IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 19, NO. 3, JUNE

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1 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL 19, NO 3, JUNE Improved Bounds on the Throughput Efficiency of Greedy Maximal Scheduling in Wireless Networks Mathieu Leconte, Jian Ni, Member, IEEE, R Srikant, Fellow, IEEE Abstract In this paper, we derive new bounds on the throughput efficiency of Greedy Maximal Scheduling (GMS) for wireless networks of arbitrary topology under the general -hop interference model These results improve the known bounds for networks with up to 26 nodes under the 2-hop interference model We also prove that GMS is throughput-optimal in small networks In particular, we show that GMS achieves 100% throughput in networks with up to eight nodes under the 2-hop interference model Furthermore, we provide a simple proof to show that GMS can be implemented using only local neighborhood information in networks of any size Index Terms Greedy Maximal Scheduling (GMS), Longest Queue First (LQF), throughput optimality, wireless networks I INTRODUCTION I N WIRELESS communication networks with limited resources, efficient resource allocation optimization play an important role in achieving high performance providing satisfying quality of service In this paper, we study link scheduling for wireless networks, where the links (node pairs) may not be able to simultaneously transmit due to transceiver constraints /or radio interference A scheduling algorithm determines which links can transmit at each time instant so that no two active links interfere with each other The performance metric of interest in this paper is throughput, we restrict our attention to MAC layer (or link-level) throughput as opposed to end-to-end throughput It is well known that the queue-length-based Maximum Weighted Scheduling (MWS) algorithm is throughput-optimal [26] in the sense that it can stabilize the queues in the network for all arrival rates in the capacity region of the network However, MWS requires the network to select a max-weight schedule in every time slot (the weight of a schedule is the sum of the weights of the scheduled links), which corresponds to finding a max-weight independent set in the interference graph This is known to be NP-hard for general interference graphs [8] In addition, MWS is not amenable to distributed implementation Manuscript received August 12, 2009; revised January 22, 2010 September 10, 2010; accepted September 27, 2010; approved by IEEE/ACM TRANSACTIONS ON NETWORKING Editor S Sarkar Date of publication November 15, 2010; date of current version June 15, 2011 This work was supported by NSF Grant CNS , AFOSR Grant FA , ARO MURI BAA , ARO MURI Subaward Z848402, DTRA HDTRA An earlier version of this paper appeared in the Proceedings of the ACM International Symposium on Mobile Ad Hoc Networking Computing (MobiHoc), New Orleans, LA, May 18 21, 2009 M Leconte is with INRIA Technicolor, Paris 75013, France ( mathieuleconte@inriafr) J Ni R Srikant are with the Coordinated Science Laboratory, University of Illinois at Urbana Champaign, Urbana, IL USA ( jianni@illinoisedu, rsrikant@illinoisedu) Digital Object Identifier /TNET These drawbacks limit the deployment of MWS in practice Even in small networks, MWS can require quite a lot of operations because its complexity is tied to the number of maximal schedules of the network As an example, it is easy to see that an 8-cycle a complete graph of eight nodes have respectively distinct maximal schedules if we consider directed links the 2-hop interference model Thus, although those networks are small, distributed implementation of MWS is still not practical for them Therefore, it is of interest to find simple, distributed scheduling algorithms that can achieve optimal or near-optimal performance Maximal scheduling is a low-complexity alternative to MWS that is amenable to parallel distributed implementation [1], [19] However, maximal scheduling may only achieve a small fraction of the capacity region [5], [24], [27] Greedy Maximal Scheduling (GMS), also known as Longest-Queue-First (LQF), is another natural low-complexity alternative to MWS [7], [10], [20] Its performance has been observed to be close to optimal in a variety of wireless network simulations (eg, [12], [18]) The focus of this paper is to derive new bounds on the throughput efficiency of GMS Our results are especially useful in the case of small to moderate-sized networks (the most common type of networks in today s civilian military applications) since they show the throughput-optimality of GMS in small networks improve previous bounds for moderate-sized networks Our results also provide further motivation to the recently proposed graph partitioning techniques that divide a large wireless network into smaller networks to improve scheduling efficiency [4], [25] Our contributions in this paper are as follows Ring networks were used in earlier papers to show that GMS is not throughput-optimal We complement these earlier results by deriving the exact throughput efficiency of GMS in such networks Moreover, the results on ring networks can be used to establish the tightness of some of our other results mentioned in the following paragraphs We derive bounds on the throughput efficiency of GMS for networks with arbitrary topology under the general -hop interference model In particular, for the 2-hop interference model, our bound improves previous results [14] in networks with up to 26 nodes We prove GMS is throughput-optimal in small networks under the -hop interference model In particular, we show throughput-optimality in networks with up to eight nodes under the 2-hop interference model This result is tight in the sense that we can find networks with nine nodes where GMS is not throughput-optimal with 2-hop interference As a by-product of these results, we also establish the throughput-optimality of GMS in networks with up to /$ IEEE

2 710 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL 19, NO 3, JUNE 2011 five nodes under the 1-hop interference model This analytically proves a numerical observation in [4] We show that GMS, which requires global knowledge of link weights, is equivalent to an algorithm called Local GMS (LGMS) [23], which uses only local neighborhood information The paper is organized as follows In the rest of this section, we introduce the related work We describe the network model in Section II provide some useful notions in Section III In Section IV, we derive the throughput efficiency of GMS in ring networks In Section V, we provide bounds on the throughput efficiency of GMS in arbitrary networks under the -hop interference model In Section VI, we prove GMS is throughput-optimal in small networks We discuss some implementation issues of GMS show the equivalence of GMS LGMS in Section VII The paper is concluded in Section VII A Related Work To identify sufficient conditions for GMS to be throughputoptimal, the concept of local pooling (will be formally defined later) was introduced in [7] The authors