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Maximal scheduling in wireless ad hoc networks with hyper graph interference models.bak

  1. 1. IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 1, JANUARY 2012 297 Maximal Scheduling in Wireless Ad Hoc Networks With Hypergraph Interference Models Qiao Li, Student Member, IEEE, and Rohit Negi, Member, IEEE Abstract—This paper proposes a hypergraph interference impacts [2]. It is expected that, as increasingly more vehiclesmodel for the scheduling problem in wireless ad hoc networks. are equipped with wireless communication capabilities, large-The proposed hypergraph model can take the sum interference scale VANETs will be developed in the near future.into account and, therefore, is more accurate as compared withthe traditional binary graph model. Further, different from the One central technical challenge in the development of large-global signal-to-interference-plus-noise ratio (SINR) model, the scale VANETs is the design of medium access control (MAC)hypergraph model preserves a localized graph-theoretic structure mechanisms [2], [5]–[7]. In fact, such a problem is well knownand, therefore, allows the existing graph-based efficient schedul- to be NP-hard [8] for generic wireless networks due to theing algorithms to be extended to the cumulative interference fundamental property of shared wireless spectrum for wire-case. Finally, by adjusting certain parameters, the hypergraphcan achieve a systematic tradeoff between the interference ap- less communications. This implies that the transmission ofproximation accuracy and the user node coordination complexity any link will be received as interference by any unintendedduring scheduling. As an application of the hypergraph model, receiver, thereby impairing its own communication quality. Forwe consider the performance of a simple distributed scheduling VANETs, the MAC design is more challenging, as no centralalgorithm, i.e., maximal scheduling, in wireless networks. We communication coordinator can be assumed [2], due to thepropose a lower bound stability region for any maximal schedulerand show that it achieves a fixed fraction of the optimal stability rapid changes in the network topological properties as inducedregion, which depends on the interference degree of the underlying by high user mobility. Thus, the MAC layer of VANETs has tohypergraph. We also demonstrate the interference approximation not only specify how the limited amount of bandwidth shouldaccuracy of hypergraphs in random networks and show that hy- efficiently be allocated among spatially separated contendingpergraphs with small hyperedge sizes can model the interference users but achieve such allocation in a distributed fashion, asquite accurately. Finally, the analytical performance is verified bysimulation results. well, even if the cumulative interference comes from users that http://ieeexploreprojects.blogspot.com the network. are arbitrarily located in Index Terms—Ad hoc networks, interference, scheduling, sta- In this paper, we try to address the MAC mechanism designbility, wireless networks. in VANETs by adopting a low complexity distributed schedul- I. I NTRODUCTION ing approach. The proposed scheduling algorithm can also be applied to other types of wireless ad hoc networks. The reasonW IRELESS ad hoc network technologies, in particular in choosing the scheduling approach is that, compared with vehicular ad hoc networks (VANETs), are envisioned to various types of MAC mechanisms in the past (e.g., [9]–[13]),play an important role in future transportation systems [2]. As the scheduling approach can effectively eliminate transmissionthese networks are spontaneously formed by vehicles traveling collisions and consequently reduce retransmissions. Therefore,on the road, VANETs can deliver both critical traffic accident it can significantly improve the latency, increase the overallor traffic jam messages [3] and useful commercial information throughput, and achieve higher energy efficiency. To motivate[4] to drivers with low latency using direct vehicle-to-vehicle the proposed scheduling algorithm, we first provide a briefcommunications. Further, compared with the cellular approach, literature review on the past scheduling research.VANETs do not charge any service fee, as no existing telecom- Based on the adopted interference model, the past researchmunication infrastructure is used. In the past, VANETS have on scheduling can roughly be classified into two categories:been identified as a key technology in enhancing road safety 1) the flow contention graph (also referred to as the protocoland transportation efficiency, as well as reducing environmental interference model [14]) based scheduling, and 2) the physi- cal signal-to-interference-plus-noise ratio (SINR) based (also Manuscript received February 23, 2011; revised June 29, 2011 and called the physical interference model [14]) scheduling. We firstSeptember 12, 2011; accepted October 16, 2011. Date of publicationNovember 18, 2011; date of current version January 20, 2012. This work discuss the graph-based scheduling as follows.was supported in part by the United States National Science Foundationunder Award CNS-0831973 and Award ECCS-0931978 and in part by UnitedStates Army Research Office under Award W911NF0710287. This paper was A. Scheduling With the Graph Interference Modelpresented in part at the IEEE International Conference on Communications,Beijing, China, May 2008. The review of this paper was coordinated by Prof. A flow contention graph [11], [12], [15]–[18] approximatesV. W. S. Wong. The authors are with the Department of Electrical and Computer Engi- the interference as “binary.” This means that the transmissionneering, Carnegie Mellon University, Pittsburgh, PA 15213 USA (e-mail: in a particular link fails if and only if there is a concurrentqiaoli@cmu.edu; negi@ece.cmu.edu). transmission in any neighboring link. For example, consider the Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org. four-link wireless network in Fig. 1, where a flow contention Digital Object Identifier 10.1109/TVT.2011.2176520 graph can be constructed by, for example, placing a guard 0018-9545/$26.00 © 2011 IEEE
  2. 2. 298 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 1, JANUARY 2012 where Si is the received signal power at link i, Ni is the noise power, Iki is the received interference at link i from transmitting link k, and θi is the SINR threshold for suc- cessful packet reception for link i, which is determined by physical layer modulation, detection, and coding specifications. Recently, there has been a large body of work on designing efficient SINR-based scheduling algorithms in wireless net- works, such as optimal scheduling [7], [25], computationally efficient scheduling with approximation bounds [26]–[31], and simulation-based heuristic scheduling algorithms [32]. Further,Fig. 1. Sample wireless network with four links, where square nodes are the there have also been numerous attempts to achieve decentral-transmitters, and round nodes are the receivers. The dashed circle is the guard ized implementations [33]–[35].zone associated with link 1. Despite the recent research efforts on SINR-based schedul-zone [20] with certain radius around the receiver of each link. ing, it is widely recognized that the design and analysis ofTwo links form an edge in the flow contention graph if one’s efficient scheduling algorithms under the SINR model, in par-transmitter is in the guard zone associated with the other. In ticular, low complexity distributed scheduling, is still far fromsuch a case, the flow contention graph for Fig. 1 has only one being solved. This is because of the fundamental global natureedge {1, 2}. Therefore, a transmission schedule is valid as long of the interference, namely, the transmission of any link willas links 1 and 2 are not chosen simultaneously. Based on such be received by any other link in the network as interference,a graph interference model, many interesting scheduling algo- thereby impairing its own communication quality. This impliesrithms [15]–[18], [21] have been proposed, whose performance that any scheduling algorithm under the global SINR model willlimits are often well understood, thanks to the deep and rich require coordinations among all the user nodes in the network,foundation of graph theory. This is particularly convenient for which makes it very difficult to design distributed schedulingthe important class of