Delay-Guaranteed Cross-Layer Scheduling in Multi-Hop Wireless Networks
aa r X i v : . [ c s . I T ] J a n Delay-Guaranteed Cross-Layer Scheduling inMulti-Hop Wireless Networks
Dongyue Xue, Eylem Ekici
Department of Electrical and Computer EngineeringOhio State University, USAEmail: { xued, ekici } @ece.osu.edu Abstract —In this paper, we propose a cross-layer schedulingalgorithm that achieves a throughput “ ǫ -close” to the optimalthroughput in multi-hop wireless networks with a tradeoff of O ( ǫ ) in delay guarantees. The algorithm aims to solve a jointcongestion control, routing, and scheduling problem in a multi-hop wireless network while satisfying per-flow average end-to-end delay guarantees and minimum data rate requirements.This problem has been solved for both backlogged as well asarbitrary arrival rate systems. Moreover, we discuss the designof a class of low-complexity suboptimal algorithms, effects ofdelayed feedback on the optimal algorithm, and extensions ofthe proposed algorithm to different interference models witharbitrary link capacities. I. I
NTRODUCTION
Cross-layer design of congestion control, routing andscheduling algorithms with Quality of Service (QoS) guaran-tees is one of the most challenging topics in wireless network-ing. The back-pressure algorithm first proposed in [1] and itsextensions have been widely employed in developing through-put optimal dynamic resource allocation and scheduling algo-rithms for wireless systems. Back-pressure-based schedulingalgorithms have also been employed in wireless networkswith time-varying channels [2][3][4]. Congestion controllersat the transport layer have assisted the cross-layer design ofscheduling algorithms in [5][6][7], so that the admitted arrivalrate is guaranteed to lie within the network capacity region.Low-complexity distributed algorithms have been proposed in[8][9][10][11]. Algorithms adapted to clustered networks havebeen proposed in [12] to reduce the number of queues main-tained in the network. However, delay-related investigationsare not included in these works.In this paper, we propose a cross-layer algorithm to achieve guaranteed throughput while satisfying network QoS require-ments. Specifically, we construct two virtual queues, i.e., a virtual queue at transport layer and a virtual delay queue ,to guarantee average end-to-end delay bounds . Moreover, weconstruct a virtual service queue to guarantee the minimumdata rate required by individual network flows . Our cross-layerdesign includes a congestion controller for the input rate to thevirtual queue at transport layer, as well as a joint policy forpacket admission, routing, and resource scheduling. We showthat our algorithm can achieve a throughput arbitrarily closeto the optimal. In addition, the algorithm exhibits a tradeoff of O ( ǫ ) in the delay bound, where ǫ denotes the distance fromthe optimal throughput.Our main algorithm is further extended: (1) to a set oflow-complexity suboptimal algorithms; (2) from a model withconstantly-backlogged sources to a model with sources ofarbitrary input rates at transport layer; (3) to an algorithmemploying delayed queue information; and (4) from a node-exclusive model with constant link capacities to a model witharbitrary link capacities and interference models over fadingchannels.The rest of the paper is organized as follows: SectionII discusses the related work. In Section III, the networkmodel is presented, followed by corresponding approaches forthe considered multi-hop wireless networks. In Section IV,the optimal cross-layer control and scheduling algorithm isdescribed, and its performance analyzed. In Section V, weprovide a class of feasible suboptimal algorithms, considersources with arbitrary arrival rates at transport layer, employdelayed queue information in the scheduling algorithm, andextend the model to arbitrary link capacities and interferencemodels over fading channels. We present numerical results inSection VI. Finally, we conclude our work in Section VII.II. R ELATED W ORK
