Control of Wirele Network with Flow Level Dynamic under Contant Time Scheduling Long Le and Ravi R. Mazumdar Department of Electrical and Computer Engineering Univerity of Waterloo,Waterloo, ON, Canada N2L 3G Email: longble@engmail.uwaterloo.ca, mazum@ece.uwaterloo.ca. Abtract We conider the network control problem for wirele network with flow level dynamic under the general k- hop interference model. In particular, we invetigate the control problem in low load and high load regime. In the low load regime, we how that the network can be tabilized by a regulated maximal cheduling policy conidering flow level dynamic if the offered load atifie a contraining bound condition. Becaue maximal matching i a general cheduling rule whoe implementation i not pecified, we propoe a contant-time and ditributed cheduling algorithm for a general k-hop interference model which can approximate the maximal cheduling policy within an arbitrarily mall error. Under the tability condition, we how how to calculate tranmiion rate for different uer clae uch that the long-term time average) network utility i maximized. Thi long-term network utility would capture the real network performance due to the fact that under flow level dynamic, the number of uer randomly change o intantaneou network utility maximization doe not reult in ueful network performance. Our reult imply that congetion control i unneceary when the offered load i low and optimal uer rate can be determined to maximize uer long-term atifaction. In the high load regime where the network can be untable under the regulated maximal cheduling policy, we propoe the cro-layer congetion control and cheduling algorithm which can tabilize the network under arbitrary network load. Through extenive numerical analyi for ome typical network, we how that the propoed cheduling algorithm ha much lower overhead than other exiting queuelength-baed contant-time cheduling cheme in the literature, and it achieve performance much better than the guaranteed bound. In addition, uing congetion control in the low load condition reult in much lower average utility compared to that due to the optimal tranmiion rate derived in the paper. Index Term Flow level dynamic, capacity region, contanttime cheduling, network tability, maximal matching, k-hop interference model I. INTRODUCTION Reource allocation in communication network ha been an active reearch topic for the lat everal year. While optimal rate control in wired network can be achieved by a ditributed algorithm []-[4], olving thi problem in wirele network i much more challenging. In fact, the bottleneck of the reource allocation problem in wirele network lie in the cheduling ub-problem [5]-[6]. The difficulty of the cheduling ubproblem come from the interference coupling of imultaneou tranmiion from different wirele link in the network. In general, interference coupling in wirele network depend on the communication technologie employed at the phyical layer. For example, the node excluive interference model can be aumed for Bluetooth network or FH-CDMA network [7], [8]. Thi interference model i alo referred to a one-hop interference model. Alo, for the 802. WLAN with four-way handhake i.e., with RTS/CTS), the two-hop interference model i implicitly aumed in the MAC protocol. Moreover, it ha been hown in [9] that in certain network etting and QoS requirement, neither one-hop nor two-hop interference model i optimal to achieve the maximum number of imultaneou tranmiion in the network. In [9] the author propoed a general interference model called k-hop interference model which i determined by a ingle parameter k. For thi interference model, wirele link k + or more hop away from one another can be cheduled to tranmit data at the ame time. Developing a joint reource allocation and cheduling algorithm for thi k-hop interference model i, therefore, much more deirable than working with a pecial cae of thi general interference model. Thi i indeed what we will purue in thi paper. Regarding the cheduling problem in wirele network, there are everal optimal and uboptimal cheme propoed in the literature. In a eminal paper [0], Taiula and Ephremide propoed an optimal back-preure policy which achieve the maximum network throughput. Thi cheduling policy i, however, centralized and computationally expenive. In [], a randomized linear-complexity cheduling algorithm wa propoed where a tranmiion chedule in time lot t wa contructed by chooing the chedule with larger total weight between the chedule in time lot t and a newlygenerated one in time lot t. Thi idea wa ued to develop ditributed throughput-optimal cheduling policie in [2]- [4] for one-hop and two-hop interference model. Note that thee cheduling algorithm achieve full utilization of wirele network with repect to what remain in the data tranmiion phae only. Specifically, a large amount of bandwidth ha been wated to exchange control information in the chedule contruction phae which would otherwie be ued for data tranmiion. In general, the amount of cheduling overhead grow with the network ize for thee throughput-optimal cheduling policie. Due to implementation contraint, the time lot interval i uually limited to a few milliecond a in mot current wirele ytem. Therefore, developing a cheduling algorithm with low and contant-time overhead would be very deirable. In fact, ome queue-length-baed contant-time cheduling algorithm were propoed for one and two hop interference
2 model [5]-[7] recently in the literature. Thee cheduling algorithm only achieve a guaranteed faction of the capacity region but they have contant time overhead. For practical implementation, collecting queue length information may be difficult, and it will create further overhead. A more general maximal cheduling policy wa conidered in [9], [20] where everal throughput performance bound were invetigated. However, implementation of thi general cheduling policy and invetigation of it actual performance in typical wirele network were not conducted in thee paper. In practice, it i deired that each wirele node only communicate with it neighbor e.g., thoe whoe tranmiion interfere with that of the underlying node) to contruct a tranmiion chedule in each time lot. Alo, cheduling algorithm hould work for a general cla of interference model e.g., k-hop interference model [9]). Another apect which wa ignored by mot exiting work in the literature i that no conflict-free chedule i available to exchange control information at the beginning of each time lot. Therefore, control information can only be exchanged by uing contentionbaed tranmiion which render information exchange more than one hop away a time-conuming operation. Alo, it i important to quantify amount of time/overhead ued to contruct the chedule and to develop explicit procedure to exchange control information in each time lot. In thi paper, we how the performance guarantee of the regulated maximal cheduling policy in wirele network conidering flow level dynamic. In fact, regulated maximal cheduling i a combination of the maximal cheduling policy [9], [20] and a traffic regulator implemented at each wirele link. Becaue regulated maximal cheduling i a general rule, we propoe a contant-time and ditributed algorithm to implement it in each time lot. We how that the propoed cheduling algorithm can approximate the regulated maximal cheduling policy within an arbitrarily mall error. The propoed cheduling algorithm work for a general k- hop interference model and doe not require queue length information. Moreover, we explicitly decribe how wirele