low complexity distributed scheduling, algorithms.such as maximal scheduling [15] and the longest queue first(LQF) scheduling [21], [22], where the throughput performance C. Summary of Contributionsis often specified by the metrics of the underlying interference In this paper, we consider a new interference model, i.e., the http://ieeexploreprojects.blogspot.comgraph, such as the “interference degree” and the “local pooling flow contention hypergraph, which can combine the advantagesfactor.” In the literature, many key insights about the throughput of both the graph model and the physical SINR model whileperformance of such schedulers in practical wireless networks avoiding their drawbacks. First, compared with the binarywere obtained by conducting graph-theoretic analysis under graph model, the hypergraph achieves better approximationdifferent physical layer specifications [15], [23]. accuracy by modeling cumulative interference constraints as However, the graph model has also been recognized as a “hyperedges,” where each hyperedge is a set of links that arerigid model, which oversimplifies the interference constraints not allowed to transmit simultaneously due to the resulting in-in wireless networks [19], [24]–[26] since it does not take into tolerable cumulative interference experienced at certain links inaccount the cumulative effect of interference. For example, in the hyperedge. Further, compared with the global SINR model,Fig. 1, it is possible that link 1 fails when links {1, 3, 4} are where distributed scheduling is particularly challenging, thescheduled due to the sum interference from both links 3 and hypergraph allows much easier scheduling design and analysis4. In such a case, the graph model can only guarantee that the by extending the existing rich body of work on graph-basedtransmission at link 1 is successful when only one of the other scheduling due to the structural similarities between these twotwo links is transmitting due to its binary nature. On the other models. Finally, as a combination of the graph model and thehand, if one conservatively builds the graph by increasing the SINR model, the hypergraph can achieve a systematic trade-size of the guard zone such that two additional edges {1, 3} off between the interference accuracy and user coordinationand {1, 4} are included (note that both links 3 and 4 have the complexity. Such tradeoff can be observed by inspecting thesame distance to link 1 in this example), the network capacity construction of hyperedges, which is as follows. As a majoris reduced, because when link 1 transmits, neither link 2 nor portion of the total interference is contributed by only a fewlink 3 is allowed to transmit, although there is no collision if nearby transmitting links in typical wireless networks, we canonly one of them transmits. approximate the SINR locally with very good accuracy as Si SiB. Scheduling With the Physical SINR Interference Model ≈ (2) Ni + j∈σ Iji Ni + j∈σ (Iji · 1j∈Li ) Unlike the approximation-based graph model, the physicalSINR model can accurately describe the interference constraint where 1{·} is an indicator function, i.e., 1true = 1 and 1false =in wireless networks. Under the SINR model, a transmission 0, and Li is the set of “local” links around link i, i.e.,schedule σ is valid if the SINR at any transmitting link i Si j ∈ Li if < βi (3)satisfies Ni + Iji Si where βi is a properly chosen threshold. Based on the forego- ≥ θi (1) Ni + k∈σ Iki ing SINR approximation, a hyperedge e = {i, i1 , i2 , . . . , ik−1 }
  3. 3. LI AND NEGI: MAXIMAL SCHEDULING IN WIRELESS NETWORKS WITH HYPERGRAPH INTERFERENCE MODELS 299Fig. 2. (a) Star shaped network with nine links and (b) its flow contention graph under the “unidirectional equal power model” [15].with cardinality k can be constructed if result in significant protocol overhead [O(N ) in the worst case] Si and nonrobust behavior in the presence of high user mobility. < θi (4) On the other hand, low complexity maximal scheduling can Ni + k−1 Iis i s=1 easily be implemented and is robust against user mobility duewhere the links {i1 , i2 , . . . , ik−1 } are selected only if they are to its local coordination.in Li so that the MAC coordination can be restricted only For a given hypergraph, we characterize the throughput per-to local links. Thus, by adjusting {βi } and the maximum formance of maximal scheduling by proposing a lower boundallowed hyperedge cardinality K, the interference accuracy can stability region that can be achieved by any maximal scheduler,gradually be improved from the binary graph model (small βi which includes the analysis in [15] as a special case. Further, (large βi and K = N , we show that maximal scheduling can achieve at least 1/Δ ofand K = 2) to the accurate SINR modelhttp://ieeexploreprojects.blogspot.comwhich is the total number of links in the network). Note that the optimal throughput, where Δ is defined as the “interferenceour model is different from the hypergraph model proposed degree” of the hypergraph, a metric generalized from the graphin [19], which is essentially the physical SINR model, due to case. For a large class of hypergraph interference models, wherethe global hyperedge construction. Thus, their model faces the the interference is locally approximated, maximal schedulingsame challenges in designing distributed scheduling algorithms can achieve a constant fraction of the optimal stability region.in wireless networks as the global SINR model. Further, their This is in contrast to the pessimistic conclusions drawn fromanalysis is restricted to cellular networks. To the best of the certain graph models. For example, consider the star shapedauthors’ knowledge, this paper is the first application of the wireless network in Fig. 2(a), whose contention graph is shownhypergraph model in general wireless ad hoc networks. in Fig. 2(b), which is based on the “unidirectional equal power As another contribution of this paper, we analyze the per- model” [15]. That is, two links cannot transmit together if one’sformance of a class of low complexity distributed scheduling transmitter is within distance d of the other’s receiver, where dalgorithms, namely, maximal scheduling [15], under the hy- is the uniform link length. It has been proved [15] that maximalpergraph model. During scheduling, the only requirement is scheduling can achieve 1/(N − 1) of the optimal stability re-that, if a link i has packets to transmit, either it transmits, or gion, where N is the number of links in the network. Now, it canthere is a set of transmitting neighbors {i1 , i2 , . . . , ik−1 } such easily be shown that the number of (outside) links can grow ar-that {i, i1 , i2 , . . . , ik−1 } form a hyperedge. The scheduling is bitrarily large while still maintaining the star shape. Thus, in theotherwise arbitrary. The maximal scheduling is simple and has worst case, maximal scheduling cannot guarantee any positivelow complexity, which requires O(log N ) in the worst case fraction of the optimal stability region. This pessimistic result[16] for the graph interference model. It can even be approx- is due to the binary interference assumption, which allows animately implemented with constant protocol overhead [36]. infinite number of transmitting links in a fixed area, since noThe simplicity and localized nature of maximal scheduling transmission fails from a single concurrent transmitting link.make it very attractive in wireless ad hoc networks, particu- However, if we consider the cumulative interference, one canlarly VANETs, where local coordination is required to achieve easily see that the number of transmitting links in this networkrobustness against various sources of uncertainties, such as is at most a constant since the area can be upper bounded.topology changes by high user mobility. Note that there are In such a case, a hypergraph model with a certain maximumother scheduling algorithms that can achieve better throughput hyperedge size can accurately model the interference in thisguarantees, such as the LQF scheduling [21]. However, the network. Therefore, we conclude that maximal scheduling canperformance of LQF scheduling is achieved with global coor- still achieve a constant fraction of the optimal stability regiondination among users in the network, which, despite the recent in the worst case, which is fundamentally different from theefforts in achieving localized coordinations [22], [31], can still pessimistic result given by the binary graph model.