Delay issues in single-hop wireless networks have beenaddressed in [13]-[21]. Especially, the scheduling algorithm in[18] provides a throughput-utility that is inversely proportionalto the delay guarantee. Authors of [19] have obtained delaybounds for two classes of scheduling policies. A randomaccess algorithm is proposed in [20] for lattice and torusinterference graphs, which is shown to achieve order-optimaldelay in a distributed manner with optimal throughput. Butthese works are not readily extendable to multi-hop wirelessnetworks, where additional arrivals from neighboring nodesand routing must be considered. Delay analysis for multi-hop networks with fixed-routing is provided in [22]. Delay-related scheduling in multi-hop wireless networks have beenproposed in [23][24][25][26][27]. However, none of the above-mentioned works provide explicit end-to-end delay guarantees.There are several works aiming to address end-to-enddelay or buffer occupancy guarantees in multi-hop wirelessnetworks. Worst-case delay is guaranteed in [28] with apacket dropping mechanism. However, dropped packets are not compensated or retransmitted with the algorithm of [28],which may lead to restrictions in its practical implementations.A low-complexity cross-layer fixed-routing algorithm is de-veloped in [29] to guarantee order-optimal average end-to-enddelay, but only for half of the capacity region. A schedulingalgorithm for finite-buffer multi-hop wireless networks withfixed routing is proposed in [30] and is extended to adaptive-routing with congestion controller in [31]. Specifically, thealgorithm in [31] guarantees O ( ǫ ) -scaling in buffer size witha ǫ -loss in throughput-utility, but this is achieved at the expenseof the buffer occupancy of the source nodes, where an infinitebuffer size in the network layer is assumed in each sourcenode. This leads to large average end-to-end delay since thenetwork stability is achieved based on queue backlogs at thesesource nodes.Compared to the above works, the algorithm presented inthis paper develops and incorporates novel virtual queue struc-tures. Different from traditional back-pressure-based algo-rithms, where the network stability is achieved at the expenseof large packet queue backlogs, in our algorithm, “the burden”of actual packet queue backlogs is shared by our proposedvirtual queues, in an attempt to guarantee specific delay per-formances. Specifically, we design a congestion controller for a virtual input rate and assign weights in the scheduling policyas a product of actual packet queue backlog and the weightedbacklog of a designed virtual queue, which will be introducedin detail in Section IV. As such, the network stabilization isachieved with the help of virtual queue structures that do notcontribute to delay in the network. Since all packet queues in the network, including those in source nodes, have finitesizes, all average end-to-end delays are bounded independentof length or multiplicity of paths.III. N ETWORK M ODEL
A. Network Elements
We consider a time-slotted multi-hop wireless networkconsisting of N nodes and K flows. Denote by ( m, n ) ∈ L alink from node m to node n , where L is the set of directedlinks in the network. Denoting the set of flows by F andthe set of nodes by N , we formulate the network topology G = ( N , L ) . Note that we consider adaptive routing scenario,i.e., the routes of each flow are not determined a priori , whichis more general than fixed-routing scenario. In addition, wedenote the source node and the destination node of a flow c ∈ F as b ( c ) and d ( c ) , respectively.We assume that the source node for flow c is alwaysbacklogged at the transport layer. Let the scheduling parameter µ cmn ( t ) denote the link rate assignment of flow c for link ( m, n ) at time slot t according to scheduling decisions andlet µ cs ( c ) b ( c ) ( t ) denote the admitted rate of flow c from thetransport layer of flow to the source node, where s ( c ) denotesthe source at the transport layer of flow c . It is clear thatin any time slot t , µ cs ( c ) n ( t ) = 0 ∀ n = b ( c ) . For simplicityof analysis, we assume only one packet can be transmittedover a link in