showed that if the network satisfies the local pooling condition, then GMS is throughput-optimal In particular, if the interference graph of the network is a tree, then the local pooling condition is satisfied In addition to tree (interference) graphs, [3], [4], [28] identified several classes of graphs that satisfy the local pooling condition Independently, [14] [28] showed that the local pooling condition is satisfied in tree networks 1 under the -hop interference model In [4] [28], the local pooling condition was tested for small interference graphs via exhaustive numerical search They found that local pooling is satisfied in networks with up to five links under the 1-hop interference model in networks with up to seven links under the -hop interference model In [13] [14], the notion of local pooling was generalized to -local pooling the concept of local-pooling factor was introduced The authors showed that the efficiency of GMS is the same as the local-pooling factor of the network proposed a recursive procedure to bound the local-pooling factor of a network Our results primarily build upon the results in [7] [13] In [17], the authors extended -local pooling to link -local pooling, which provides a refined throughput characterization of GMS Performance of GMS under the SINR interference model has been recently studied in [15] LGMS implements GMS using information available in a local neighborhood [2], [11], [23] It has been shown previously [23] that LGMS GMS achieve the same approximation ratio compared to MWS In this paper, we show a stronger result namely, the sets of schedules produced by GMS LGMS are the same II NETWORK MODEL We model a wireless network by a graph, where is the set of nodes is the set of links Nodes are wireless transmitters/receivers, there exists a link between two nodes 1 Note that the interference graph of a tree network may not be a tree if they can directly communicate with each other For any link, we can define the set of its interfering links as interferes with (1) We assume that if interferes with, then also interferes with Associated with a network graph, we can define an interference graph, where there is an edge in from to if This graph contains all the information regarding the interference model used Throughout this paper, we will always assume that a node can only be part of one active link at a time We consider a time-slotted system A schedule of is a subset of links that can be activated/scheduled at the same time according to the interference constraint, ie, no two links in interfere with each other We assume that all links have unit capacity, ie, a scheduled link can transmit one packet in one time slot Associated with a schedule is a rate vector The th element of is equal to 1 if link is scheduled ; otherwise Then, a convex combination of schedules will be an element of A schedule is said to be maximal if no link can be added to it without violating the interference constraint Denote by the set of the rate vectors of all maximal schedules of, by its convex hull A scheduling algorithm is a procedure to decide which schedule should be used (ie, which subset of links should be activated) in every time slot The capacity region of the network is the set of all arrival rates for which there exists a scheduling algorithm that can stabilize the queues, ie, the queues are bounded in some appropriate stochastic or deterministic sense depending on the arrival model used For the purposes of this paper, we will assume that if the arrival process is stochastic, then it is a stationary, ergodic process with finite first second moments, the resulting queue length process admits a Markovian description Alternatively, one can also assume that the arrival process is deterministically bounded such as the well-known process [6] In this paper, we focus on the MAC layer, thus we only consider one-hop traffic It is known from [26] that the capacity region is given by When dealing with vectors, inequalities are interpreted componentwise We say that a scheduling algorithm is throughputoptimal, or achieves the maximum throughput, if it can keep the network stable for all arrival rates in the capacity region Suppose each link is associated with a nonnegative weight Let be the weight of schedule Amaximum-weight schedule of is a schedule that has the maximum weight among all schedules of Note that if all link weights are strictly positive, then a maximum-weight schedule must be a maximal schedule It is well known that the MWS algorithm, which selects a max-weight schedule in every time slot, with the link weights equaling the link queue lengths, is throughput-optimal [26] However, as we mentioned in Section I, finding a max-weight (2)

3 LECONTE et al: IMPROVED BOUNDS ON THROUGHPUT EFFICIENCY OF GREEDY MAXIMAL SCHEDULING IN WIRELESS NETWORKS 711 schedule of a network is equivalent to finding a max-weight independent set of the associated interference graph, which is known to be NP-hard for general interference graphs In addition, MWS is centralized in nature is not amenable to distributed implementation The GMS algorithm, which is a natural low-complexity alternative to MWS, proceeds as follows In each time slot, the schedule that will be used is built sequentially Start with an empty schedule At each step, choose a link with maximum weight among the nondisabled links add it to the current schedule, then disable all links that interfere with Continue until all the remaining links are disabled Note that any schedule obtained by GMS is maximal Greedy Maximal Scheduling (GMS) Algorithm WHILE Pick a globally heaviest link : ENDWHILE Since GMS is a greedy algorithm, in general, it may not achieve the full capacity region of the network The efficiency ratio of a scheduling algorithm is the largest fraction of the capacity region that is stabilized by the algorithm Definition 1: (Subset Local Pooling SLoP): A set of links satisfies subset local pooling if, such that, Note that the interference constraints used to compute maximal schedules of are given by the interference graph of Remark 1: One way to underst the subset local pooling condition is to view as the utility obtained when receiving a unit of service on link Then, is the total utility associated with the service rate vector If the total utility for any rate vector is a constant, then there is no way a vector can strictly dominate another vector because otherwise would provide a strictly larger total utility than Definition 2: (Overall Local Pooling OLoP): A graph satisfies overall local pooling if all subsets of satisfy subset local pooling It was proved in [7] that GMS is throughput-optimal in a network if the network graph satisfies overall local pooling Although local pooling is useful when trying to determine whether GMS is throughput-optimal in a network, it is also of interest to get a sense of how well GMS performs when it is not optimal To answer this question, the local-pooling factor of a graph was introduced in [13] This factor gives a bound on how bad the schedules picked by GMS can be, compared