  4. 4. 300 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 1, JANUARY 2012 Finally, we demonstrate the approximation accuracy of thehypergraph model by analyzing its outage probability in ran-dom infinite networks, where the nodes are described by aspatial Poisson point process (PPP), and the channels are sub-ject to Rayleigh fading. We obtain a closed-form expressionof the outage probabilities under the hypergraph interferencemodel with different maximum hyperedge sizes. Numericalresults show that a small hyperedge size is often quite accuratein approximating the total interference in wireless networks.The performance results of maximal scheduling with differenthypergraph models are verified by simulation results. Fig. 3. Generalized star-shaped hypergraph with five links. There The remaining sections of this paper are organized as follows. are a total of six hyperedges in the figure: E = {{1, 2, 3}, {1, 3, 4}, {1, 4, 5}, {1, 2, 5}, {1, 2, 4}, {1, 3, 5}}. Only two hyperedges {1, 2, 3} andIn Section II, we introduce the system model, and in Section III, {1, 2, 4} are shown.we analyze the performance of the scheduling algorithmswith hypergraph models. Section IV demonstrates the outage where the hyperedge e has to include both links i and j (Δij =probabilities of the hypergraph models, Section V shows the 0 if i and j are not neighbors). Now, define the interferencesimulation results, and finally, Section VI concludes this paper. degree of link i as follows: ⎧ ⎫ II. S YSTEM M ODEL ⎨ ⎬ In this section, we describe the system model and define Δi = max max Δij , 1 (6) ⎩σ⊆Ni ,σ∈M ⎭the scheduling problem at the MAC layer in wireless ad hoc j∈σnetworks. where maxσ⊆Ni ,σ∈M j∈σ Δij is the weight of the max-A. Hypergraph Interference Model weight independent set formed by links in Ni , and M We assume that the wireless network consists of N links, is the family of all independent sets. Intuitively, the termwhich are denoted as the set V. The hypergraph interference maxσ⊆Ni ,σ∈M j∈σ Δij gives an estimate of the maximummodel is defined as H = (V, E), where V is the link set, number of “active hyperedges” with respect to link i, whereand E is the set of hyperedges such that each hyperedge a hyperedge e = {i, i1 , i2 , . . . , ik−1 } is “active” with respect http://ieeexploreprojects.blogspot.come = {i1 , i2 , . . . , ik } ∈ E consists of a subset of links, which to link i if all links in e except link i are scheduled. Inare not allowed to transmit together, due to the packet the graph case, this is equivalent to the maximum number ofreception failure at one or more links in e from the strong sum “active edges” or simply the maximum number of concurrentinterference. For example, in Fig. 1, if links 1, 3, and 4 transmit transmissions in a link i’s neighborhood [15] since Δij = 1together, then link 1 will fail due to the interference from both for all j ∈ Ni . For general hypergraphs, we have Δij < 1links 3 and 4. Thus, one can form a hyperedge {1, 3, 4} so due to the fundamental property of cumulative interference.that such a transmission mode is not allowed. Note, however, Finally, define Δ = maxi∈V Δi as the interference degree of thethat {1, 3} is not a hyperedge, and therefore, links 1 and 3 can hypergraph H. It will be shown later that 1/Δ is a lower boundindeed transmit together. This distinction between the effect on the scheduling efficiency of maximal scheduling algorithmsof sum interference and interference from a single link cannot with the hypergraph model, which will be defined later in thisbe captured by the conventional graph model. We require that section.the hyperedges should be constructed to be minimal, i.e., strict Example: Consider the hypergraph H = (V, E) in Fig. 3supersets of hyperedges are not included in E, and therefore, with five links, where V = {1, 2, 3, 4, 5} is the link set, and E =the hypergraph representation is not redundant. For example, {{1, 2, 3}, {1, 3, 4}, {1, 4, 5}, {1, 2, 5}, {1, 2, 4}, {1, 3, 5}} isfor the hypergraph in Fig. 1, {1, 2, 3, 4} is also an invalid the set of six hyperedges (in the figure, only two hyperedges aretransmission schedule, but is not a hyperedge, since it already shown). We have N1 = {2, 3, 4, 5}, and Δ12 = Δ13 = Δ14 =includes two hyperedges {1, 2} and {1, 3, 4} as subsets. Δ15 = 1/2, since all the hyperedges have cardinality 3. Thus,Finally, note that the links in each hyperedge e are chosen Δ1 = 2, since {2, 3, 4, 5} is the max-weight independentlocally, in the sense that there is a link i ∈ e such that all the set in N1 with total weight 2. Similarly, we have Δ2 = 3/2,other links in e are in Li , as described in (3). because {3, 4, 5} is the max-weight independent set in N2 with In each time slot, the scheduling algorithm has to choose a set weight 3/2. By symmetry, we have Δ2 = Δ3 = Δ4 = 3/2,σ of independent links for transmission, where the term “inde- and therefore, the interference degree for the hypergraph ispendent” implies that no subset of σ can form a hyperedge. For Δ = 2. We will prove in Section III that maximal schedulingeach link i in the hypergraph, define Ni = {k ∈ V : {i, k} ⊆ can achieve at least 1/2 of the optimal stability region.e for some e ∈ E} as the set of neighbors of link i. Thus, wecan define the interference degree Δi for a link i as follows. Wefirst associate each neighboring link j ∈ Ni with weight Δij B. Queueing Modelas follows: We assume that time is slotted. For each link i ∈ V, we 1 associate an external packet arrival process Ai (n), which is Δij = max (5) e∈E,{i,j}⊆e |e| − 1 the total number of arrived packets during the first n time
  5. 5. LI AND NEGI: MAXIMAL SCHEDULING IN WIRELESS NETWORKS WITH HYPERGRAPH INTERFERENCE MODELS 301slots. The assumptions on the packet arrival processes are III. M AXIMAL S CHEDULING W ITHas follows. H YPERGRAPH M ODELS 1) The maximum number of arrived packets in each time The popularity of maximal scheduling, to a large extent, is slot is uniformly bounded by a constant, i.e., with attributable to its simple specification, which allows distributed probability 1 (w.p.1), we have (random) algorithms with low overhead [15], [18]. Similar to the graph case, the maximal scheduling algorithm with the Ai (n) − Ai (n − 1) ≤ Amax ∀i ∈ V, n ≥ 1 (7) hypergraph model is also quite simple. A conceptual maximal scheduler can work as follows. In each time slot, the scheduler where Amax is a positive constant. considers the set of backlogged links in an arbitrary order. 