one slot, so ( µ cmn ( t )) takes values in { , } ∀ ( m, n ) ∈ L . We also assume that µ cs ( c ) b ( c ) ( t ) is boundedabove by a constant µ M ≥ : ≤ µ cs ( c ) b ( c ) ( t ) ≤ µ M , ∀ c ∈ F , ∀ t, (1)i.e., a source node can receive at most µ M packets from thetransport layer in any time slot. To simplify the analysis,we prevent looping back to the source, i.e., we impose thefollowing constraints X m ∈N ( µ cmb ( c ) ( t )) = 0 ∀ c ∈ F , ∀ t. (2)We employ the node-exclusive model in our analysis, i.e., eachnode can communicate with at most one other node in a timeslot. Note that our model is extended to arbitrary interferencemodels with arbitrary link capacities and fading channels inSection V.D.We now specify the QoS requirements associated witheach flow. The network imposes an average end-to-end delaythreshold ρ c for each flow c . The end-to-end delay periodof a packet starts when the packet is admitted to the sourcenode from the transport layer and ends when it reaches itsdestination. Note that the delay threshold is a time-averagedupper-bound, not a deterministic one. In addition, each flow c requires a minimum data rate of a c packets per time slot. B. Network Constraints and Approaches
For convenience of analysis, we define L c , L ∪{ ( s ( c ) , b ( c )) } , where the pair ( s ( c ) , b ( c )) can be considered asa virtual link from transport layer to the source node. We nowmodel queue dynamics and network constraints in the multi-hop network. Let U cn ( t ) be the backlog of the total amountof flow c packets waiting for transmission at node n . For aflow c , if n = d ( c ) then U cn ( t ) = 0 ∀ t ; Otherwise, the queuedynamics is as follows: U cn ( t + 1) ≤ [ U cn ( t ) − X i :( n,i ) ∈L µ cni ( t )] + + X j :( j,n ) ∈L c µ cjn ( t ) , if n ∈ N \ d ( c ) , (3)where the operator [ x ] + is defined as [ x ] + = max { x, } .Note that in (3), we ensure that the actual number of packetstransmitted for flow c from node n does not exceed itsqueue backlog, since a feasible scheduling algorithm maynot depend on the information on queue backlogs. The terms P i :( n,i ) ∈L µ cni ( t ) and P j :( j,n ) ∈L c µ cjn ( t ) represent, respec-tively, the scheduled departure rate from node n and thescheduled arrival rate into node n by the scheduling algorithmwith respect to flow c . Note that (3) is an inequality since thearrival rates from neighbor nodes may be less than P j µ cjn ( t ) if some neighbor node does not have sufficient number ofpackets to transmit. Since we employ the node-exclusivemodel, we have ≤ X c ∈F [ X i :( n,i ) ∈L µ cni ( t )+ X j :( j,n ) ∈L µ cjn ( t )] ≤ , ∀ n ∈ N . (4) From (1)(2), we also have X j :( j,n ) ∈L c µ cjn ( t ) ≤ µ M , if n = b ( c ) , (5)if it is ensured that no packets will be looped back to thesource.Now we construct three kinds of virtual queues, namely,virtual queue U cs ( c ) ( t ) at transport layer, virtual service queue Z c ( t ) at sources, and virtual delay queue X c ( t ) , to later assistthe development of our algorithm: (1) For each flow c at transport layer, we construct a virtualqueue U cs ( c ) ( t ) which will be employed in the algorithmproposed in the next section. We denote the virtual input rateto the queue as R c ( t ) at the end of time slot t and we upper-bound R c ( t ) by µ M . Let r c denote the time-average of R c ( t ) .We update the virtual queue as follows: U cs ( c ) ( t + 1) = [ U cs ( c ) ( t ) − µ cs ( c ) b ( c ) ( t )] + + R c ( t ) , (6)where the initial U cs ( c ) (0) = 0 . Considering the admitted rate µ cs ( c ) b ( c ) ( t ) as the service rate, if the virtual queue U cs ( c ) ( t ) is stable, then the time-average admitted rate µ c of flow c satisfies: µ c , lim t →∞ t t − X τ =0 µ cs ( c ) b ( c ) ( τ ) ≥ r c , lim t →∞ t t − X τ =0 R c ( τ ) . (7) (2) To satisfy the minimum data rate constraints, we constructa virtual queue Z c ( t ) associated with flow c as follows: Z c ( t + 1) = [ Z c ( t ) − R c ( t )] + + a c , (8)where the initial Z c (0) = 0 . Considering a c as the arrival rateand R c ( t ) as the service rate, if queue Z c ( t ) is stable, we have: r