to the optimal ones As for local pooling, this notion is better explained when broken down into parts Definition 3: ( -Local Pooling): A set of links satisfies -local pooling if,, Definition 4: (Local-Pooling Factor): The local-pooling factor of a graph is the supremum of all such that every subset of satisfies -local pooling the system is stable under arrival rate (3) Throughout this paper, we will often focus on the -hop interference model This model says that two links interfere with each other if only if the shortest path in between them is of length at most The 1-hop 2-hop interference models are of special practical interest Under the 1-hop interference model (also known as the node-exclusive or primary interference model), any two links sharing a common node cannot be active simultaneously This can be used to describe wireless networks that use FH-CDMA with only one transceiver per node [9] The 2-hop interference model is well suited to model networks that use RTS/CTS link-level ACK, which many practical communication schemes do (such as the IEEE MAC protocol [5], [27]) III PRELIMINARIES In [7], the notion of local pooling was introduced Local pooling is better explained when broken down into two parts [4]: subset local pooling (SLoP) overall local pooling (OLoP) It was proved in [13] that the efficiency ratio of GMS in a network is equal to the local-pooling factor of the network graph Therefore, we will use interchangeably in the rest of the paper Remark 2: Note that if a set of links satisfies subset local pooling, then it satisfies -local pooling for all, but the converse is not true As a consequence, if a graph satisfies overall local pooling, then its local-pooling factor is 1, equivalently GMS is throughput-optimal in that graph However, the converse does not hold true: GMS can be throughputoptimal in graphs that do not satisfy overall local pooling In that sense, overall local pooling is a more stringent condition (it is sufficient for GMS to be throughput-optimal, but not necessary) compared to (which is both sufficient necessary) If GMS is not throughput-optimal in a network, we know that, under some feasible arrival rate, there exist links whose queues will go up to infinity To analyze those links, [13] introduces the notion of unstable subset of links, which is a subset of links whose queues can all go to infinity under GMS under some feasible arrival rate We use the following slightly different definition

4 712 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL 19, NO 3, JUNE 2011 Definition 5: (Unstable Subset of Links): Let be a network graph We say that is an unstable subset of links of if, such that Note that if only if such that Thus, the unstable subsets of links of are the subsets of that fail -local pooling for some This implies that the local-pooling factor ( thus the efficiency ratio of GMS ) of a graph is strictly less than 1 if only if there exists at least one unstable subset of links of Thus, to prove that GMS is throughput-optimal in a graph, we only need to show that has no unstable subsets of links Also, if a subset of links is unstable, then there exists a feasible arrival rate a specific arrival pattern with this arrival rate such that, for all, the average service rate of under GMS is strictly less than the average arrival rate of In other words, there exists a feasible arrival rate such that the queues of all the links of go to infinity Such a feasible arrival pattern can be constructed as in [13] Moreover, it is clear that if a set of links satisfies subset local pooling, then it cannot be an unstable subset of links because otherwise, such that, then for all nonzero A Properties of Unstable Subset of Links We establish here some useful properties of unstable subsets of links that are valid under very general interference models More precisely, as long as the interference model can be defined using an interference graph, the following propositions apply Proposition 1: Let be a network graph be a subset of links of If any two maximal schedules of sharing a common link have the same size, then satisfies subset local pooling hence cannot be unstable Proof: For all, let be the size of the maximal schedules containing, take Let be the set of maximal schedules in, where denotes a particular maximal schedule Let, ie, such that,, Then For all, we have, because,, is equal to the size of Thus which is the definition of subset local pooling We now present some intuition behind the above proof As GMS uses only maximal schedules, when a link is scheduled for an amount of time, some service is also provided to some of the other links in the network The sum of the service provided to other links while scheduling is proportional to because the size of the maximal schedules containing is a constant More precisely, let be the set of links with maximal schedules of size Then, if is a maximal schedule containing a link in, is of size Therefore,, belongs to because all maximal schedules containing must have the same size as Thus,, where is the fraction of time we use a maximal schedule of size It means that the only way to provide a larger amount of service to all the links in is to spend a larger amount of time serving links in However, we cannot spend more time serving links in for all s at the same time, so one vector in cannot strictly dominate another one In other words, to have, with,, we must have for all In particular,, thus for all, which is not possible as From this sufficient condition for a set of links to satisfy subset local pooling, we can obtain a necessary condition that unstable subsets of links must satisfy Proposition 2: Any unstable subset of links must contain at least three links that can be simultaneously scheduled Proof: Let be an unstable subset of links of Suppose that the maximal schedules of have size at most 2 Using Proposition 1, we will simply check that all the maximal schedules of that share a common link have the same size Let Consider all the maximal schedules in that contain If one of them has size 2, which means that once is scheduled, it is still possible to schedule another link, then all those maximal schedules must have size 2; otherwise, all the maximal schedules containing are of size 1 ( actually is the only such maximal schedule) Proposition 1 implies that satisfies subset local pooling, so it cannot be unstable, a contradiction Thus, there must exist a schedule of of size 3 IV THROUGHPUT EFFICIENCY OF GMS IN RING NETWORKS In this section, we derive the exact efficiency ratio of GMS in ring networks under the -hop interference model Ring networks were among the first known network graphs for which GMS was shown to be not throughput-optimal (eg, the 6-cycle network under the 1-hop interference model analyzed in [7] [13]), so it is interesting to know the efficiency ratio of GMS in this class of networks Furthermore, the results in this section provide examples showing the tightness of some of the results we will derive in later sections Consider a ring network with an arbitrary number of nodes under the -hop interference model for arbitrary We will use to denote a ring network of size We can label the links with the integers from 1 to, such that consecutive links are labeled with consecutive integers We also refer to maximal schedules of as vectors of size with 0 s 1 s, where a 1 at position means that the link labeled is in the schedule a 0 that it is not is the set of all maximal schedules of, is its convex hull The size of a schedule refers to the number of links it contains Let denote the -norm of : In particular, if is the rate vector associated with a schedule, then is the number of links in Lemma 1: The maximum size of a maximal schedule of is The minimal size is Proof: Let be maximal schedules of with the maximum minimum size Because of the -hop interference model, two consecutive active links of a maximal schedule must be separated by at least at most inactive links,