link coordinate their contention to contruct the chedule in each time lot. The afore-mentioned performance guarantee reult implie that we do not need to perform congetion control even with flow level dynamic if the traffic load lie within a region which can be tabilized by the underlying cheduling algorithm. Thi i an intereting finding given the fact that there are exiting work which employ congetion control algorithm to tabilize the network [5], [2] under low network load. Given the fact that the cheduling algorithm i a randomized one and the number of uer dynamically change, intantaneou network utility maximization would not reult in good network performance. Becaue of thi, we are intereted in finding tranmiion rate for each uer cla which achieve maximum long-term time average) utility. In fact, we how that there exit optimal tranmiion rate for all uer clae to achieve uch maximum long-term utility. When the network load i high, we propoe a cro-layer congetion control algorithm which can tabilize the network for arbitrary network load. The reult preented in thi paper have everal important implication for ytem implementation. Firt, we do not need to perform congetion control in low network load even with flow level dynamic. Hence, communication overhead a well a implementation iue uch a aynchronou [8] and noiy [24] feedback due to congetion control operation can be completely avoided. Second, the problem of network utility maximization can be decoupled from that of tabilizing the network. Specifically, the network can be tabilized by implementing traffic regulator at wirele link together with a uitable cheduling mechanim. In fact, we how via numerical example that uing congetion control algorithm to tabilize the network in low load actually degrade the long-term network utility coniderably. The remaining of thi paper i organized a follow. We decribe the ytem model and performance bound in ection II. Performance guarantee of the regulated maximal cheduling policy i preented in ection III. In ection IV, we preent the ditributed cheduling algorithm to approximate the maximal cheduling policy. We derive the optimal tranmiion rate to achieve long-term utility maximization in ection V. The cro-layer congetion control and cheduling algorithm in the high load regime i decribed in ection VI. Some numerical reult are preented in ection VII and ection VIII tate the concluion. For notational convenience, we will put element of different meaure into the correponding vector. For example, the vector of tranmiion rate will be denoted by x where x i it -th element which i the tranmiion rate of cla- uer. II. SYSTEM MODELS AND PERFORMANCE BOUND We model a wirele network a a directed graph G = V, E) where V i the et of wirele node and E i the et of wirele link. A wirele link from node i to node j exit if node j can correctly receive information tranmitted by node i. In practice, exitence of uch a link depend on tranmiion power, path lo, fading, interference, deired bit error rate and other factor. We aume that there are S clae of uer each of which aociate with a fixed routing path from a ource node to a detination node. The uer route are tored in an incidence matrix [H k ] where H k = if link k i on the route of cla- uer and H k = 0 otherwie. Uer of cla arrive to the network with rate λ and each bring a file for tranfer whoe ize i exponentially ditributed with mean /µ. The offered load by cla- uer i, therefore, ρ = λ /µ. The vector of offered load will be denoted a ρ = [ρ, ρ 2,, ρ S ]. We aume that uer of each cla tranmit at the ame rate. Interference contraint are denoted by a contention matrix [C ij ] i,j E. Specifically, link i i aid to interfere with link j if C ij = and C ij = 0 otherwie. Thi general notation of the interference relationhip i ued to decribe the capacity region and to derive the performance guarantee of the regulated maximal matching policy in thi ection and ection III only. The k-hop interference model i a pecial cla repreenting thi general interference relationhip. The k-hop interference model i aumed in all remaining ection of thi paper. Time i divided into lot of unit duration. Link l can tranmit at
3 rate if it interfering link are not cheduled to tranmit in a ame time lot. In the following, we provide ome important definition which will be ued in the equel. Definition I: Interference et I l of link l i the et of link which interfere with link l, i.e., I l = {k E : C kl = }. ) Definition II: Interference degree d I l) of link l i the maximum number of link in it interference et which do not interfere with each other. Definition III: Interference degree d I G) of graph G i the maximum interference degree of it link, i.e., d I G) = max l E d I l). Capacity region i defined to be the et of traffic load uch that the network can be tabilized by ome cheduling policy. In [0], capacity region for wirele network i well characterized. In particular, capacity region i given by the et { [ S ] } H l Ω = ρ : ρ CoR) 2) R l l E where CoR) i the convex hull of all link chedule R that atify Wirele the linkregulator buffer RegulatorTranmiion queue contraint impoed by the underlying interference Wirele link model. A cheduling policy i aid be throughput optimal if it tabilize the network for all offered load within the capacity region Ω. Fig.. Regulator implementation at each wirele link. We aume that a new chedule i contructed in the firt phae of each time lot which i ued to tranmit data in the econd phae of each time lot. Traffic flow in the network may travere one hop or multiple hop. For the multiple-hop cae, we aume that traffic of each uer cla i regulated before entering a tranmiion queue for being tranfered over each wirele link. The employment of regulator wa previouly propoed by Hume to tabilize manufacturing ytem [22] which have been hown to be untable in ome cae due to cycle of material flow [23]. Regulator were recently ued in wirele network [8], [9]. A λ-regulator aociated with link l generate packet to tranmiion queue of link l with a maximum rate of λ. A regulator can be implemented a follow. In each time lot, a λ-regulator aociated with link l check the correponding regulator queue. If the queue length i greater than link capacity then it tranfer unit of data to the tranmiion queue with probability λ/. Otherwie, it tranfer nothing. The regulator implementation i illutrated in Fig.. In thi paper, we aume a maximal cheduling policy which wa invetigated in [9], [20]. The maximal cheduling rule can be decribed a follow. For any link l E with tranmiion queue length larger than the link capacity in any time lot, it i required that at leat one link in it interference et I l be cheduled. Specifically, if Q l / where Q l i the queue length of tranmiion queue for link l), we require π k 3) where π k = if link k i cheduled and π k = 0 otherwie. Due to the combination of maximal cheduling and regulator implementation, the reulting cheduling will be called regulated maximal cheduling in the equel. Note that maximal cheduling i a general cheduling rule without pecific implementation. We will preent a contant time and ditributed cheduling algorithm which approximate the maximal cheduling in ection IV. The following performance bound of the maximal cheduling policy wa proved in [20], [2], it i retated here for completene. Lemma : For all traffic load ρ within the capacity region defined in 2), we have S H k ρ d I l), l E. 4) Thi upper bound will be ued to quantify the throughput guarantee of the regulated maximal cheduling policy in the next ection. III. PERFORMANCE