c ≥ a c . Additionally, if U cs ( c ) ( t ) is stable, then according to(7), the minimum data rate for flow c is achieved. (3) To satisfy the end-to-end delay constraints, we constructa virtual delay queue X c ( t ) for any given flow c as follows: X c ( t + 1) = [ X c ( t ) − ρ c R c ( t )] + + X n ∈N U cn ( t ) (9)where the initial X c (0) = 0 . Considering the packets keptin the network in time slot t , i.e., P n ∈N U cn ( t ) , as the arrivalrate and ρ c R c ( t ) as the service rate, and according to queueingtheory, if queue X c ( t ) is stable, we have lim t →∞ t t − X τ =0 X n ∈N U cn ( τ ) ≤ ρ c lim t →∞ t t − X τ =0 R c ( τ ) = ρ c r c . Furthermore, if U cs ( c ) ( t ) is stable, then according to (7), wehave: µ c lim t →∞ t t − X τ =0 X n ∈N U cn ( τ ) ≤ ρ c . (10)In addition, by Little’s Theorem, (10) ensures that the averageend-to-end delay of flow c is less than or equal to the threshold ρ c with probability (w.p.) .From the above description, we know that the networkis stable (i.e., each queue at all nodes is stable) and the average end-to-end delay constraint and minimum data raterequirement are achieved if queues U cn ( t ) and the three virtualqueues are stable for any node and flow, i.e., lim sup t →∞ t t − X τ =0 E { X c ( τ ) } < ∞ , ∀ c ;lim sup t →∞ t t − X τ =0 E { U cn ( τ ) } < ∞ , ∀ n ∈ N ∪ { s ( c ) : c ∈ F} ;lim sup t →∞ t t − X τ =0 E { Z c ( τ ) } < ∞ , ∀ c. Now we define the capacity region of the considered multi-hop network. An arrival rate vector ( z c ) is called admissible if there exists some scheduling algorithm (without congestioncontrol) under which the node queue backlogs (not includingvirtual queues) are stable. We denote Λ to be the capacityregion consisting of all admissible ( z c ) , i.e., Λ consists ofall feasible rates stabilizable by some scheduling algorithm without considering QoS requirements (i.e., delay constraintsand minimum data rate constraints). To assist the analysis inthe following sections, we let ( r ∗ ǫ,c ) denote the solutions to thefollowing optimization problem: max ( r c ):( r c + ǫ ) ∈ Λ X c ∈F r c s.t. r c ≥ a c , ∀ c ∈ F . where ǫ is a positive number which can be chosen arbitrarilysmall. For simplicity of analysis, we assume that ( a c ) is in theinterior of Λ and without loss of generality, we assume thatthere exists ǫ ′ > such that r ∗ ǫ,c ≥ a c + ǫ ′ ∀ c ∈ F . Accordingto [32], we have lim ǫ → X c ∈F r ∗ ǫ,c = X c ∈F r ∗ c , where ( r ∗ c ) is the solution to the following optimization: max ( r c ):( r c ) ∈ Λ X c ∈F r c s.t. r c ≥ a c , ∀ c ∈ F . IV. C
ONTROL S CHEDULING A LGORITHM FOR M ULTI -H OP W IRELESS N ETWORKS
Now we propose a control and scheduling algorithm
ALG for the introduced multi-hop model so that
ALG stabilizesthe network and satisfies the delay constraint and minimumdata rate constraint. Given ǫ , the proposed ALG can achievea throughput arbitrarily close to P c ∈F r ∗ ǫ,c , under certainconditions related to delay constraints which will be later givenin Theorem 1.The optimal algorithm ALG consists of two parts: a conges-tion controller of R c ( t ) , and a joint packet admission, routingand scheduling policy. We propose and analyze the algorithmin the following subsections. A. Algorithm Description and Analysis
Let q M ≥ µ M be a control parameter for queue length.We first propose a congestion controller for the input rate ofvirtual queues at transport layer:
1) Congestion Controller of R c ( t ) : min ≤ R c ( t ) ≤ µ M R c ( t )( ( q M − µ M ) U cs ( c ) ( t ) q M − X c ( t ) ρ c − Z c ( t ) − V ) (11)where V > is a control parameter. Specifically, when q M − µ M q M U cs ( c ) ( t ) − X c ( t ) ρ c − Z c ( t ) − V > , R c ( t ) is setto zero; Otherwise, R c ( t ) = µ M .After performing the congestion control, we perform thefollowing joint policy for packet admission, routing andscheduling (abbreviated as scheduling policy ):