5 LECONTE et al: IMPROVED BOUNDS ON THROUGHPUT EFFICIENCY OF GREEDY MAXIMAL SCHEDULING IN WIRELESS NETWORKS 713 which implies that However, these numbers might not be integers, so We can easily construct maximal schedules that achieve those values For example, consider the schedule such that the active links are the links labeled,, with, the resulting schedule is of size, it is maximal because Thus, Likewise, if the active links are the links labeled,, with, the resulting schedule is of size, we have Thus, we can add exactly one more link to that schedule so that it will be maximal Hence, Theorem 1: The efficiency ratio of GMS in under the -hop interference model is Proof: We will prove that is both an upper bound lower bound of, which then implies that As any strict subgraph of is a tree, we know that GMS has an efficiency of 1 in those, so we need only worry about the whole ring Again, let be maximal schedules of of the maximum minimum size, respectively Also let be the operator that shifts one link of a schedule, ie, Then, we have will use to say that We first prove the following lemma Lemma 2: Let be a network graph,, then satisfies -local pooling for all Proof: Suppose that does not satisfy -local pooling for some Then,, such that componentwise We can write where is the vector with a 1 at position 0 s everywhere else Writing, we also have with,,,,, let Then, for all,wehave Exping using the notations introduced above, we obtain the following inequalities: (4) where is the all-1 vector of size Using Lemma 1, the first of those two vectors dominates the second one by a factor, so Moreover,, there exist two vectors in such that componentwise However, then As are convex combinations of maximal schedules of, wehave Thus, Letting completes the proof From the theorem we immediately get the following corollary Corollary 1: The efficiency ratio of GMS in a ring network of any size under the -hop interference model is at least For example, the efficiency ratio of GMS in any ring networks is at least 2/3 under the 1-hop interference model 3/5 under the 2-hop interference model V PERFORMANCE BOUNDS OF GMS IN ARBITRARY NETWORKS For arbitrary network graphs in general, it is very difficult to compute the exact efficiency ratio of GMS In this section, we derive bounds on the throughput efficiency of GMS for networks with arbitrary topology under the -hop interference model We Combining (5) (6), we get, which means that satisfies -local pooling for all Lemma 3: If is an upper bound of the sizes of the maximal schedules of, then Proof: If, it means that there exists a maximal schedule that includes only one link, ie, link, no link can be added to it further Then, we know that also, thus On the other h if, then since, we get Based on the observations above, we can obtain a lower bound on the throughput efficiency of GMS Theorem 2: Under the -hop interference model, let be a network graph of size for some The local-pooling factor of is at least equal to (5) (6) (7)

6 714 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL 19, NO 3, JUNE 2011 Proof: We want to prove that any unstable subset of links of have TABLE I EFFICIENCY OF GMS IN SMALL TO MODERATE-SIZED NETWORKS (2-HOP INTERFERENCE MODEL) (8) We need only consider unstable subsets of links of because all other subsets of links satisfy -local pooling for all Then, by Lemma 2 since the local-pooling factor of is the supremum of all such that every subset of satisfies -local pooling, it will follow that Let be an unstable subset of links of Let us suppose that for some Let be the size of a maximal schedule of By Lemma 3, it is enough to check that Under the -hop interference model, we can observe that a node can be within hops of only one active link at a time Indeed, otherwise, the distance between the two active links would be at most We will say that the nodes within hops of an active link are disabled by that link It is then clear that any active link disables at least nodes, since itself consists of two nodes, we have which implies (8) holds Hence, Remark 3: The results for the efficiency of GMS in ring networks show that the smallest cycle in which GMS is not throughput-optimal under the -hop interference model is of size The theorem above shows that at least nodes are required for GMS to fail to achieve throughput-optimality Hence, there is at most a factor 2 in size between the smallest graph for which GMS is not throughput-optimal under the -hop interference model the smallest cycle for which we know GMS is not throughput-optimal for that same interference model A Improved Bounds for 2-Hop Interference Model In Section VI, we will show that under the 2-hop interference model, a maximal schedule of size 2 or more of an unstable subset of links will disable at least three nodes in the network (see Proposition 3) Using this result, we can improve the performance bounds of GMS in Theorem 2 for the 2-hop interference model Theorem 3: Under the 2-hop interference model, let be a network graph of size or for some The local-pooling factor of is at least equal to Proof: The proof is similar to the proof of Theorem 2 We want to prove that (8) holds for any unstable subset of links of Let s suppose that for some Let be the size of a maximal schedule of with It is enough to check that From Proposition 3, the maximal schedule will disable at least three nodes, each link in the maximal schedule consists of two nodes, so we must have As are integers, it follows that Then,, which completes the proof As an example, for, which is the case in many applications of wireless ad hoc mesh networks, we get that Recall that the best lower bound so far is (it is derived in [14] applies to geometric network graphs under the 2-hop interference model) Thus, the lower bound we provide here strictly improves on previous results for networks of up to 26 nodes Table I shows the current best lower bound on the efficiency ratio of GMS, with 2-hop interference, as a function of the number of nodes in the network VI THROUGHPUT-OPTIMALITY OF GMS IN SMALL NETWORKS In this section, we analyze the performance of GMS in small networks Indeed, many wireless ad hoc networks that one may encounter in