OF THE REGULATED MAXIMAL MATCHING SCHEME In thi ection, we how that the network i table under the regulated maximal cheduling when the offered load atifie a pecific condition. We aume that a ρ + kɛ)-regulator i employed at k-th hop on the route of cla- uer. It i worth to mention that the following tability reult i imilar in pirit to that in [9], although there i an important difference here. In fact, we capture flow dynamic in thi paper while the author in [9] only conidered dynamic at the packet level. In [5] and [2], the author captured flow dynamic but their tability reult were achieved by a crolayer congetion control algorithm. In thi paper, network tability i achieved by a imple regulator implementation o communication overhead involved in the congetion control operation can be avoided. The tability reult i tated in the following propoition. Propoition : If the traffic load atifie S H k ρ <, l E 5) then the network i table under the regulated maximal cheduling policy. Thi condition will be called a contraining bound in the equel. Proof: The proof i in Appendix A. Thi reult wa alo tated in [2] where the network wa tabilized by a cro-layer congetion control algorithm. Note that thi contraining bound i tight in the ene that the network can become untable with any arbitrarily mall increae of the bound i.e., the right hand ide of 5) become
4 + κ for any κ > 0). Thi fact wa proved in everal paper for example, ee [9], [20]). In the following, we tate the performance guarantee for the regulated maximal cheduling policy. Lemma 2: The regulated maximal cheduling policy achieve /d I G) capacity region. Proof: The proof follow directly by comparing the upper and contraining bound on capacity region in 4) and 5), repectively, and uing the definition of d I G). IV. DISTRIBUTED SCHEDULING ALGORITHM A mentioned before, maximal cheduling i a general rule whoe implementation i not pecified. In thi ection, we preent a ditributed cheduling algorithm which approximate the maximal cheduling policy in each time lot within an arbitrarily mall error. In fact, the propoed algorithm will include the BP-SIM cheduling algorithm [7] propoed for the node excluive i.e., one-hop) interference model a a pecial cae. Our propoed algorithm work with the general k- hop interference model. Alo, in contrat to the exiting queuelength-baed cheduling algorithm [6], [7], in our algorithm each node with incident backlogged link doe not require queue length information of other link in it neighborhood to contruct the tranmiion chedule. In addition, the propoed algorithm i fully ditributed and it ha contant time overhead which doe not grow with the network ize. Our propoed algorithm i, therefore, much more flexible and general than exiting one in the literature. For eae of reference, we will refer to our cheduling algorithm a random approximate maximal matching RAMM) cheduling in the equel. Link are added to the chedule in each round through a matching requet and matching acknowledgment meage exchange a follow. At the beginning of each round, each active node the notion of active/inactive node will be clarified hortly) decide to be left or right with probability /2. Node becoming right wait to receive matching requet from their neighboring node. Backlogged link are added to the chedule in each round a follow. Each left node with at leat one backlogged outgoing link i.e., a link from thi node to one of it neighbor) will chooe a random backoff in [, B]. When the backoff expire, a left node will chooe one of it backlogged neighbor randomly to end a matching requet if it ha not heard any matching requet tranmitted by other node o far in the round. A right node which receive a matching requet will reply with a matching acknowledgment meage C2 C and the correponding link i added to the chedule. We aume that if two or more matching requet are tranmitted in one mini-lot, colliion occur and no matching acknowledgment C3 A C6 B C5 meage C4 i tranmitted. A. Algorithm Decription The RAMM algorithm i run in the firt phae of each time lot. Specifically, we divide each time lot into two phae: a cheduling phae and a data tranmiion phae. The tranmiion chedule i contructed in the cheduling phae, and it i ued to tranmit data in the data tranmiion phae. The cheduling phae i further divided into K round each of which contain B mini-lot. In each round, new link are added to the current tranmiion chedule. The tranmiion chedule obtained at the end of the K-th round will be ued to tranmit data in the data tranmiion phae. In addition, only wirele link whoe queue length are larger than the link capacity Scheduling phaetranmiion Time lot are cheduled phae Time lot by the algorithm in each time lot. The time lot tructure of the RAMM algorithm i illutrated in Fig. 2.. Scheduling round Fig. 2. Timing diagram of the RAMM cheduling algorithm. Fig. 3. Illutration of the contention reolution of RAMM cheduling under two-hop interference model. In each round, we require that if a link i added to the tranmiion chedule, all wirele link in it interference et be not added to the tranmiion chedule in ubequent round. Thi requirement guarantee that we will obtain a conflictfree tranmiion chedule at the end of the cheduling phae. It i oberved that thi requirement can be eaily achieved by one-hop and two-hop interference model. Specifically, for the one-hop interference model after a link i added to the chedule, both it tranmitting and receiving node will not tranmit and reply any matching requet. For the two-hop interference model after a link i added to the chedule, all their one-hop neighboring node i.e., one hop away) of both tranmitting and receiving node will be aware of thi through hearing the matching requet or matching acknowledgment) and will not tranmit or reply any matching requet until the end of the cheduling phae. For a general k-hop interference model with k 3, we aume that a large enough power level i ued to tranmit matching requet/acknowledgment meage in the cheduling phae o that if a link i added to the chedule in one round, all node within k hop from both the tranmitting and receiving node of the link are aware of thi o they will not
5 tranmit or reply any matching requet in ubequent round. Node within k hop from the tranmitting and receiving node of any link in the chedule are called inactive node. All other node are active one. Note that any inactive node will remain inactive until the end of the cheduling phae. In general, the number of node participating in the chedule contruction proce reduce rapidly over conecutive round. Becaue new link are kept added to the exiting chedule in each round, the tranmiion chedule in the lat round would approximate well the maximal chedule if B and K are large enough. We will how the performance guarantee of the propoed cheduling algorithm in the next ubection. The contention reolution of the RAMM cheduling algorithm under two-hop interference model i illutrated in Fig. 3. In thi figure, if link AB i added to the chedule, all node C i i =, 2,, 6) will be aware of thi through hearing either the matching requet or matching acknowledgment meage from A or B. Hence, after link AB i added to the chedule, thee node C i, i =, 2,, 6) will become inactive. Conequently, all the link in thi figure except AB which are conflict with link AB will never be added to the chedule in the ubequent round. B. Analyi Now, let degree d i of node i be the number of node having link directly connecting to node i i.e., one-hop neighbor of node i). Let d be the maximum of d i for all node in the network i.e., d = max i V d i ). In addition, a matching requet tranmitted by one node may collide with thoe tranmitted by other node. Let I i be the number of node whoe tranmitted matching requet may collide with that of node i if node i and one or more of thee node tranmit imultaneouly under the correponding power level ued in the cheduling phae. Let I be the maximum of I i i.e., I = max i V I i ). Alo, let I0 be the maximum number of node which are at mot k hop away from either A or B including A and B for any link AB in the network. We have the following reult. Propoition 2: For any µ 0, ), we can chooe the number of cheduling round K which depend only on B, d, I, I 0, and µ but independent of network ize uch that for any backlogged link l, the probability that at leat one backlogged link in it interference et I l i cheduled after K round i larger than or equal to µ. Proof: The proof i in Appendix B. Uing RAMM cheduling algorithm together with regulator implementation a decribed in ection II, we have the following tability reult. Propoition 3: If the traffic load atifie S H k ρ < µ, l E 6) and under the condition tated in propoition 2, the network will be table when RAMM algorithm i ued together with the regulator implementation a decribed in ection II. Proof: The proof follow the ame line with that of Propoition. However, the right hand ide of the contraining bound become µ intead of one due to the performance bound achieved by RAMM cheduling cheme. The reult in thi propoition mean that we can achieve the performance bound of the regulated maximal matching tated in propoition within an arbitrarily mall error by uing the RAMM cheduling algorithm with contant-time overhead. V. LONG-TERM UTILITY MAXIMIZATION UNDER LOW LOAD CONDITION Propoition 3 alo mean that when regulator and RAMM cheduling algorithm are implemented and traffic load atifie the condition tated in 6), the network i table a long a uer rate are bounded away from zero. A a conequence of thi reult, it i clear that we do not need any congetion control algorithm a long a the traffic load in the network i low. Hence, communication overhead due to meage exchange of the congetion control algorithm can be avoided if regulator are implemented in the network. In addition, the number of uer for each cla change dynamically due to the flow level dynamic, o intantaneou network utility maximization may not lead to good network performance. Therefore, under thi tability condition, it i natural to ak: how to chooe uer rate uch that optimal long-term time average) network utility can be achieved? Specifically, our objective i to maximize the long-term network utility which can be explicitly tated a max xt) lim τ τ τ t=0 [ S ] n t t)u x t)) dt 7) where n t t) and x t) are the number of cla- uer tranmitting in time lot t and their tranmiion rate, repectively; U x ) i the utility function, which can, for example, reflect the level of atifaction for cla- uer. We aume that uer arriving during time lot t can only tranmit from time lot t + onward. Suppoe that the queueing proce at each ource node i ergodic thi fact wa jutified in [25]). Let f n t, x) denote the joint probability denity function of n t and x in equilibrium. Becaue element [ of n t are pairwie S ] independent, we have f n t, x) = fnt x) f x). Thu, we can rewrite 7) a S max n t U x )fn t x) f x)d x. 8) x X n t =0 Let u define g x) = = = = S n t U x )fn t x) n t =0 S U x ) n t fn t x) n t =0 S U x )E [ N x t ] S ρ x U x )
6 where we have ued Little law in deriving E [N x] t in the above equation. Specifically, the expected waiting time for a cla- uer i /µ x ), uing Little law we have E [N x] t = λ /µ x ) = ρ /x. Thu, we can rewrite 8) a max x g x)f x)d x 9) X Now, uppoe we wih to find optimal uer rate x [0, M ]. Let x be the maximum of g x ) = ρ /x U x ) in [0, M ] and the correponding optimum rate vector i x. Then, it i eay to ee that chooing f x) = δ x x ) will maximize 9) where δ.) i the delta function. Thu, the longterm utility maximization can be achieved by allowing uer of cla to tranmit at the optimal rate x. In ummary, when the traffic load atifie 6), the network i table under the propoed cheduling policy and no congetion control i needed. In addition, we can decouple the long-term utility maximization from tability under thi tability condition. Example: When the utility function i U x ) = lnx ) which correpond to proportional fair rate allocation among uer, we have g x ) = ρ /x lnx ). The global maximum of g x ) i x = e. Thu, if M > e, the optimal tranmiion rate to achieve maximum long-term utility i x = e. We will compare long-term utility under thi olution and for the cae where cro-layer congetion control algorithm i ued [2]. VI. CONGESTION CONTROL UNDER HEAVY LOAD In thi ection, we conider the heavy load regime where the bound tated in 5) i violated. Thi may be the cae when there are many long-lived flow in the network. A. Cro-Layer Congetion Control Algorithm In thi ubection, we preent a cro-layer congetion control algorithm which can tabilize the network under any offered load. We aume that the flow dynamic are low compared to the time cale of a congetion control algorithm. Hence, we aume that there are fixed number of uer of each cla in the network which will be denoted by N. Our propoed cro-layer congetion control algorithm work a follow. where U.) i the invere of derivative of utility function U.) and [x] + = max[x, 0]. The implicit cot are updated by q l t + ) = [ q l t) + α l S H k N x + κ )] + where κ > 0 i a mall number and α l i the tep-ize. Tranmiion cheduling: The regulated maximal cheduling policy i employed in each time lot. We aume that the utility function U x ) i increaing, trictly concave, twice differentiable. The propoed congetion control algorithm implicitly olve the following optimization problem. S maximize N U x ) ubject to S H k N x κ, l E 0) Note that contraint of thi optimization problem come from the contraining bound tated in 5). Here, the tranmiion rate of cla- uer i ued intead of it offered load. We introduce a mall κ > 0 on the right hand ide of thee contraint to enure that feaible olution of thi optimization problem trictly atify the contraining bound. By introducing Lagrange multiplier q l for the contraint in 0), we have the following Lagrangian L x, q) = S N U x ) l E q l [ S H k N x + κ Uing the tandard dual decompoition technique, we can calculate the uer rate from x t) = argmax N U x ) N x q l t) 0 x M l E :H k= which give the uer rate update rule preented in the algorithm. Alo, implicit cot update tep of the propoed algorithm come from the dual update of the underlying optimization problem. Now, let x be the optimal olution of the preented algorithm, we implement a x + kɛ)-regulator in the k-th hop of cla uer. We have the following reult on the tability of the preented algorithm. Propoition 4: If the tepize i choen to be mall enough, the preented algorithm can tabilize the network under any offered load. Proof: The proof i in Appendix C. It i expected that a congetion control algorithm hould tabilize the network under any offered load. In thi repect, the propoed congetion control algorithm doe a good job compared to thoe in [5], [2]. Note that in practice where the RAMM cheduling algorithm i implemented to approximate Uer rate i determined by the maximal matching policy RAMM intead of maximal x t) = min U + matching i ued in tep 3 of thi algorithm), we can modify q l t), M the propoed congetion control algorithm by replacing on the right hand ide of the contraint in 0) by µ and modify l E :H k= the algorithm accordingly. B. Some Implementation Iue In practice, there i no imple way to know whether or not the contraining bound in 5) i atified. Therefore, it i unknown when the congetion control algorithm propoed in ection VI.A hould be activated. Moreover, it would be wie to avoid performing congetion control if it i unneceary. Alo, uer in the network hould tranmit at their optimal rate a calculated in ection V to maximize the long-term ]