2) Scheduling Policy : In each time slot, with the constraintsof the underlying interference model as described in SectionIII including (1)(2)(4), the network solves the following opti-mization problem: max ( µ cmn ( t )) X m,n µ c ∗ mn ( t ) mn ( t ) w mn ( t ) (12)s.t. µ cmn ( t ) = 0 ∀ c = c ∗ mn ( t ) , ∀ ( m, n ) ∈ L c ,µ cmn ( t ) = 0 if n = s ( c ) , ∀ c ∈ F , where c ∗ mn ( t ) and w mn ( t ) are defined as follows: c ∗ mn ( t ) = arg max c ∈F w cmn ( t ) ,w mn ( t ) = [max c ∈F w cmn ( t )] + , with weight assignment as follows w cmn ( t ) = U cs ( c ) ( t ) q M [ U cm ( t ) − U cn ( t )] , if ( m, n ) ∈ L ,U cs ( c ) ( t ) q M [ q M − µ M − U cb ( c ) ( t )] , if ( m, n ) = ( s ( c ) , b ( c )) , , otherwise. (13)In addition, when w mn ( t ) = 0 , without loss of optimality, weset µ cmn ( t ) = 0 ∀ c ∈ F to maximize (12).Note that L ∪ { ( s ( c ) , b ( c )) : c ∈ F} forms the ( m, n ) pairsin ( µ cmn ( t )) over which the optimization (12) is performed.Thus, the optimization is a typical Maximum Weight Matching(MWM) problem. We first decouple flow scheduling from theMWM. Specifically, for each pair ( m, n ) , the flow c ∗ mn ( t ) isfixed as the candidate for transmission. We then assign theweight as w mn ( t ) . Note also that although similar productform of the weight assignment (13) have been utilized in[30][31], no virtual queues are involved there. Whereas in ALG , we assign weights as a product of weighted virtual queuebacklog ( U cs ( c ) ( t ) q M ) and the actual back-pressure, in an aim toshift the burden of the actual queue backlog to the virtualbacklog. To analyze the performance of the algorithm, we firstintroduce the following proposition. Proposition 1:
Employing
ALG , each queue backlog in thenetwork has a deterministic worst-case bound: U cn ( t ) ≤ q M , ∀ t, ∀ n ∈ N , ∀ c ∈ F . (14) Proof:
We use mathematical induction on time slot in theproof. When t = 0 , U cn (0) = 0 ≤ q M ∀ n, c . In the inductionhypothesis, we suppose in time slot t we have U cn ( t ) ≤ q M ∀ n, c . In the induction step, for any given n ∈ N and c ∈ F ,we consider two cases as follows: (1) We first consider the case when n = b ( c ) , i.e., when n is the source node of flow c . Since U cn ( t ) ≤ q M from theinduction hypothesis, we further consider two subcases: • In the first subcase, U cb ( c ) ( t ) ≤ q M − µ M . Then accordingto the queue dynamics (3) and the inequality (5), U cb ( c ) ( t +1) ≤ U cb ( c ) ( t ) + µ M ≤ q M ; • In the second subcase, q M − µ M < U cb ( c ) ( t ) ≤ q M .According to the weight assignment (13), we have w cs ( c ) b ( c ) ( t ) < which leads to µ cs ( c ) b ( c ) ( t ) = 0 . Hence, U cb ( c ) ( t + 1) ≤ U cb ( c ) ( t ) ≤ q M by (2)(3). (2) In the second case, n = b ( c ) , i.e., n is not the source nodeof flow c . Similar to the first case, we further consider thefollowing two subcases: • In the first subcase, U cn ( t ) < q M . Then, since we employnode-exclusive model, U cn ( t + 1) ≤ U cn ( t ) + 1 ≤ q M by(3)(4). • In the second subcase, U cn ( t ) = q M . According to theweight assignment (13) we have w cmn ( t ) ≤ ∀ m :( m, n ) ∈ L . Now, for any given node m : ( m, n ) ∈ L ,we have:(i) If c = c ∗ mn ( t ) , then by (12), µ cmn ( t ) = 0 ;(ii) Otherwise, c = c ∗ mn ( t ) , which induces w mn ( t ) =[ w cmn ( t )] + = 0 and by the scheduling policy, µ cmn ( t ) =0 .Hence µ cmn ( t ) = 0 ∀ m : ( m, n ) ∈ L , and U cn ( t + 1) ≤ U cn ( t ) = q M by the queue dynamics (3).The above analysis holds for any given n ∈ N and c ∈ F .Therefore the induction step holds, i.e., U cn ( t + 1) ≤ q M ∀ n, c ,which completes the proof.Now we present our main results in Theorem 1. Theorem 1:
Given that q M > N − µ M ǫ + µ M and ρ c > N q M r ∗ ǫ,c ∀ c ∈ F , (15) ALG can achieve a throughput lim inf t →∞ t t − X τ =0 X c ∈F E { R c ( τ ) } ≥ X c ∈F r ∗ ǫ,c − BV , (16)where B , N Kq M µ M + K q M − µ M q M µ M + µ M P c ∈F ρ c + KN q M + Kµ M + K P c ∈F a c .In addition, ALG ensures that the virtual queues have atime-averaged bound: lim sup t →∞ t t − X τ =0 X c ∈F E { U cs ( c ) ( τ )+ X c ( τ )+ Z c ( τ ) } ≤ B ′ δ , (17) where B ′ , B + V B R , with B R and δ constant positivenumbers given in the next subsection. Remark 1 (Network Stability) : The inequalities (14) fromProposition 1 and (17) from Theorem 1 indicate that