reality are small More precisely, we want to find the maximum size of networks under which GMS is guaranteed to achieve full capacity We consider connected network graphs because the links in different components of disconnected graphs do not interfere under the -hop interference model Using the performance bounds we have derived in Section V (Theorem 2), one immediately knows that GMS is throughputoptimal in networks up to a certain size under the -hop interference model This is summarized in the following corollary Corollary 2: Under the -hop interference model, GMS is throughput-optimal (achieves the full capacity region) in all network graphs with up to nodes Proof: In Theorem 2, let Then if is a network graph of size, the local-pooling factor of is at least equal to 1, ie, GMS is throughput-optimal in When, the corollary says that GMS is throughput-optimal in networks with five or fewer nodes This result confirms an observation obtained by exhaustive numerical search in [4] Furthermore, the result is tight in the sense that we can find network graphs with six nodes (eg, the 6-cycle ring network) where GMS is not throughput-optimal When, the corollary also says that GMS is throughputoptimal in networks with up to five nodes We will further improve this result in the next theorem Theorem 4: Under the 2-hop interference model, GMS is throughput-optimal in all network graphs with eight nodes or fewer

7 LECONTE et al: IMPROVED BOUNDS ON THROUGHPUT EFFICIENCY OF GREEDY MAXIMAL SCHEDULING IN WIRELESS NETWORKS 715 To prove that GMS is throughput-optimal in graphs with eight nodes or fewer, our approach is to prove that such graphs cannot have any unstable subsets of links The proof of the above result is quite complicated, so we provide an outline first The proof of the theorem follows rather easily from some intermediate propositions The propositions essentially show that unstable subsets of links have certain properties that imply there must be at least some minimum number of nodes in the network graph The first proposition we will use is Proposition 2 in Section III-A, which says that an unstable subset of links contains at least three active links The next proposition we will use is the following Proposition 3: Under the 2-hop interference model, if has an unstable subset of links, then a maximal schedule of of size 2 or more disables at least three nodes of We say that a node is disabled if it is exactly one hop away from a scheduled link Since the minimum distance between two scheduled links is two hops in the 2-hop interference model, a disabled node cannot be scheduled The proof of Theorem 4 is quite straightforward based on the above two results Proof: (Theorem 4): Let be a network graph be an unstable subset of links of Let Because of Proposition 2, we know there exists a maximal schedule of that contains at least threelinks When using that particular maximal schedule, as each link involves two different scheduled nodes, at least six nodes are scheduled Moreover, Proposition 3 tells us that this maximal schedule disables at least three other nodes of The total number of nodes must be larger than or equal to the number of scheduled nodes + the number of disabled nodes, hence It implies that any network graph of eight nodes or less cannot have any unstable subset of links, hence GMS is throughout-optimal in those graphs Now we prove Proposition 3 We will use the following lemma, its proof is included in the Appendix Lemma 4: We consider the 2-hop interference model Let be an unstable subset of links of For any link of, there are at least 2 nodes of exactly 1-hop away from This implies that any maximal schedule of disables at least two nodes thus, using the same line of proof as for Theorem 4, it follows that GMS achieves full capacity in any network of seven nodes or fewer under 2-hop interference Note that, using Lemma 9 in the Appendix, similar bounds can be obtained for the -hop interference model with any However, it is not clear if these bounds are tight Establishing that these bounds are either tight or not is a topic for future work Proof: (Proposition 3): We consider the 2-hop interference model Let be an unstable subset of links of We want to prove that any maximal schedule of of size 2 or more disables at least three nodes of It follows from Lemma 4 that at least two nodes are disabled We will show that all the links of a maximal schedule of of size 2 or more cannot disable the same two nodes By contradiction, suppose, with, such that disables exactly two nodes Denote by these two nodes Let be the set of links of containing, respectively We will show that Indeed, suppose there exists another Fig 1 Two examples of networks of nine nodes with (b) 8-cycle with an open link < 1 (a) 9-cycle link One of its extreme nodes has to be either disabled or scheduled, otherwise could be scheduled, which would contradict the fact that is maximal If one of the extreme nodes of is disabled, then it must be or is in Thus, one of the extreme nodes of must be scheduled, then the other one cannot be scheduled because otherwise is not feasible If that second node is not scheduled, then, by definition, it is disabled, which cannot be, as seen earlier Thus, there is no other link in Now for any, we will show that the maximal schedules of containing have the same size If, then all the s interfere with, either all the links in interfere with then is the only maximal schedule containing, or we can schedule exactly one additional link of, then all maximal schedules containing have size 2 The case is completely similar If, then all the links in interfere with, thus is the only maximal schedule containing We have considered all the possible cases Thus, using Proposition 1, satisfies subset local pooling, which cannot be 1) What About 9 Nodes?: Now we show that the result in Theorem 4 is tight in the sense that there are networks with nine nodes in which GMS is not throughput-optimal We provide two examples of networks with nine nodes in which the efficiency of GMS is bounded away from 1 The first example is a 9-cycle ring network as shown in Fig 1(a) Indeed, as we have proved in Section IV (Theorem 1), the local-pooling factor of this ring network is Note that this result also implies that the bound of Theorem 3 is tight for 10 The second example we analyze here is a size-8 ring network with one additional open link Let us call it label the links as in Fig 1(b) We will construct two