2) The strong law of large numbers (SLLN) applies, i.e., When a link is considered, it is put onto the schedule if no w.p.1, we have hyperedge constraint is violated. Thus, compared with the scheduling algorithm with the graph interference model [15], the only change is that, instead of the edge constraints, one lim Ai (n)/n = ai ∀i ∈ V (8) now needs to check hyperedge constraints, which corresponds n→∞ to sum interference. Consequently, similar to the graph model, where ai is defined as the arrival rate of link i. maximal scheduling with locally constructed hypergraphs can be implemented in a distributed fashion with low computational Note that our assumptions on the packet arrival processes are complexity. Note that this is in contrast with the global SINRquite mild since they can be correlated over multiple time slots model, where the addition of any link to the schedule has toas well as across different links. Now, the queueing dynamics require coordinations of all the scheduled links, which canof the wireless network can be described as follows: arbitrarily be located in the network. Qi (n) = Qi (n − 1) − σi (n) + Ai (n) − Ai (n − 1) (9) A. Stability Regionwhere Qi (n) represents the queue length of link i at the endof time slot n, and σi (n) is the indicator function of whether We next consider the stability region of maximal schedul-link i is scheduled at time slot n, i.e., σi (n) = 1 if link i ing. In the following theorem, we provide a lower bound onis scheduled at time slot n, otherwisehttp://ieeexploreprojects.blogspot.com any maximal scheduler. Let the set σi (n) = 0. Note that the stability region ofσi (n) ≤ Qi (n − 1) since the queues cannot be negative. With W consists of all N × N matrices that satisfy the followingan abuse of notation, we also refer to the vector σ as an properties.independent set. We are interested in the throughput performance of a sched- 1) W is symmetric, and 0 ≤ Wij ≤ 1 for all i and j.uler π, which is represented by its stability region Aπ = {a ∈ 2) Wii = 0 for all i, and Wij = 0 if j ∈ Ni .RN : a is stable under π}. This implies that any arrival process 3) For any hyperedge e that includes link i, j∈e Wij ≥ 1. +satisfying (7) and (8) with an average arrival rate a ∈ Aπ is We have the following theorem.stable under the scheduler π. The stability in this paper is Theorem 1: Let a maximal scheduler π with a hypergraphdefined as the rate stability [41] H be given. Then, the network is stable for an arrival rate a if n there is a matrix W ∈ W such that (I + W )a 1, where I is 1 lim σi (k) = ai , w.p.1 ∀i ∈ V. (10) the identity matrix, and 1 = (1, 1, . . . , 1)T . n→∞ n k=1 Note that if the hypergraph H is indeed a graph, the ma- trix W is the graph incidence matrix: Wii = 0, Wij = 1 ifThus, if the network is rate stable, we can guarantee an average j ∈ Ni , otherwise Wij = 0. Therefore, the stability region ofthroughput of ai for each link i. It has been shown [10] that (I + W )a 1 reduces to that proved in [15]. Thus, the lowerthe optimal stability region is A = Co(M), where Co(·) bound region in Theorem 1 is a generalization of the lowerdenotes convex hull. For a suboptimal scheduling algorithm bound for the graph model to the hypergraph models.π, one is usually interested in its scheduling efficiency [18] Proof: The analysis of stability regions associated withγ = sup{κ : κA ⊆ Aπ }, which is the largest fraction of A general arrival processes is quite hard since to show that anthat can be stabilized by the algorithm π. Maximal schedulers arrival rate a is stable, one has to guarantee stability for allare of particular interests. It has been shown [15] that with stochastic arrival processes with the same rate a. In this paper,the graph interference model, any maximal scheduler π can the analysis is done in the framework of fluid limits [15], [41],achieve at least a scheduling efficiency of 1/Δ, where Δ is the which are introduced in Appendix A. Appendix B proves theinterference degree of the underlying graph. In this paper, we theorem using fluid limits.are interested in the hypergraph interference model. In the next Thus, Theorem 1 proposes a more general form of lowersection, we will analyze the stability region maximal scheduling bound on the stability region under maximal scheduling. Thisalgorithms with the hypergraph model and prove that it achieves can be used as a sufficient condition to check the feasibility of aa similar scheduling efficiency with a generalized definition of given arrival rate a. We next analyze the scheduling efficiencyinterference degree. of maximal scheduling with the hypergraph model.
  6. 6. 302 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 1, JANUARY 2012B. Scheduling Efficiency scope of this paper, we note that for hypergraphs constructed locally, 1/Δ can be bounded by a constant, since the number Based on the foregoing analysis on the stability region, we of transmitting nodes in a fixed area should be bounded, duenext show that maximal scheduling with hypergraph interfer- to sum interference constraint in (4). This is fundamentally dif-ence models can achieve a scheduling efficiency of at least 1/Δ. ferent from the graph model, where the binary interference as- Theorem 2: Suppose a wireless network with hypergraph sumption may result in an infinite number of transmitting linksinterference model H is given. Consider an arbitrary arrival in a network, even if the network is restricted in a fixed area.process satisfying (7) and (8) with arrival rate a ∈ A . Then, An illustrating example is the star shaped network in Fig. 2(a),a/Δ ∈ Aπ is stable for any maximal scheduler π. where it has been proved in [15] that, in the worst case, maximal We first present the outline of the proof. For a backlogged scheduling cannot guarantee any positive fraction of the optimallink i, in each time slot, maximal scheduling implies that stability region since one can arrange, under the “unidirectionaleither link i transmits, or there is an “active” hyperedge equal power” model, an arbitrarily large number of independent{i, i1 , . . . , ik−1 } with respect to i (the definition of active transmitting links along the dashed circle, in which case, thehyperedge is in Section II-A). In both cases, it can be shown that scheduling efficiency 1/Δ can arbitrarily be small. On the σi (n) + Δij σj (n) ≤ Δi ≤ Δ (11) other hand, under hypergraph interference models, the number j∈Ni of transmitting links in this network can be effectively upper bounded due to the sum interference constraint in (4). Thus,according to the definition of {Δij }. Thus, for any feasible the performance analysis of maximal scheduling in hypergrapharrival rate a ∈ A , we have models is more accurate than the graph model due to the consid- eration of sum interference. In the next section, we explore the ai + Δij aj ≤ Δ. (12) tradeoff between interference approximation accuracy and user j∈Ni node coordination complexity during scheduling by analyzing the outage performance of the hypergraph model in randomFurther, we will prove that the set of weights {Δij } ∈ W, infinite networks.along with (12), imply that a/Δ is in the lower bound regiondefined in Theorem 1. Proof: See Appendix C. IV. O UTAGE P ERFORMANCE OF THE We next discuss the tightness of the preceding bound on the H YPERGRAPH M ODEL http://ieeexploreprojects.blogspot.comscheduling efficiency. Note that if Δ = 1, it is obvious that the While the hypergraph model is more complex than a graphscheduling efficiency is tight. We now assume that Δ > 1 and model, it allows more accurate modeling (and thus control) ofshow a tightness result in the following theorem. interference. In this section, we demonstrate the modeling accu- Theorem 3: Let a hypergraph H be given such that racy of the locally constructed hypergraph model by analyzingany link i ∈ V with Δi = Δ > 1 satisfies the following its outage probability in random infinite networks, where thecondition. The set of independent links in Ni , which nodes form a homogeneous PPP [37]. We first describe theachieve an integer interference degree Δ, can be written as random network model.