tunisia. He received the Ph.D. pp. 309Ð312, Feb. 2019. degree in electrical engineering from the California Institute of Technology (Caltech), Pasadena, CA, Giovanni Chisci (S’15–M’18) received the Laurea USA, in 1998. He served as a Faculty Member degree (summa cum laude) in electrical engineering at the University of Minnesota, Minneapolis, MN, from the University of Florence, Italy, in 2014, and USA, and then at the Texas A&M University at the Ph.D. degree in information engineering from Qatar, Education City, Doha, Qatar, before joining the University of Ferrara, Italy, in 2018, where he is the King Abdullah University of Science and Tech- currently enrolled as a Postdoctoral Fellow. nology, Thuwal, Makkah province, Saudi Arabia, He was with the Wireless Information and Net- as a Professor of electrical engineering in 2009. work Sciences Laboratory at the Massachusetts His current research interests include the modeling, design, and performance Institute of Technology, and with the Computer, analysis of wireless communication systems. Electrical and Mathematical Sciences and Engi- neering Division at the King Abdullah University of Science and Technology in 2016 and 2018, respectively. His research encompasses fundamental theories and algorithm design for wireless networks and complex systems. His current research topics include intrinsic network secrecy, low latency networks, and cyber physical systems. He served as a Moe Z. Win (S’85–M’87–SM’97–F’04) is a Pro- TPC member for IEEE conferences and as a reviewer for IEEE journals. fessor at the Massachusetts Institute of Technology (MIT) and the founding director of the Wireless Hesham ElSawy (S’10–M’14–SM’17) received the Information and Network Sciences Laboratory. Prior Ph.D. degree in electrical engineering from the Uni- to joining MIT, he was with AT&T Research Labo- versity of Manitoba, Canada, in 2014. He joined ratories and NASA Jet Propulsion Laboratory. the Electrical Engineering Department, King Fahd His research encompasses fundamental theories, University of Petroleum and Minerals, Saudi Arabia, algorithm design, and network experimentation for as an Assistant Professor in Fall 2018. Prior to that, a broad range of real-world problems. His current he was a Post-Doctoral Fellow at the King Abdullah research topics include network localization and University of Science and Technology, Saudi Arabia, navigation, network interference exploitation, and a Research Assistant at TRTech, Winnipeg, MB, quantum information science. He has served the IEEE Communications Canada, and a Telecommunication Engineer at the Society as an elected Member-at-Large on the Board of Governors, as elected National Telecommunication Institute, Egypt. His Chair of the Radio Communications Committee, and as an IEEE Distinguished research interests include statistical modeling of wireless networks, stochastic Lecturer. Over the last two decades, he held various editorial positions for geometry, and queueing analysis for wireless communication networks. He IEEE journals and organized numerous international conferences. Currently, received several academic awards including the NSERC Industrial Postgrad- he is serving on the SIAM Diversity Advisory Committee. uate Scholarship from 2010 to 2013, and the TRTech Graduate Students Dr. Win is an elected Fellow of the AAAS, the IEEE, and the IET. He was Fellowship from 2010 to 2014. He coauthored three award-winning papers honoured with two IEEE Technical Field Awards: the IEEE Kiyo Tomiyasu that are recognized by the IEEE COMSOC Best Survey Paper Award, the Award (2011) and the IEEE Eric E. Sumner Award (2006, jointly with Best Scientific Contribution Award to the IEEE International Symposium on R. A. Scholtz). Together with students and colleagues, his papers have received Wireless Communication Systems 2017, and the Best Paper Award in Small several awards. Other recognitions include the IEEE Communications Society Cell and 5G Networks (SmallNets) Workshop of the 2015 IEEE International Edwin H. Armstrong Achievement Award (2016), the International Prize Conference on Communications. He is recognized as an Exemplary Reviewer for Communications Cristoforo Colombo (2013), the Copernicus Fellowship by the IEEE TRANSACTIONS ON COMMUNICATIONS for the three conces- (2011) and the Laurea Honoris Causa (2008) from the Universitá degli Studi sive years (2014Ð2016), and by the IEEE TRANSACTIONS ON WIRELESS di Ferrara, and the U.S. Presidential Early Career Award for Scientists and COMMUNICATIONS in 2017. Engineers (2004). He is an ISI Highly Cited Researcher.