8 716 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL 19, NO 3, JUNE 2011 vectors, such that componentwise, with The maximal schedules used for are: {1, 4, 7} twice, {3, 6}, {5, 8}, {3, 8} Therefore, The maximal schedules used for are: {1, 5}, {1, 6}, {4, 8} twice, {3, 7} twice Therefore, Wehave, thus Note that this second example also shows that the bound provided by Lemmas 9 10 is tight for VII IMPLEMENTATION OF GMS USING LOCAL INFORMATION ONLY GMS requires global information of the link weights (queue lengths) as it sequentially adds a globally heaviest link to the schedule removes the interfering links of the added link from consideration In this section, we discuss some implementation considerations We will show that GMS is equivalent to an algorithm called LGMS, which uses only local neighborhood information Here, local refers to the exchange of information among neighbors in the inference graph Note that two neighboring links in the interference graph can be at most hops away under the -hop interference model LGMS was introduced by Preis [23] as a low-complexity approximation algorithm to solve the maximum weighted matching problem LGMS produces a (maximal) schedule as follows In each step, it adds a locally heaviest link (a link that has the largest weight compared to its nondisabled interfering links) to the schedule; it then disables all the interfering links of the added link The process is repeated until the schedule is maximal Local Greedy Maximal Scheduling (LGMS) Algorithm WHILE Pick a locally heaviest link : ENDWHILE Remark 4: Note that LGMS is different from the Local Greedy Scheduling algorithm presented in [12], which schedules only those links that are locally heaviest at the beginning does not proceed iteratively to produce a maximal schedule The throughput analysis involving local greedy scheduling in [12] is incomplete, hence it is worthwhile to study LGMS as an alternative The key difference between LGMS GMS is that, in each step, LGMS picks a locally heaviest link instead of a globally heaviest one Since LGMS only requires local neighborhood information, it is more amenable to distributed implementation In [11], a distributed implementation of LGMS was developed for the 1-hop model A possible algorithm to implement LGMS is as follows Each time slot is divided into a scheduling period a transmission period At the beginning of the scheduling period, each link will send its weight (ie, queue length) to its interfering links Then, every link will determine whether it will be scheduled (to transmit data in the transmission period) or not according to the messages received from its interfering links We assume the link weights are distinct (each link can add a rom number uniformly selected in [0, 1] to its weight so that, with probability 1, the link weights will be distinct) Algorithm to Implement LGMS (At Link ) WHILE ( ) If, :, send to all links in If received from :, send to all links in If received from : ENDWHILE : We can verify that after the execution of the above algorithm, every link will have either (which means that is included in the schedule) or (which means that is disabled because one of its interfering links is scheduled) During the transmission period, any link with can transmit data First, we note that a complete protocol for implementing LGMS would require a reliable mechanism for the links to send control messages to their interfering links For example, we can color the links so that no two links interfering with each other have the same color, then the links can send their control messages in a round-robin fashion based on their colors The signaling overhead (size of control messages) will depend on the actual values of the link weights which, in theory, can be unbounded While the computational complexity of LGMS is low, the signaling time overhead can increase with the size of the network in the worst case (depending on the network topology the link weights) For a bounded-degree graph under the -hop interference model for some constant, both the signaling time complexities are for a network of links A distributed approximation of GMS (called D-GMS) with constant overhead was developed in [21], which is based on rom backoff mechanism RTS/CTS-type message exchanging However, it is hard to characterize the throughput performance of such an approximation At first glance, one may think that LGMS GMS are different algorithms can return different schedules because LGMS proceeds even more greedily than GMS While

9 LECONTE et al: IMPROVED BOUNDS ON THROUGHPUT EFFICIENCY OF GREEDY MAXIMAL SCHEDULING IN WIRELESS NETWORKS 717 LGMS-type heuristics have been used in earlier papers (eg, [2], [10], [11]) to develop parallel distributed algorithms for solving the maximum weighted independent set problem, the known results on GMS LGMS only show that the loss of optimality under both algorithms is the same For example, for the weighted matching problem, Preis [23] showed that, like GMS, LGMS returns a matching with a weight of at least 1/2 of the weight of a maximum weighted matching However, a stronger result can be obtained We will show that the sets of schedules produced by GMS LGMS are identical for arbitrary interference graphs with general link weights While this fact may appear to be intuitive, surprisingly it does not seem to have been recognized previously A On the Equivalence of LGMS GMS 1) Networks With Distinct Link Weights: Proposition 4: For networks with distinct link weights, LGMS GMS produce the same schedule Let be the schedule produced by GMS LGMS, respectively, when applied to a set of links We first prove two lemmas Lemma 5 says that if we know that a certain link is included in, then an alterative way to generate is to add first then apply GMS on the remaining links that do not interfere with Lemma 5:, Proof: GMS is a sequential algorithm Let be the sequence of links selected by GMS when applied on 2 For any in the schedule produced by GMS, consider a two-step procedure to produce a maximal schedule as follows First, add to the schedule disable its interfering links Then, apply GMS on the remaining links Because do not interfere with link (otherwise they cannot form a schedule), we know that will be in, ie, selecting in the first step will not affect the eligibility of other s in the second step Therefore, since has the largest weight among the links in, GMS will select disable its interfering links Similarly, GMS will select next, so on Thus, this two-step procedure will produce exactly the same schedule as, ie, Lemma 6: Let be the first link selected by LGMS must also be selected by GMS Proof: is a locally heaviest link in, ie, its weight is larger than the weight of the links in Suppose is not selected by GMS, then at least one link in will be selected by GMS, otherwise the schedule produced by GMS is not maximal Let be the first link in selected by GMS At the time instant when GMS decides to select, is also eligible has a larger weight than, a contradiction Therefore, must be selected by GMS From the two lemmas, we can prove Proposition 4 2 It is easy to verify that, for networks with distinct link weights, GMS produces a unique schedule Proof: (Propostion 4): We prove the proposition by induction on the cardinality of (the number of links contained in ) Step 1) The proposition is clearly true for all such that, because in this case the globally heaviest link coincides with the (unique) locally heaviest link that will be the only link selected by both GMS LGMS Step 2) Assume the proposition is true for all with Now consider any with Let be the first link selected by LGMS By Lemma 6, Then, by Lemma 5, Since,wehave by induction