{e1 /{i}, e2 /{i}, . . . , eΔ /{i}}, where the hyperedges {ek } aredisjoint except a common link i. Then, for any > 0, there is afeasible arrival rate a ∈ A and an arrival process with rate a , A. Random Network Modelwhich is arbitrarily close to a in the sense that We consider the well-known Poisson random network model aj ≤ (1/Δ)aj + ∀j ∈ V. (13) [39], where the set of contending nodes form a homogeneous PPP on an infinite 2-D plane. This model is widely used in theFurther, there is a maximal scheduler π such that the network is literature of wireless network analysis since it is tractable, al-unstable under π with this arrival process. lowing valuable insights into the behavior of large networks. By Essentially, the theorem assumes that the hypergraph H Slivnyak’s theorem [37], we assume, without loss of generality,includes a generalized “star” shaped hypergraph, where the that there is a receiver placed at the origin. We further assumeindependent set is a set of disjoint hyperedges (excluding link that all transmitting nodes transmit with equal power ρ, as isi). An example is the hypergraph in Fig. 3, where Δ1 = Δ = 2, common in 802.11 networks (power adaptation is a challengingwhich is achieved by two disjoint hyperedges (excluding the question, which is outside the scope of this paper). In thisintersection at link 1), i.e., {{2, 3}, {4, 5}}. section, to obtain some insights into the impact of hyperedge Proof: See Appendix D. sizes on the outage performance in large (infinite) wireless Thus, the scheduling efficiency 1/Δ is a tight lower bound networks, we assume that the channel is subject to Rayleighon the performance of maximal scheduling algorithms with fading. Note that for many wireless systems, multipath fadinghypergraph models. According to (6), Δ can be interpreted can often be effectively handled by diversity techniques. Here,as the estimation of the number of active hyperedges in a the assumption of Rayleigh fading is largely due to its tractabil-link’s neighborhood. Similar to the graph case, the performance ity in allowing insightful closed-form solutions in the outageof maximal scheduling under different physical layer specifi- performance analysis [39]. Under the Rayleigh fading model,cations can be analyzed using geometry and graph theoretic the received signal power at the center receiver can be expressedtechniques [15], [23]. Whereas the exact analysis is out of the as S0 = ρh0 d−α , where h0 is the power fading coefficient, 0
  7. 7. LI AND NEGI: MAXIMAL SCHEDULING IN WIRELESS NETWORKS WITH HYPERGRAPH INTERFERENCE MODELS 303which is exponentially distributed with mean 1, d0 is the length TABLE I PARAMETERS FOR N UMERICAL C ALCULATIONSof the center link, and α is the path loss exponent. We assumethat SINR is an appropriate metric of performance and allow thesystem to be direct-sequence spread spectrum (DSSS) due to itscapability in handling nontrivial levels of multiuser interferencein wireless networks. Thus, a packet is received successfully atthe center receiver if ρh0 d−α θ its approximation accuracy with respect to the true outage 0 −α ≥ (14) probability at the center receiver, which is defined as N0 + j∈σ ρhj xj M ρh0 d−α θwhere N0 is the received noise power over the entire bandwidth, Pout = P ∞ 0 −α < . (16) N0 + i=1 ρh[i] x[i] Mσ is the set of transmitting links, xj is the location of thetransmitter of a transmitting link j, and M is the spreading The key result of this section is the following theorem, whichfactor of DSSS (M = 1 in nonspread spectrum systems). l gives closed-form solutions to the outage probabilities Pout and Due to the interference constraint, the set of actual scheduled Pout .transmitters in σ is a subset of the contending node set. In l Theorem 4: The outage probabilities Pout and Pout can befact, the distribution of transmitting nodes is quite compli- expressed as follows:cated, which depends on various factors, such as interferencemodel, stochastic packet arrival processes, channel fading, and ∞ θ 2(λπx2 )K Ψ(x)e−λπx dx (17) 2scheduling algorithms. In this paper, to make the analysis Pout = 1 − exp − ltractable, we apply an approximation by assuming that the set Mη xΓ(K) 0of transmitting nodes is also a PPP with a smaller density λ, 2 θ θ α 2π/αwhich is obtained by proper “thinning” of the original PPP. Pout = 1 − exp − − λπd2 0 (18)Note that, strictly speaking, the set of transmitting nodes should Mη M sin(2π/α)be separated by a certain distance, in which case, a hardcore −αpoint process [37] is more suitable. However, it has been where η = ρd0 /N0 is the signal-to-noise ratio (SNR) at x α+1observed that the PPP model can still achieve very accurate the center receiver, and Ψ(x) = ( 0 (2M r /x2 (M rα + α K−1 http://ieeexploreprojects.blogspot.comapproximation [39] on the distribution of the interference, d0 θ))dr) .particularly when the guard zone sizes are relatively small. This Proof: See Appendix E.has also been verified by simulation results in the case of graph Having obtained the analytical results, we will illustrate theinterference models (see details in [20]). We next analyze the interference approximation accuracy of the hypergraph modeloutage performance under the approximate PPP model. by comparing the foregoing two outage probabilities using numerical calculations in the next section.B. Outage Analysis V. N UMERICAL R ESULTS To explore the interference modeling limit of the hypergraphmodel, we assume that the transmission density λ under the In this section, we demonstrate the numerical results on thehypergraph model is as follows. A hypergraph with maximum performance of the hypergraph model in wireless networks.hyperedge size K can always guarantee that the following ap- We first consider the random infinite network described in lproximate “local” outage probability Pout at the center receiver Section IV and explore the interference approximation accuracyis bounded by of the hypergraph model by numerical calculations. ρh0 d−α θ A. Infinite Random Networks Pout = P l K−1 0 −α < ≤ (15) N0 + i=1 ρh[i] x[i] M We next calculate the two outage probabilities in Theorem 4.where is a positive constant, x[i] is the location of the ith near- By choosing parameters as in Table I, we plot the numericalest transmitting node, and h[i] is its corresponding power fading results in Fig. 4, where the outage probabilities are shown incoefficient. This is because the hypergraph model approximates both cases, as functions of the transmission density λ, accordingthe total interference by the sum interference from the nearest to (17) and (18), under different path loss exponents. In the fig-K − 1 transmitters. Further, note that such an outage bound can ure, “K-Hypergraph” refers to the hypergraph whose maximumeasily be achieved by a hypergraph with maximum hyperedge hyperedge size is K. Note that since the hypergraph modelsize K, since if (15) fails to hold for a particular transmitting always underestimates the interference, we have Pout ≤ Pout lset consisting of K links, one can simply form a hyperedge for all transmission densities.to exclude such a transmission scenario. Finally, note that Remarks:by choosing an appropriate outage bound , the transmission 1) Approximation accuracy: Compared with the graphdensity λ can be controlled. model, the hypergraph model always approximates the l Since the approximated “local” outage probability Pout only true outage probability with better accuracy. For example,considers a subset of interfering links, we are interested in when λ = 10−3 and α = 3, the true outage probability is