utility if the network i known to be table by the regulated maximal cheduling policy. To achieve uch goal, uer would attempt to tranmit at their optimal rate auming that the offered load i low and the network i table. If thi i not the cae, the network hould be equipped with an appropriate mechanim to detect and inform all uer about the ongoing congetion and requet them to activate the congetion control algorithm. Congetion can be detected at network node through oberving frequent buffer overflow and/or large end-to-end delay. Upon detecting network congetion, network node hould end appropriate feedback to inform all uer in the network and requet them to run the congetion control algorithm. Thee operation enure that the bet performance can be achieved and the mot appropriate operation are performed at all time. Fig. 5. 3 7 4 28 5 39 6 0 4 7 5 8 2 6 9 25 3 20 26 32 2 27 33 22 28 34 23 29 35 24 30 36 Grid network of 36 node with 30 one-hop flow or 6 multihop flow. 7 VII. NUMERICAL RESULTS In thi ection, we how ome illutrative numerical reult for the propoed cheduling algorithm and the long-term utility maximization. We conider grid network with one-hop and multihop flow. We aume that tranmiion rate on each wirele link equal = 0 unit/time lot, average length of each file brought by any uer cla i /µ = 0 unit. Uer of each cla arrive according to Poion proce with arrival rate λ. We vary arrival rate to adjut the traffic load ρ = λ /µ 3 2 3 4 5 6. Here, a unit of data i a block of information bit of uitable 9 7 4 ize. 20 8 5 We aume 2 9 6 0 2 all 22 7 flow 23 8 have 24 the ame load ρ in all the reult. 25 3 26 32 27 33 28 34 29 35 30 36 Fig. 4. Grid network of 36 node with 60 one-hop flow or 2 multihop flow. A. Performance of Scheduling Algorithm In Fig. 7, we how the minimum probability of achieving a maximal chedule due to the RAMM algorithm veru the number of round K under different maximum backoff value B for the grid and random topologie hown in Fig. 4, Fig. 6 under the one-hop interference model. For both topologie, we aume there are two flow in two different direction on each link i.e., for any link AB, there are one flow from A to B and one flow from B to A). We aume that all the flow are alway backlogged which i the wort cae cenario. The probability of achieving a maximal chedule for a backlogged 00 90 80 70 60 50 40 30 20 0 0 0 0 20 30 40 50 60 70 80 90 Fig. 6. Random network of 36 node with 98 bidirectional link 99 undirectional link), d = 9, I = 24. link i the probability that it i cheduled or at leat one link in it interference et i cheduled. The minimum matching probability i the mallet probability for all link in the network. We have obtained thi probability by averaging over 0 4 time lot. It i oberved that with B = 3, to achieve a minimum matching probability of 0.9, we need K = 4 and 5 round for the grid and random topologie, repectively. Alo, with B = 3 and K = 9, i.e., the total number of minilot required i M = B K = 27), the minimum matching probability i almot. Similarly, we how the minimum probability of achieving a maximal chedule due to the RAMM algorithm veru the number of round K under different maximum backoff value B for the grid and random topologie under two-hop interference model in Fig. 8. Thi figure how that with B = 4, we only need K = for the grid topology and K = 3 for the random topology to achieve minimum matching probability very cloe to. It i alo hown that even the ize of the node interference et i very large I = 22 for the grid topology and I = 24 for the grid topology), the improvement of minimum matching probability i very marginal when the maximum backoff B i larger than 8. In fact, it i more beneficial in term of overhead i.e., M = B K) to ue mall value of B e.g., B = 4 or 5 i a good choice).
8 Minimum matching probability 0.9 0.8 0.7 0.6 0.5 0.4 Grid, B = 2 Random, B = 2 Grid, B = 3 Random, B = 3 Grid, B = 4 Random, B = 4 2 4 6 8 0 2 Number of round K Total average queue length 9 x 05 8 7 6 5 4 3 2 RAMM, M = 20 RAMM, M = 28 RAMM, M = 36 RAMM, M = 44 GMM P&C, M = 44 0 0 0.5.5 2 2.5 3 3.5 Load ρ Fig. 7. Minimum probability of achieving the maximal cheduling veru number of round for a grid topology in Fig. 4 with 20 one-hop flow or random topology in Fig. 6 with 98 one-hop flow under one-hop interference model. Fig. 9. Performance of RAMM cheduling algorithm under two-hop interference model for maximum backoff value B = 4, grid network with 36 node and 30 one-hop flow in Fig. 5). Minimum matching probability 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0. Grid, B = 4 Random, B = 4 Grid, B = 5 Random, B = 5 Grid, B = 8 Random, B = 8 Grid, B = 0 Random, B = 0 2 4 6 8 0 2 4 6 Number of round K Total average queue length 9 x 05 8 7 6 5 4 3 2 Policy W, M = 44 Policy W, M = 28 Policy W, M = 52 Policy W, M = 024 GMM P&C, M = 44 0 0 0.5.5 2 2.5 3 3.5 Load ρ Fig. 8. Minimum probability of achieving the maximal cheduling veru number of round for a grid topology in Fig. 4 with 20 one-hop flow or random topology in Fig. 6 with 98 one-hop flow under two-hop interference model. B. Comparion of Different Scheduling Algorithm In thi ub-ection, we compare performance of the propoed RAMM cheduling algorithm with the other two queuelength-baed contant-time cheduling algorithm in [6], namely policy V and policy W for one-hop and two-hop interference model, repectively. For eae of reference, we briefly decribe thee two cheduling algorithm here. Before doing o, let u introduce ome neceary notation. Let l) and rl) denote the tranmitting node and receiving node of link l, repectively. Alo, let Ei) denote the et of link connected to node i, and Nl) denote the et of neighboring link haring a common node with link l i.e., Nl) = El)) Erl))\{l}. In addition, let Nl) + denote the union of Nl) and {l}, and n l and n + l denote the number of link in Nl) and Nl) +, repectively. For the comparion purpoe, we aume ingle-hop flow o data traffic will be directly buffered in the tranmiion queue for each link i.e., no regulator implementation). In the cheduling policie V and W, each time lot alo conit of cheduling phae and data tranmiion phae. There are Fig. 0. Performance of policy W [6] under two-hop interference model for M = 44, 28, 52, 024, grid network with 36 node and 30 one-hop flow in Fig. 5). M mini-lot in the cheduling phae. The two cheduling algorithm work a follow. Policy V: In each mini-lot of the cheduling phae, each backlogged link contend with probability α x l M where α = M )/2 and Q l / x l = { max k El)) Q k/, } k Erl)) Q k/ where Q l i the queue length and i the tranmiion rate of link l, repectively. Policy W: In each mini-lot of the cheduling phae, each backlogged link contend with probability β y l M where M β = n ) where n = max l E n + l y l = max k Nl) + and Q l /. h Nk) Q + h /R h In Fig. 9, we how the performance of RAMM cheduling algorithm in a grid network with 36 node and 30 one-hop