assumption Therefore By induction argument, we show that holds for any 2) Networks With General Link Weights: In Section VII-A1, we assumed that the link weights are distinct We can generalize the result to networks where two or more links may have the same weight One approach to hle this is to introduce a deterministic tie-breaking mechanism For example, we can associate each link with a distinct label 3 For the cidate links (globally or locally heaviest links) with the same weight, preference is given to the link with the smallest (or largest) label If both GMS LGMS use the same deterministic tie-breaking mechanism, then we can show that they will again produce the same schedule Formally, let denote a deterministic tie-breaking mechanism (eg, the labeling mechanism described above) Let denote the schedule produced by GMS LGMS on using the tie-breaking mechanism, respectively We can show the following Proposition 5: For networks with general link weights under the tie-breaking mechanism, The proof is similar to the proof of Proposition 4 On the other h, if GMS/LGMS apply a romized mechanism to break the tie, ie, they romly select a link among the cidate links with the same weight, then GMS/LGMS may produce different schedules Nevertheless, we can show the following Theorem 5: For networks with general link weights under a romized tie-breaking mechanism, LGMS GMS produce the same set of possible schedules Let be the set of possible schedules produced by GMS LGMS on Theorem 5 consists of the following two lemmas Lemma 7:, respectively The proof of 3 Each link can generate a rom real number uniformly in [0, 1] as its label With probability 1, these labels are distinct

10 718 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL 19, NO 3, JUNE 2011 Proof:, suppose is the sequence of links selected by GMS when generating In step 1 of LGMS, is a globally heaviest link because it can be scheduled by GMS, thus it is also a locally heaviest link could be scheduled by LGMS Similarly, in step of LGMS, given have been scheduled, is a globally heaviest link among the remaining eligible links because it can be scheduled by GMS, thus it is also a locally heaviest link could be selected by LGMS Then, by induction, LGMS could also select in sequence under a romized tie-breaking mechanism, so Lemma 8: Proof:, suppose is the sequence of links selected by LGMS when generating We label the links in as follows is labeled 1, the interfering links of in are labeled (suppose ); is labeled, the interfering links of in are labeled (suppose ), so on Let denote the deterministic tie-breaking mechanism using those link labels, ie, when a tie occurs, preference is given to the link with the smallest label Let denote the schedule produced by GMS LGMS on under the tie-breaking mechanism, respectively Based on Proposition 5, we know Note that in every step of GMS, the selection made by is one of the possible choices of the romized tie-breaking rule, thus can be selected by GMS under the romized tie-breaking mechanism, hence VIII CONCLUSION GMS is a low-complexity scheduling algorithm that has been observed to achieve very good performance in a variety of wireless network simulations However, theoretical bounds to date on the performance of GMS only show that it can achieve a fraction of the capacity region In this paper, we have derived new bounds on the throughput efficiency of GMS for networks with arbitrary topology Our results are especially useful for networks of small to moderate size, which is the case in many practical wireless ad hoc mesh networks In particular, we have established the throughput-optimality of GMS in networks with up to eight nodes under the 2-hop interference model We have also provided a lower bound on its throughput efficiency in larger networks, which improves upon previously known bounds Furthermore, we have shown that GMS is equivalent to LGMS, which uses only local information Greedy approximations to the max-weight algorithm have also been studied in the context of fair resource allocation for wireless networks [22], it will be an interesting avenue for future work to investigate the performance of GMS in terms of other performance metrics such as delay fairness APPENDIX PROOF OF LEMMA 4 In order to prove Lemma 4, we need to further characterize unstable subsets of links Indeed, in a general network graph, a scheduled link may disable only one node or even no node at all However, not any link can be unstable, ie, belong to an unstable subset of links For example, [14] [28] proved that tree networks cannot be unstable under the -hop interference model, which means that isolated links cannot be unstable, thus, a scheduled unstable link must disable at least one node To be able to prove that a scheduled unstable link disables at least two nodes, we will need to determine additional properties of unstable subsets of links More precisely, we will prove the following lemma Lemma 9: Under the -hop interference model, the maximum depth of an unstable link in an open tree of cliques is We will now define all the notions required In order to describe what an open tree of cliques is, we will first explain the notions of tree of cliques open tree Recall that a clique is a complete graph, which means that it has an edge between any two vertices Note that we can define a tree as a connected graph without cycles We use a similar definition for trees of cliques: Definition 6: A tree of cliques is a connected graph such that, for any cycle in, there is a link between any two nodes of that cycle A tree of cliques can be viewed as a tree, whose nodes can also be replaced by cliques In particular, a tree is a tree of cliques Definition 7: An open tree of a network graph is a subgraph of that is a tree such that only one node in is linked to nodes in We say that is the root of In other words, an open tree is a tree attached by its root to a graph In particular, a tree is an open tree The definition of an open tree of cliques follows naturally Definition 8: An open tree of cliques of a network graph is a subgraph of that is a tree of cliques such that only one node in is linked to nodes in We say that is the root of As mentioned earlier, we are interested in the depth of links in such structures To define depth, we will use the following function: In a graph, the distance between is the minimum number of links of needed to connect to Note that this can apply as well to two nodes, two links, or a node a link Definition 9: Let be an open tree of cliques of root The depth of a link is In other words, if the shortest path in between is of length, then we say that is of depth Note that if is the link connecting nodes, then We will first state prove a weaker version of Lemma 9 because this proof should convey the main intuition Then, proving Lemma 9 is only technical detail Lemma 10: Under the -hop interference model, the maximum depth of an unstable link in an open tree is Proof: Let be an unstable subset of links of There exists a feasible arrival rate in the interior of such that the queues of all links in increase to infinity Let be an open tree of, let Let be a link in of maximum depth with respect to Assume, toward a contradiction, that, so in fact