  8. 8. 304 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 1, JANUARY 2012Fig. 4. Numerical results of outage calculations for the infinite 2-D random wireless networks with Rayleigh fading. (a) Case with the path loss exponent α = 3and (b) the case with α = 4. around 0.22. However, the outage probability approxima- tion by the graph model is only around 0.15. Therefore, roughly speaking, around 30% of the outage events are ignored by the graph due to its binary interference nature. In this case, 4-Hypergraph has a better approximation of around 0.19. Thus, by considering the sum interference, the hypergraph model can effectively capture more out- age events and therefore reduces the outage probability. 2) Accuracy versus complexity: The approximation accu- http://ieeexploreprojects.blogspot.com racy of the hypergraph has a “diminishing returns” prop- erty, which can be seen by observing the fact that the outage approximation error improvement decreases as the maximum hyperedge size K increases. On the other hand, the construction complexity of the hypergraph ex- ponentially increases in K. Thus, one can tradeoff some Fig. 5. Topology of the random network with 40 links used for simulation. The square nodes are transmitters, and the round nodes are receivers. approximation accuracy by only considering properly small hyperedge sizes so that the sum interference is are receivers. We assume that the link lengths are equal. We approximated with low complexity. further assume that the effect of fading is properly handled 3) Effect of path loss exponent α: The modeling accuracy using diversity techniques, such that all random coefficients of both graph and hypergraph models improves when the {hi } in (14) take the constant value 1. The SNR at the receivers pass loss exponent gets larger. In particular, when α = 4 are the same for all links. The parameters used in the simulation and Pout = 0.3, all hypergraphs can capture above 95% are shown in Table I. outage events, and the graph can model around 80% The hypergraph is generated according to the description in outage events, as can be seen by computing the ratio Section I-C, with the modification that (4) is replaced with l Pout /Pout . On the other hand, for the case α = 3 and Si θ Pout = 0.3, the ratio is only about 85% for hypergraphs k−1 < (19) Ni + s=1 Iis i M and about 70% for the graph. Such improvement with larger values of α is because, when α grows larger, due to the DSSS physical layer. Further, we assume that no the contributions of the far-away interferers are much superset of a hyperedge is a hyperedge, and therefore, the smaller as compared with the nearby interferers, and hypergraph specification is not redundant. therefore, the local interference approximation becomes In the simulation, we assume that the packet arrival processes more accurate. are independent identically distributed with Bernoulli distribu- tion and a uniform arrival rate. We simulate both hypergraph (which includes graph) and the global SINR-based schedulingB. Finite Random Network algorithms. We consider random maximal scheduling for the We next consider the performance of hypergraph model in a hypergraph case, which adds the links to the schedule accordingfinite random network, where the topology is shown in Fig. 5. to a randomly generated order in each time slot, such that aIn the figure, 40 links are uniformly distributed over a 2-D backlogged link is scheduled if there is no hyperedge constraintplane. The square nodes are transmitters, and the round nodes violation when it is being considered. We also consider the
  9. 9. LI AND NEGI: MAXIMAL SCHEDULING IN WIRELESS NETWORKS WITH HYPERGRAPH INTERFERENCE MODELS 305 http://ieeexploreprojects.blogspot.comFig. 6. Simulation results of the maximum total queue lengths in the 40-link random wireless network, with the path loss exponent α and the threshold β valuesshown in each figure.random maximal scheduling under the global SINR, which network-wide user node coordination due to the global nature ofwe denote as SINR-MS. As an upper bound, we simulate the the SINR model. In this sense, the hypergraph-based schedulersperformance of SINR-based LQF scheduling (SINR-LQF) in are more attractive due to the localized user coordination during[31], which adds the links according to the queue lengths order, scheduling. The better throughput performance of SINR-LQFsubject to the physical SINR constraint (14). Finally, the packet over SINR-MS is also expected since the queue lengths orderreception at a transmitting link i is assumed to fail if the true information is used in the former case, which requires higherSINR constraint (14) is violated. scheduling complexity. 1) Throughput: The total queue lengths under different ar- In all cases, the hypergraph-based scheduling algorithmsrival rates are shown in Fig. 6. The results are averaged over achieve larger throughput than graph-based scheduling. Note30 independent simulations, where each simulation consists that due to the construction procedure in (19), the set ofof 104 time slots. One can detect the boundary of the stabil- hyperedges is always a superset of the set of edges, andity region (the maximum uniform throughput) by identifying therefore, it may seem like the hypergraph “should” achieve athe point at which the total queue length begins to increase smaller capacity than the graph due to the more restricted rules.sharply. For example, in the case of α = 3 and β = 4 dB, the However, since the graph model is a binary approximation, thegraph model achieves a maximum uniform rate of 0.22, the actual throughput is reduced by the packet collisions caused byhypergraph models achieve about 0.24, the SINR-MS achieves its “aggressive” transmissions. Thus, although the hypergrapharound 0.26, and the SINR-LQF has the largest rate, which models are relatively “conservative” since they place more con-is around 0.30. In the case of random maximal scheduling, straints according to the sum interference, overall they can stillthe SINR-based scheduling algorithms achieve around 10% achieve a better throughput, as compared with the graph model.throughput gain over hypergraph-based algorithms when α = 3 Finally, by comparing the throughput results under differentand β = 4 dB. The gain is around 5% when α = 4 and β = β, one can easily observe the “accuracy versus complexity”4 dB. Such throughput gain is mainly because of the perfect tradeoff for hypergraph models. In the simulation results,accuracy of the SINR model, which results in zero packet the throughput performance for hypergraph schedulers im-collision. On the other hand, this is achieved at the expense of proves with larger β due to the more accurate interference