9 Total average queue length.8.6.4.2 0.8 0.6 0.4 0.2 2 x 05 RAMM, M = 20 RAMM, M = 28 RAMM, M = 36 RAMM, M = 44 GMM 0 0 0.5.5 2 2.5 3 3.5 Load ρ Fig.. Performance of RAMM cheduling algorithm under two-hop interference model for B = 4, grid network with 36 node and 6 multihop flow in Fig. 5). Total average queue length.8.6.4.2 0.8 0.6 0.4 0.2 2 x 05 Policy V, M = 28 Policy V, M = 52 Policy V, M = 024 RAMM, M = 27 GMM 0 0 0.5.5 2 2.5 3 3.5 Load ρ Fig. 2. Performance of RAMM with B = 3, K = 9) and policy V [6] cheduling algorithm under one-hop interference model for grid network with 36 node and 60 one-hop flow in Fig. 4). flow a in Fig. 5. For the RAMM algorithm, we how the performance with different tranmiion round K while we fix the maximum backoff value of B = 4 the number of mini-lot in the cheduling phae i M = B K). We alo preent performance of Pick-and-Compare P&C) [], [3] and greedy maximal matching GMM) [5] cheduling cheme. For the P&C cheme, a new chedule i generated in each time lot by uing RAMM algorithm. Then the total weight of the old chedule and the newly generated chedule are compared where the weight of a link i the length of it tranmiion queue, and the chedule with larger weight i choen for tranmiion in the time lot. In fact, the huge complexity of the P&C cheme incur in the compare tep. For the GMM cheme, the chedule i contructed by adding one feaible link with the larget weight to the chedule and removing all conflicting link with the added link in each tep. Thi procedure i repeated until no further link can be added to the chedule. Note that both P&C and GMM cheduling cheme are either centralized or require huge overhead to implement in a ditributed manner. Alo, it i known that thee two cheduling cheme achieve almot the capacity region. It i confirmed in thi figure that P&C and GMM cheme achieve imilar performance although GMM ha a bit maller total queue length for traffic load cloe to the boundary of the capacity region. It i evident that when we increae the number of round for the RAMM algorithm, we achieve better performance. Moreover, in the conidered network d I G) = 4, o the performance guarantee tated in propoition i jut /4 capacity region. However, the actual performance achieved by the propoed RAMM cheduling algorithm i much better than the performance bound a can be een in Fig. 9. In Fig. 0, we preent the performance of the queue-lengthbaed contant time cheduling policy W for the two hop interference model [6]. We how the performance of thi cheduling cheme for different number of mini-lot ued in the cheduling phae M the firt phae of thi cheduling policy correpond to the firt phae of our RAMM cheduling algorithm). A i evident, with M = 44 mini-lot, policy W achieve much lower performance than the RAMM algorithm. Even with M = 024 mini-lot, performance of policy W i till wore than that of RAMM algorithm with only M = 44 mini-lot. Note that due to practical implementation contraint, the time lot duration i uually limited to few milliecond a in mot current wirele ytem. If the duration of a mini-lot i 20 µ a in the WLAN tandard, M = 00 correpond to 2 m which i already quite large. Our cheduling algorithm, therefore, preent a ignificant advantage compared to policy W becaue we cannot make the time lot interval arbitrarily large in practice. In Fig., we how performance of the RAMM cheduling algorithm for the grid network with 6 multihop flow in Fig. 5. Similar performance to the ingle-hop cae preented in Fig. 9 i oberved for thi etting. It i evident that although multihop flow contribute more traffic to the network becaue each flow travere multiple link, cheduling algorithm i till the key to determine performance of the network. In Fig. 2, we compare performance of RAMM and policy V [6] under one-hop interference model for the grid network with 60 one-hop flow in Fig. 4. It i evident that RAMM cheduling with only M = 27 mini-lot achieve imilar performance with policy V for M = 024 mini-lot. The RAMM cheduling algorithm, therefore, ha ignificantly lower overhead compared to policy V. Note that the RAMM cheduling algorithm doe not require queue length information, o it i much eaier to implement compared to policy V and W. In addition, collecting queue length information will create further overhead which may alo be very ignificant, epecially for the two-hop interference model becaue each node need to forward queue length information of it incident link two hop away). C. Long-term Utility We compare the long-term time average) utility under the optimal tranmiion rate derived in ection V and under the cae where a cro-layer congetion control algorithm i ued. Specifically, we will conider the cro-layer congetion control algorithm propoed in [2] which i the extenion of
0 that for one-hop interference model in [5]. The cro-layer congetion control algorithm work a follow: Congetion price for each link l i updated a q l t + ) = [q l t) + α l q l t)] + 2) where α l i the tep ize and q l t) = [ S k Il) H k t+ t n t)x t) k St) and St) denote the et of link belonging to the chedule in time lot t,.) i the indicator function. Tranmiion rate of cla- uer i updated a l E q lt + ) H k k Il) x t + ) = min, M. 3) Tranmiion cheduling: The network i cheduled in each time lot by the correponding cheduling algorithm GMM or RAMM algorithm). Average utility 50 0 50 00 50 Congetion control, RAMM Congetion control, GMM Optimal tranmiion rate 200 0.5.5 2 2.5 Load ρ Fig. 3. Average utility of the propoed optimal tranmiion rate and with congetion control algorithm under one-hop interference model for grid network with 36 node and 2 multihop flow, GMM and RAMM cheduling with B = 3, K = 9 in Fig. 4). We conider the utility function U x ) = lnx ). Hence, for the propoed approach the optimal tranmiion rate for each uer i x = e i.e., we aume that M > e). We will illutrate performance of the cro-layer congetion control algorithm under both GMM and RAMM cheduling algorithm. Long-term average utility i obtained by averaging the utility over 5.0 4 time lot. For the cro-layer congetion control algorithm, we fixed the tranmiion rate at x = e for the firt 0 3 time lot while till updating the congetion price and generating a new chedule in each time lot. Thi initial period provide time for the congetion price to converge to better value. The tep ize i initialized a α = 0. and it i updated a α = max { α/t, 0 3}. We how the average utility achieved by the optimal tranmiion rate and by the congetion control algorithm in Fig. 3, 4 for one-hop and two-hop interference model, repectively. It i evident that the propoed approach achieve higher average utility then thoe uing congetion control ] Average utility 50 0 50 00 50 Congetion control, RAMM Congetion control, GMM Optimal tranmiion rate 200 0.5.5 2 2.5 Load ρ Fig. 4. Average utility of the propoed optimal tranmiion rate and with congetion control algorithm under two-hop interference model for grid network with 36 node and 6 multihop flow, GMM and RAMM cheduling with B = 4, K = in Fig. 5). algorithm for all traffic load under both interference model. In fact, the average utility with congetion control decreae ignificantly when the traffic load i cloe to the boundary of the region which can be tabilized by the underlying cheduling algorithm. Thi obervation confirm the argument that performing congetion control i unneceary if the network can be tabilized by the underlying regulated cheduling algorithm. VIII. CONCLUSIONS We invetigated the network control problem uing contanttime cheduling under the k-hop interference model. With flow dynamic conideration, we have hown that the network can be tabilized by uing a regulated maximal cheduling policy if the offered load atifie the contraining bound. We have preented a contant-time and ditributed