11 LECONTE et al: IMPROVED BOUNDS ON THROUGHPUT EFFICIENCY OF GREEDY MAXIMAL SCHEDULING IN WIRELESS NETWORKS 719 Fig 2 Tree of cliques the three possible cases Consider the set of all interfering links of in call it We want to prove that the queues of all the links in cannot increase to infinity By definition of, it is clear that, at every time instant, at least one link of is scheduled Indeed, all the links that are not in have no packet to transmit Since all the links of that interfere with are in, if none of them is scheduled, then should be scheduled Furthermore, we will check that any two links of interfere with each other If, then we have, similarly, Thus, interfere with each other If, then, again interfere with each other Finally, if, the proof in [14] applies, we get that interfere with each other Therefore, any two links of interfere with each other, thus it is not possible to schedule more than one link in at every time instant Since is in the interior of the capacity region, As one link of is scheduled at every time instant, it is not possible that all the queues of the links of increase to infinity, so cannot be true We will now prove Lemma 9 Proof: (Lemma 9): The proof is identical to that of Lemma 10 except when we want to prove that two links interfere with each other To that end, define a nearest common ancestor of as a node with smallest depth in that belongs to a shortest path in between As is a tree of cliques, may have many nearest common ancestors, but they will all belong to the same clique In that case, we say that is the nearest common ancestor of in Let be the nearest common ancestor of in be that of We will consider all possible cases for the relative locations of Those cases are represented in Fig 2 The case corresponds to case 1 If, then either is an ancestor of,or is an ancestor of These last two cases are symmetrical, thus we will consider only one of them For any, denote by the node of that is closest to, ie, is such that Note that this definition is valid whether is a node or a link We consider case 3 first because it is simpler case 1 does not bring any new idea because is of maximum depth in Therefore, thus, interfere with each other In case 1, because We can switch the roles of get another expression Thus, we have the following two inequalities: (9) (10) We now check that in any case we can use one of the previous equations to prove that interfere with each other If, then so we can conclude using (10) If, then so, again, we can conclude using (10) If, then we use (9) Note that this bound is tight for according to [13] We also provide an example to show that it is also tight for We are now ready to prove Lemma 4 Proof: (Lemma 4): We consider the 2-hop interference model Let be an unstable subset of links of be a link of We want to show that there are at least two nodes of exactly one hop away from We will consider all possible cases If is an open link, then Lemma 10 implies that one of

12 720 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL 19, NO 3, JUNE 2011 the extreme nodes of is in a cycle of, so its two neighboring nodes in that cycle are disabled If is in a cycle of size 4 or more, each extreme node of disables a neighboring node in If is in a cycle of size 3, then it first disables the third node of However, since a cycle of size 3 is a clique, Lemma 9 implies that at least one of the extreme nodes of will disable a node in the rest of Finally, if is not an open link not in a cycle, then there are two subgraphs of, both with more than one node, that are connected only through will disable one node in both those subgraphs REFERENCES [1] N Alon, L Babai, A Itai, A fast simple romized parallel algorithm for the maximal independent set problem, J Algor, vol 7, no 4, pp , 1986 [2] S Basagni, Finding a maximal weighted independent set in wireless networks, Telecommun Syst, vol 18, no 1 3, pp , 2001 [3] B Bir, M Chudnovsky, B Ries, P Seymour, G Zussman, Y Zwols, Analyzing 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under slowly convergent traffic using sequential maximal scheduling, IEEE Trans Autom Control, vol 53, no 10, pp , Nov 2008 [25] S Sarkar S Ray, Arbitrary throughput versus complexity tradeoffs in wireless networks using graph partitioning, IEEE Trans Autom Control, vol 53, no 10, pp , Nov 2008 [26] L Tassiulas A Ephremides, Stability properties of constrained queueing systems scheduling policies for maximal throughput in multihop radio networks, IEEE Trans Autom Control, vol 37, no 12, pp , Dec 1992 [27] X Wu R Srikant, Scheduling efficiency of distributed greedy scheduling algorithms in wireless networks, in Proc IEEE IN- FOCOM, Apr 2006, pp 1 12 [28] G Zussman, A Brzezinski, E Modiano, Multihop local pooling for distributed throughput maximization in wireless networks, in Proc IEEE INFOCOM, Apr 2008 Mathieu Leconte graduated from Supelec, Paris, France, in 2008 He received the Master s degree in electrical computer engineering from the University of Illinois at Urbana Champaign in 2009 is currently pursing the PhD degree in electrical computer engineering at INRIA Technicolor, Paris, France His research interests include wireless networks, rom graphs, information theory, statistical physics Jian Ni (S 02 M 09) received the BEng degree in automation from Tsinghua University, Beijing, China, in 2001, the MPhil degree in electrical electronic engineering from the Hong Kong University of Science Technology (HKUST) in 2003, the PhD degree in electrical engineering from Yale University, New Haven, CT, in 2008 He is currently a Post-Doctoral Researcher with the Coordinated Science Laboratory, University of Illinois at Urbana Champaign His research interests include performance analysis, algorithm design, communication networks, statistical inference, machine learning R Srikant (S 90 M 91 SM 01 F 06) received the BTech degree from the Indian Institute of Technology, Madras, India, in 1985 the MS PhD degrees from the University of Illinois at Urbana Champaign in , respectively, all in electrical engineering He was a Member of Technical Staff at AT&T Bell Laboratories, Murray Hill, NJ, from 1991 to 1995 He is currently with the University of Illinois at Urbana Champaign, where he is the Fredric G Elizabeth H Nearing Professor with the Department of Electrical Computer Engineering a Research Professor with the Coordinated Science Laboratory His research interests include communication networks, stochastic processes, queueing theory, information theory, game theory He was an Associate Editor of Automatica, the IEEE TRANSACTIONS ON AUTOMATIC CONTROL, the IEEE/ACM TRANSACTIONS ON NETWORKING He has also served on the Editorial Boards of special issues of the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS IEEE TRANSACTIONS ON INFORMATION THEORY He was the Chair of the 2002 IEEE Computer Communications Workshop in Santa Fe, NM, a Program Co-Chair of IEEE INFOCOM 2007