  10. 10. 306 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 1, JANUARY 2012 http://ieeexploreprojects.blogspot.comFig. 7. Simulation results of average outage probability in the 40-link random wireless network, with the path loss exponent α and the threshold β values shownin each figure.approximation by adding more links in Li so that the number 2) Outage capacity: If we consider the outage capacity, i.e.,of packet collisions is reduced. Note that the throughput gain the maximum achievable rate under a certain outage prob-also depends on α, where a larger α implies a more accurate in- ability constraint, the hypergraph interference models canterference approximation for the same β since the contribution have a much larger gain as compared with the graphfrom far-away links gets smaller. model. For example, under the outage probability con- 2) Outage Probability: The simulation results of average straint of 0.02 when α = 3 and β = 4 dB, 4-Hypergraphoutage probabilities are shown in Fig. 7 with different arrival can support a maximum rate of around 0.2, whereasrates under different path loss exponents α and threshold β the graph model can only achieve about 0.15. Thus, byvalues. Note that the results of both SINR-based scheduling are considering the sum interference, the hypergraph modelnot shown in the figure, as they both have zero outage. achieves around 30% increase in the outage capacity as Remarks: compared with the graph model. 1) Outage probability: The hypergraph model achieves a 3) Accuracy versus complexity: The outage probabilities significant outage probability reduction as compared with of hypergraph models decrease when the threshold β the graph model. For example, when α = 3 and β = increases due to the improved approximation accuracy 4 dB, the average outage probability for 3-Hypergraph by considering more links in a link’s neighborhood. For is around 0.02 under arrival rate 0.2. On the other hand, example, when α = 3 and β = 0 dB, all hypergraph under the graph model, it is around 0.05. Thus, by mod- models have an outage probability of 0.03 under arrival eling the sum interference, the hypergraph model can rate 0.2, as compared with around 0.02 when β = 4 dB. reduce the packet collisions very effectively. Note that On the other hand, the threshold β has no effect on the outage probability curves have slower slopes when the graph model due to the binary interference nature. the arrival rates are large. This corresponds to the case Further, note that when β is small, all hypergraph mod- when the network is unstable. In such a case, the number els achieve very similar outage probability results, in of packet transmissions is less sensitive to the increase in which case, 3-Hypergraph is more attractive due to its arrival rates. lower coordination complexity. In general, increasing the
  11. 11. LI AND NEGI: MAXIMAL SCHEDULING IN WIRELESS NETWORKS WITH HYPERGRAPH INTERFERENCE MODELS 307 threshold β can effectively reduce the outage probability where the function f (·) can be Qi (·), Ai (·), or Di (·). It can be by considering the interference from farther transmitting verified that these functions are uniformly Lipschitz continu- links but at the expense of more coordination overheads ous, i.e., for any t > 0 and δ > 0, we have among links. Finally, the reduction in outage proba- bility decreases as the hypergraph sizes increase. This |f r (t + δ) − f r (t)| ≤ Kδ (22) “diminishing marginal returns” property again confirms our observation that in wireless networks, the majority where the positive constant K is Amax for functions Ai (·) of interference is from a few nearby transmitting links. and Qi (·) and 1 for function Di (·). Thus, these functions are Therefore, a hypergraph with a small hyperedge size (e.g., equi-continuous. According to the Arzéla–Ascoli theorem [40], 4-Hypergraph) can model the interference with good any sequence of functions {f rn (t)}∞ contains a subsequence n=1 accuracy. {f rnk (t)}∞ such that w.p.1 k=1 4) Effect of path loss exponent: When the path loss exponent ¯ lim sup f rnk (τ ) − f (τ ) = 0 (23) α gets larger, both graph and hypergraph models have k→∞ τ ∈[0,t] smaller outage probabilities. Further, the gap between these two models is also smaller. This is the same conclu- ¯ where f (t) is a uniformly continuous function (and therefore sion as the numerical results for infinite random networks differentiable almost everywhere [40]). Define any such limit in Section V-A. The intuition is that the interference ¯ ¯ ¯ (Q(t), A(t), D(t)) as a fluid limit. signals get more attenuation as α gets larger and there- Properties of Fluid Limits: The following properties hold for fore is more likely to be dominated by a few nearby any fluid limit: transmitting links since the faraway transmissions are ¯ Ai (t) = ai t (24) attenuated more severely than the nearby transmissions. Thus, the accuracy of both graph and hypergraph models d ¯ ¯ become better, and the difference between the two is also Q(t) = 0, if Q(t) = 0 (25) dt smaller. where (24) occurs because of the (functional) SLLN, and (25) ¯ is because any regular point t with Q(t) = 0 achieves the VI. C ONCLUSION ¯ minimum value (since Q(t) ≥ 0) and, therefore, has zero deriv- We have proposed a hypergraph interference model for the ative. We further have the following lemma, which provides a http://ieeexploreprojects.blogspot.comscheduling problem in wireless networks. The proposed hyper- sufficient condition about rate stability (see the proof in [41]).graph is quite flexible in modeling sum interference constraints, Lemma 1: The network is rate stable if any fluid limit with ¯ ¯which can include both the graph model and the physical Q(0) = 0 has Q(t) = 0 ∀t ≥ 0.SINR model as special cases. We investigated the performanceof the scheduling algorithms with the hypergraph model andproposed a lower bound on the stability region for any maximal A PPENDIX Bscheduler. We further proposed a lower bound on its scheduling P ROOF OF T HEOREM 1efficiency, which depends on a certain definition of the interfer- Proof: Consider the following Lyapunov function:ence degree of the underlying hypergraph. Finally, we analyzed ⎛ ⎞the outage probability of the hypergraph model in random 1networks and showed that hypergraphs with small hyperedge L(t) = Qi (t) ⎝Qi (t) + ¯ ¯ Wij Qj (t)⎠ . ¯ (26) 2sizes can model the interference quite accurately. i∈V j∈Ni ¯ Now, assuming that Q(0) = 0, we have that L(0) = 0. Further, A PPENDIX A its derivative is F LUID L IMITS ˙ ¯ ˙ ¯ 1 ¯ ˙ ¯ We briefly introduce the framework of fluid limits, which is L(t) = Qi (t)Qi (t) + (Wij + Wji )Qi (t)Qj (t) 2used to prove the rate stability. For details, see [41] and the i∈V i∈V j∈Nireferences therein. ⎛ ⎞ Definition of Fluid Limits: We first rewrite the queueing Qi (t) ⎝Qi (t) + Wij Qj (t)⎠ (a) ˙ ˙ = ¯ ¯ ¯equation in (9) as follows: i∈V j∈Ni Qi (n) = Qi (0) − Di (n) + Ai (n) (20) ⎛ ⎛ ⎞⎞where Di (n) = n σi (k) is the cumulative packet depar- (b) = ¯ ⎝ Qi (t) ai + ⎝¯ ˙ Wij aj− Di (t)+ Wij Dj (t) ⎠ ¯ ⎠ ˙ k=1tures from link i during the past n time slots. Given the network i∈V j∈Ni j∈Nidynamics (Q(n), A(n), D(n))∞ , we first extend the support t=0from N to R+ using linear interpolation. For a fixed sample path where (a) is because the matrix W is symmetric, and (b) is ¯ because of SLLN in (24). Note that if Qi (t) = 0 for all i ∈ Vω, define the following fluid scaling: ˙ and t ≥ 0, from (27), we conclude that L(t) = 0 for all t ≥ 0, f r (t, ω) = f (rt, ω)/r (21) ¯ and therefore, L(t) = 0, and Q(t) = 0 for all t ≥ 0, from which