cheduling algorithm for a general k-hop interference model. The cheduling algorithm doe not require queue length information and ha overhead not growing with network ize. Our propoed cheduling algorithm achieve performance arbitrarily cloe to that of the regulated maximal cheduling. Under the tability condition, we have derived optimal tranmiion rate which achieve the maximum long-term network utility. For the high load regime, we have propoed a cro-layer congetion control algorithm which can tabilize the network for any offered load. Numerical reult have hown that the propoed cheduling algorithm achieve much better performance than the exiting contant-time cheduling algorithm in the literature, and it ha much better performance than it performance guarantee. Alo, performing congetion control under low load condition actually degrade performance in term of long-term utility ignificantly compared to the optimal tranmiion rate. APPENDIX I PROOF OF PROPOSITION Let Q l t) and Q lt) be tranmiion queue length for cla and for all uer clae at link l in time lot t, repectively. Similarly, let P l t) and P lt) be regulator queue length
for cla and for all uer clae at link l in time lot t, repectively. Let u denote by b l and a l the previou link and next link of link l on the route of cla- uer. Alo, let Cl t) and Dl t) be the number of packet tranmitted from regulator and tranmiion queue in time lot t, repectively. We have the following queue update equation Q l t + ) = Q l t) Dl t) + Cl t) 4) Pl t + ) = Pl t) Cl t) + Db t). l 5) Thu, we have Q l t+)+pa t+) = Q l l t)+pa t)+c l l t) Ca t). 6) l We will ue the following Lyapunov function for the ytem where V P, Q) = V Q) + ξv 2 P, Q) 7) V Q) = l V 2 P, Q) = l Q l t) Q k t) P a l + Q l ) 2. In fact, thi Lyapunov function wa alo ued in [9]. Now, let u conider V Qt + )) V Qt)) = 2 ) Q l t) Q k t + ) Q k t)) l + ) Ql t + ) Q l t)) l ) Q k t + ) Q k t)) = 2 l + l Q l t) ) Ck t) D k t)) C l t) D l t)) C k t) D k t)) ) 8).9) Since the number of packet tranmitted from regulator and tranmiion queue in each time lot are bounded, the econd term in 9) can be bounded by a contant B. Thu, we have [ E V Qt + )) V Qt)) P t), Qt) ] 2E [ l Q l t) 2E Q l t) l:q l t) C k t) D k t)) C k t) ) P t), Qt) ] ) D k t) + B P t), Qt) ] + B 2. Let L be the larget number of hop travered by any uer cla, we have E Q l t) Ck t) P R t), Qt) l:q l t) k Q l t) H k ρ + Lɛ. 20) R l:q l t) k Alo, due to the definition of maximal cheduling policy, we have Q l t) D k t) Q l t). 2) l:q l t) l:q l t) For traffic load atifying 5), we can find ɛ and δ mall enough uch that S H k ρ + Lɛ + δ, l E. 22) From 20), 2), 22), we have [ E V Qt + )) V Qt)) P t), Qt) ] Q l t) δ + B 2 δ l:q l t) l Q l t) + B 3. 23) Now, uing the procedure a in [9], we can obtain [ E V 2 P t + ), Qt + )) V 2 P t), Qt)) P t), Qt) ] Q l t) η Pa t) + B 4. 24) l l l Combining the reult in 23), 24), we have [ E V P t + ), Qt + )) V P t), Qt)) P t), Qt) ] [ ] δ ξ Q l t) ξη Pa R t) + B 5. 25) l l l l δ We can chooe ξ mall enough uch that ξ > 0. Thu, the drift will be negative if the regulator and/or tranmiion queue become large enough. Therefore, the tability reult follow by uing theorem 2 of [26]. Note that the choen Lyapunov function doe not take regulator queue on the firt hop of each uer cla into account. Thee regulator queue are, however, table becaue their output rate i ρ + ɛ which i larger than the average input load i.e., ρ ). APPENDIX II PROOF OF PROPOSITION 2 Conider any link AB between node A and B. We will find the probability that at leat one link in the interference et I AB i cheduled. Thi event will be denoted a M AB in the equel. A mentioned before, RAMM cheduling algorithm include BP-SIM cheduling algorithm for the one-hop node excluive) interference model propoed in [7] a a pecial cae. In the following we prove propoition 2 for k 2. For the pecial cae of k =, we refer the reader to [7] for the proof and the correponding analyi. Note that the proof for the cae k 2 i very challenging and completely different from that for the
2 cae k = in [7] due to the more complicated interference relationhip. We will illutrate ome important definition ued in the proof in Fig. 5. Let I 0 be the et of node which i at mot k hop away from either A and B including A, B. For the grid network and link AB hown in Fig. 5 under the twohop interference model, I 0 = {A, B, C,, C 6 }. Alo, the interference et for node C 6 i.e., I C6 ) conit of all node in I 0 and all blank node. Note that the notion of node interference et i different from that of link interference et provided in definition I. Here, all node in I C6 are at mot 3 hop away from C 6. We oberve that all link incident to any node in I 0 will belong to I AB becaue they are within k hop from link AB. In the following, we will find the lower bound for the probability of M AB by conidering ub-cae in which there are i left node in the et I 0. For convenience, we will ue I 0 to denote both the et itelf and the correponding number of node in I 0. Let L i be the event that there are i left node in the et I 0. The probability that at leat one link in the interference et of link AB i cheduled can be lower bounded a P m I 0 i= PrM AB L i )PrL i ). 26) Recall that for any particular node A, there are at mot I node whoe matching requet can collide with that tranmitted from node A with the power ued in the cheduling phae. Note that thee I node will be at mot k + hop away from mode A for the interference model with k 2 where the ame tranmiion power level i ued in both cheduling and tranmiion phae of each time lot. To find the lower bound for the probability of M AB, we will aume the wortcae cenario where each node ha I interfering node. For convenience, we alo ue I A to denote the et of thee interfering node whoe tranmiion can collide with that from node A. We now C conider C6 the following cae. C2 C3 A C4 B C5 Fig. 5. Interference et I C6 under two-hop interference model. A. There i only one left node in I 0 We conider the following two ub-cae. If thi left node i either A or B, then the left node will have at leat one neighbor which i a right node. Thi cae occur with probability 2/2) I0. For eae of reference, we will refer to thi left node a node C i.e., C i either A or B). Alo, there are at mot I I 0 node whoe matching requet can collide with that from node C. To find the lower bound for PrM AB ), we aume that there are I I 0 uch interfering node. Now, uppoe there are i left node among thee I I 0 interfering node. The matching requet tranmitted by node C will be uccefully received if the backoff value of thee i left node are larger than that of node C. Specifically, the matching requet from node C will be uccefully received and the correponding link will be cheduled with a probability which i lower bounded by F = I I 0 i=0 I I0 i ) ) I I0 2 B B m= m ) i B where we have broken the event into ub-cae where there are i left node among I I 0 interfering node and thee i left node have backoff value larger than that of node C. If thi left node i any node other than A and B then it can be any node among I 0 2) node. Again, we refer to thi left node a node C. Note that all node which are within one hop from A or B including A and B are at mot k + hop away from C o they all belong to the interfering et I C. Let x be the total number of node within one hop from A and B including A and B, then there are at mot I x node whoe matching requet can collide with that from node C. Thi i becaue thee x node belong to I 0 and they are all right node except C if C i in I 0. We will aume thi wort-cae cenario to calculate the lower bound of the matching probability in the following. Note that node C will have at leat one neighbor which i a right node becaue it i the only left node in I 0. Again, uppoe there are i left node among potential interfering node in I C. The matching requet tranmitted by node C will be uccefully received if the backoff value of thee i left node are larger than that from node C and the matching requet from node C i tranmitted to a right node. Now, we conider the following two ubcae. For the firt cae, if C i one of x node i.e., x one-hop neighbor of A or B) but not A and B. Then, thi cae occur with probability x 2)/2) I0. In thi cae, the matching requet from node C will be uccefully received and the correponding link will be cheduled with a probability which i lower bounded by F 2 = I x i=0 I x i ) 2 ) I x B B m= m ) i B where i i the number of left node. Alo, in calculating the lower bound for P m we aume that C tranmit it