  12. 12. 308 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 1, JANUARY 2012the proposition holds following Lemma 1. Now, assuming that Further, for any hyperedge e, we have ¯there exists link i and t0 ≥ 0 such that Qi (t0 ) > 0, we now 1show that Δij ≥ =1 (31) |e| − 1 ⎛ ⎞ j∈e j∈e,j=i ai + Wij aj − ⎝Di (t0 ) + ˙ ¯ Wij Dj (t0 )⎠ ≤ 0 (27) ˙ ¯ due to the definition in (5). Thus, we conclude that the matrix j∈Ni j∈Ni {Δij } belongs to W, and therefore, the lemma holds according to Theorem 1. ˙from which one can still conclude that L(t) ≤ 0, following We next prove the following lemma, which proposes anwhich the theorem holds. upper bound on the weighted departure in the neighborhood Now, consider any converging subsequence {rnk }∞ , which {i} ∪ Ni for any link i. k=1 ¯ ¯ ¯is associated with the fluid limit (Q(t), A(t), D(t)). Since Lemma 3: For any feasible arrival rate a ∈ A , we have ¯ ¯Qi (t0 ) > 0, there is δ > 0 such that Qi (t0 ) > δ > 0. Further, ¯since the function Qi (t) is uniformly continuous, there exists ai + Δij aj ≤ Δi ≤ Δ ∀i ∈ V. (32) ¯τ > 0 such that Qi (t0 ) > δ/2 for all t ∈ (t0 − τ, t0 + τ ). Thus, j∈Nifor sufficiently large k, the convergence in (23) implies that Proof: Let a link i ∈ V be given. Since a ∈ A , we assume rnQi k (t) > δ/4 for all t ∈ (t0 − τ, t0 + τ ), i.e., that there is a scheduler π such that a ∈ Aπ . For any such scheduler π, consider the following “Lyapunov” function:Qi (t) > rnk δ/4 ≥ 1 ∀t ∈ (rnk (t0 −τ ), rnk (t0 +τ )) . (28) Vi (n) = Qi (n) + Δij Qj (n) (33) j∈NiThus, for sufficiently large k, link i always has a nonemptyqueue during time slots (rnk (t0 − τ ), rnk (t0 + τ )). Finally, = Qi (0) + Ai (n) − Di (n) (34)according to maximal scheduling, in each time slot, either linki transmits, or there is an active hyperedge e with respect to link + Δij (Qj (0) + Aj (n) − Dj (n)) (35)i. In both cases, we have j∈Niσi (t)+ Wij σj (t) ≥ 1 where Di (n) = n σi (k) is the cumulative departures dur- ∀t ∈ (rnk (t0 − τ ), rnk (t0 + τ )) k=1 j∈Ni ing the first n time slots. http://ieeexploreprojects.blogspot.com Since the network is rate stable under the scheduler π, we havewhich is due to the fact that the matrix W satisfies j∈e Wij ≥1 for any hyperedge e. Summing up the departures, we have lim Vi (n)/n = 0, w.p.1 (36) n→∞ rnk rnk Di (t0 + τ ) − Di (t0 − τ ) which implies that w.p.1 rnk rnk Ai (n) + j∈Ni Δij Aj (n) + Wij Dj (t0 + τ ) − Dj (t0 − τ ) ≥ 2τ. lim inf n→∞ n j∈Ni Di (n) + j∈Ni Δij Dj (n) (a) ≤ lim sup ≤ Δi ≤ ΔFrom which we conclude that, since τ can be arbitrarily small, n→∞ nin the fluid limit, we have where (a) is because of the upper bound on the total departures ˙ ¯ Di (t) + ˙ ¯ Wij Dj (t) ≥ 1 (29) in each time slot in (11). Therefore, the lemma follows from j∈Ni the SLLN in (8) and the fact that the scheduler π is chosen arbitrarily.and therefore, (27) holds due to (I + W )a 1. Proof of Theorem 2: We can now prove Theorem 2. From the result in Lemma 3, we conclude that if a ∈ A , then (1/Δ)a must satisfy (30), and therefore, according to Lemma 2, is A PPENDIX C stable under any maximal scheduler π. P ROOF OF T HEOREM 2 To prove the theorem, we first need to prove two lemmas. We A PPENDIX Dstart with Lemma 2. P ROOF OF T HEOREM 3 Lemma 2: An arrival rate a is stable under any maximalscheduler π if Proof: Consider the following arrival rate vector a: aj = 1 if and only if j ∈ {e1 , e2 , . . . , eΔ }/{i}; otherwise, aj = 0. ai + Δij aj ≤ 1 ∀i ∈ V. (30) It is easily seen that a ∈ A since the set of links j∈Ni {e1 , e2 , . . . , eΔ }/{i} is an independent set. Now consider the arrival rate a such that aj = aj /Δ if j = i, and ai = . Thus, Proof: Note that for the set of weights {Δij } defined in we have aj − (1/Δ)aj = for all j ∈ V. We next show that(5), we have Δij ≥ 0, Δii = 0, and Δij = Δji for all i, j ∈ V. there exists an arrival process with such rate a , which makes
  13. 13. LI AND NEGI: MAXIMAL SCHEDULING IN WIRELESS NETWORKS WITH HYPERGRAPH INTERFERENCE MODELS 309the network unstable under a maximal scheduler π that assigns Step (b) happens because the fading coefficient h is exponen-link i the lowest priority. That is, link i is always considered last tially distributed with mean 1. Further, according to [38], theby the scheduler π during scheduling. The arrival process is as distribution of RK is as follows:follows. In each kth time slots out of every Δ time slots, there isa packet arriving at each link in the set of links ek /{i}. Then, it 2(λπx2 )K −λπx2 fRK (x) = e (40)is immediately transmitted in the next time slot, because these xΓ(K)links have higher priority than link i and form an independent and therefore, the lemma holds after taking the expectation withset. Further, it is easily seen that there is no departure from link respect to fRK (x).i, since in each time slot, the transmitting links form an “active” Similarly, denote the total interference received at the centerhyperedge with respect to link i. As far as link i is concerned, receiver as Itot = ∞ l( x[i] ). The following lemma de- i=1we assume that in each time slot, there is a packet arriving scribes the distribution of Itot .at link i with probability so that ai = . Thus, since link i Lemma 5: The MGF of the total interference Itot isnever gets a chance to transmit, it is starved, and the network isunstable. 2 2π/α ΦItot (s) = exp −λπ(−sρ) α . (41) sin(2π/α) A PPENDIX E Proof: This is a standard result. See, for example, [39]. P ROOF OF T HEOREM 4 Based on the foregoing lemmas, we are now able to prove the To prove the theorem, we first need to prove two lemmas. theorem.Define the “local” interference contributed by the nearest K −1 Proof of Theorem 4: By definition, we calculate the localtransmitting nodes Iloc (K − 1) as follows: outage probability as ρh0 d−α θ K−1 Pout = P l 0 < Iloc (K − 1) = ρh[i] l x[i] . (37) N0 + Iloc (K − 1) M i=1 dα θ = P h0 < 0 (N0 + Iloc (K − 1))We have the following lemma describing the distribution of MρIloc (K − 1). (a) dα θ = EIloc (K−1) P 0 http://ieeexploreprojects.blogspot.comh0 < M ρ Lemma 4: The moment generating function (MGF) ofIloc (K − 1) can be expressed as follows: ∞ 2(λπx2 )K × (N0 +Iloc (K −1)) |Iloc (K −1) Ψ(x)e−λπx dx 2 ΦIloc (K−1) (s) = (38) xΓ(K) 0 (b) dα θ = 1 − EIloc (K−1) exp − 0 Iloc (K − 1)where Γ(K) is the standard Gamma function Mρ ∞ dα θ × exp − N0 Γ(K) = xK−1 e−x dx. (39) Mρ 0 (c) dα θ dα θ = 1 − ΦIloc (K−1) − 0 exp − 0 N0 Mρ Mρ Proof: Denote Rk = x[k] as the short-hand notation forthe distance of the kth nearest transmitting node. The MGF of where step (a) follows from the law of total probability, step (b)Iloc (K − 1), conditioned on the event that RK = rK , can be happens because the random variable h0 is exponentially dis-expressed as follows: tributed with mean 1, and step (c) follows from the definition of K−1 the MGF. Thus, the claim holds from noting that η = ρd−α /N0 0E esIloc (K−1) |RK =rK =E es i=1 ρh[i] l(Ri ) |RK = rK and applying the result in (38). Now, using a similar argument, the true outage probability ⎛r ⎞K−1 K Pout can be expressed as follows: 2r = ⎝ )dr⎠ (a) sρhr −α 2 E(e rK ρh0 d−α θ 0 Pout = P 0 < N0 + Itot M ⎛r ⎞K−1 dα θ K 2rα+1 = P h0 < 0 (N0 + Itot ) = ⎝ dr⎠ (b) Mρ rK (rα − ρs) 2 dα θ dα θ 0 = 1 − EItot exp − 0 Itot exp − 0 N0 Mρ Mρwhere step (a) happens because, conditioned on the fact that dα θ dα θthere are K − 1 nodes in the disk centered at the origin with = 1 − ΦItot − 0 exp − 0 N0 Mρ Mρradius rK , these K − 1 nodes are independently and uniformlydistributed inside the disk due to the property of PPP [37]. from which the claim holds after applying (41).

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