3 matching requet to a right node which i not an one-hop neighbor of A or B. Hence, there are at mot I x left node which can collide with the matching requet from C. For the econd cae, if C i not one of x node i.e., x one-hop neighbor of A or B). Then, thi cae occur with probability I 0 x)/2) I 0. In thi cae, the matching requet from node C will be uccefully received and the correponding link will be cheduled with a probability which i lower bounded by F 3 = I x 2 i=0 I x 2 i ) ) I x 2 B B m= m ) i B where in calculating the lower bound for P m we aume that C tranmit it matching requet to a right node which i not an one-hop neighbor of A or B. Hence, there are at mot I x 2 left node which can collide with the matching requet from C. B. There are two or more left node in I 0 Suppoe node C in the et I 0 become left and win the contention. Then node C hould have the mallet backoff value among all the node whoe matching requet can collide with the matching requet from C. Alo, node C hould end the matching requet to a node which i a right one. For eae of reference, we will refer to thi right node a node D in the equel. In general, D can belong to et I 0 or not. However, to find the lower bound of P m, we aume that D belong to I 0 ; therefore, there are at mot I 0 2 other left node beide C and D in I 0. A before, we aume the wort-cae cenario where there are I node whoe tranmiion can collide with that of node C. Recall that all x node which are one-hop neighbor of A or B belong to the et I. Similar to the previou cae, we conider the following two ub-cae. For the firt cae, C i one of x node i.e., x one-hop neighbor of A or B). In thi cae the matching probability can be lower bounded a F 4 = x I 0 2 i= I x j=0 I0 2 i ) ) I0 I x 2 j ) I x 2 B B m= ) m ) i+j B where i i the number of left node beide C and D in the et I 0. And j i the number of left node which belong to I but are not one-hop neighbor of A or B i.e., there i no link between thee node and A or B). We will denote thi et a I \ x in the equel. In general, D can belong to I \ x or not; however, to find the lower bound for P m, we only allow j take value from 0 to I x. In addition, C can be any node among x node o we have a factor of x before the um. Alo, we require that all left node i left node belonging I 0 and j left node belonging to I \ x) achieve larger backoff value than that of node C. For the econd cae, C belong to the et I \ x. In thi cae, the matching probability can be lower bounded a F 5 = I 0 x) I 0 2 i= I x 2 j=0 I0 2 i ) ) I0 I x 2 2 j ) I x 2 B B m= ) m ) i+j B where C can be one of I 0 x node o we have the factor I 0 x) before the um. Alo, to calculate the lower bound for the matching probability, we aume D alway belong to I \ x, o j can be at mot I x 2. Subtitute reult of all conidered cae into 26), the matching probability i lower bounded by P m P 0 m = 2/2) I 0 F + x 2)/2) I 0 F 2 +I 0 x)/2) I 0 F 3 + F 4 + F 5 27) where F, F 2, F 3, F 4, and F 5 are defined above. From the lower bound of the matching probability P 0 m derived above, we can calculate the lower bound of P m for any link AB a p = min x, I 0 P 0 m where we find the minimum of P 0 m over all poible x and I 0. Note that poible value of x and I 0 will be in the range of [3, 2d ] and [3, I 0 ], repectively. Here, x and I 0 are at leat three for the network to be connected i.e., A and B hould have at leat one one-hop neighbor). It i oberved that p i independent of the network ize and depend only on B, d, I, I 0. Now, we can chooe K uch that at leat one link in I l of a backlogged link l i cheduled with probability greater than µ after K round a follow: { K = min k : p ) k µ }. 28) k Thu, we can chooe K which i independent of network ize and only depend on B, d, I, I 0, µ uch that performance guarantee arbitrarily cloe to the contraining bound can be achieved. Example: For the grid network and two-hop interference model, we have I = 22, I 0 = x can take value of 5, 6, 8. With maximum backoff value B = 0, by uing the analyi preented above, the minimum number of cheduling round to achieve µ = 0.9 i K = 60. In fact, thi calculation i quite conervative becaue it conider the wort cae cenario. In practice, the ize of interference et decreae quickly over cheduling round, o the required value of K i much maller. APPENDIX III PROOF OF PROPOSITION 4 The proof i imilar to that of Propoition i.e., we ue the ame Lyapunov function and proof procedure). In particular,
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PLACE PHOTO HERE Long Le received the B.Eng. degree with highet ditinction from Ho Chi Minh City Univerity of Technology in 999, the M.Eng. degree from Aian Intitute of Technology AIT) in 2002, and the Ph.D. degree from Univerity of Manitoba in 2007. He i currently a Potdoctoral Fellow in the Department of Electrical and Computer Engineering at Univerity of Waterloo. Hi current reearch interet include cognitive radio, cooperative diverity and relay network, and cro-layer deign for communication network. He i a Member of the IEEE. Ravi Mazumdar wa born in Bangalore, India. He obtained the B.Tech. in Electrical Engineering from the Indian Intitute of Technology, Bombay, India in 977, the M.Sc. DIC in Control Sytem from Imperial College, London, U.K. in 978 and the Ph.D. in Sytem Science from the Univerity of California, Lo Angele, USA in 983. He i currently a Univerity Reearch Chair Profeor of Electrical and Computer Engineering at the Univerity of Waterloo, Waterloo, Canada and an Adjunct Profeor of ECE at Purdue Univerity. He ha erved on the facultie of Columbia Univerity NY, USA), INRS- Telecommunication Montreal, Canada), Univerity of Eex Colcheter, UK), and mot recently at Purdue Univerity Wet Lafayette, USA). He ha held viiting poition and abbatical leave at UCLA, the Univerity of Twente Netherland), the Indian Intitute of Science Bangalore); and the Ecole Nationale Superieure detelecommunication Pari). He i a Fellow of the IEEE and the Royal Statitical Society. He i a member of the working group WG6.3 and 7. of the IFIP and a member of SIAM and the IMS. He hared the Bet Paper Award with G. Sharma and N. Shroff at the INFOCOM 2006 and wa co-author with N. Likhanov for the Bet Paper runner up at INFOCOM 998. He i an Editor of the IEEE Tran. on Networking and ha erved a a guet editor for a number of journal. Hi reearch interet are in wirele and wireline network; application of game theory to networking; applied probability, queueing theory, and tochatic analyi with application to traffic engineering, tochatic filtering theory, and mathematical finance.