ALG sta-bilizes the actual and virtual queues. As an immediate result,
ALG stabilizes the network and satisfies the average end-to-end delay constraint and the minimum data rate requirement.In addition, Proposition 1 states that the actual queues are deterministically bounded by q M , which ensures finite buffersizes for all queues in the network, including those in sourcenodes. Remark 2 (Optimal Utility and Delay Analysis):
Since ( U cs ( c ) ( t )) are stable, the inequality (16) gives a lower-boundon the throughput that ALG can achieve. Given some ǫ > ,since B is independent of V , (16) also ensures that ALG canachieve a throughput arbitrarily close to P c ∈F r ∗ ǫ,c . When ǫ tends to , ALG can achieve a throughput arbitrarily closeto the optimal value P c ∈F r ∗ c with the tradeoff in queuebacklog upper-bound q M and the delay constraints ( ρ c ) , bothof which are lower-bounded by the reciprocal terms of ǫ asshown in (15) in Theorem 1. In other words, the averageend-to-end delay bound is of order O ( ǫ ) . We note that in ALG , the control parameter V , which is typically chosen to belarge, does not affect the actual queue backlog upper-boundor the average end-to-end delay bound, but only affects theupper-bound of the virtual queue backlogs (shown in (17)).In comparison, in the algorithm proposed in [31], the authorsshow that the internal buffer size is deterministically boundedwith order O ( ǫ ) , but at the expense of the buffer occupancyat source nodes which is of order O ( V ) , where V has tobe large enough for their algorithm to approach P c ∈F r ∗ ǫ,c .This design assumes an infinite buffer size at source nodes andtypically results in congestion at the source nodes as shownin the simulation results in [31], which further induces anunguaranteed and large average end-to-end delay. Moreover,one can expect that there are no buffer-size guarantees forsingle-hop flows by employing the algorithm in [31]. Incontrast, in our proposed ALG , we shift “the burden of V ”from actual queues to virtual queues and ensure that theaverage end-to-end delay constraints are satisfied with finitebuffer sizes for all actual packet queues. Remark 3 (Implementation Issues) : To update the virtualqueue X c ( t ) and perform the R c ( t ) congestion controller atthe transport layer, the queue backlog information of flow c is crucial. This information can be collected back to thesource node by piggy-backing it on ACK from each node. Inorder to account for such delay of queue backlog information,the R c ( t ) congestion controller (11) of the algorithm canemploy delayed queue backlog of X c ( t ) . Similarly, delayedqueue backlog information of U cs ( c ) ( t ) can be employed at theweight assignment (13) of the scheduling policy. The modifiedalgorithm and its validity are further discussed in SectionV.C. By employing delayed queue backlog information, wecan extend the algorithm to distributed implementation inmuch the same way as in [8][11] to achieve a fraction of the optimal throughput. In order to achieve a through- put arbitrarily close to the optimal value with distributedimplementation, we can employ random access techniques[37][38] in the scheduling policy with fugacities [39] chosenas exp { α ¯ U cs ( c ) ( t )[ U cm ( t ) − U cn ( t )] + q M } for each link ( m, n ) ∈ L ,where ¯ U cs ( c ) ( t ) is a local estimate (e.g., delayed information)of U cs ( c ) ( t ) and α a positive weight. It can be shown that thedistributed algorithm can still achieve an average end-to-enddelay of order O ( ǫ ) with the time-scale separation assumption[20][36][37]. A variation of such distributed implementationin single-hop networks can be found in our recent work [40].We prove Theorem 1 in the following subsection.
B. Proof of Theorem 1
Before we proceed, we present the following lemmas whichwill assist us in proving Theorem 1.
Lemma 1:
For nonnegative numbers A , A , A , Q ∈ R such that Q ≤ [ A − A ] + + A , we have Q ≤ A + A + A − A ( A − A ) . The proof of Lemma 1 is trivial and omitted. We will lateruse Lemma 1 to simplify virtual queue dynamics.
Lemma 2:
For any feasible rate vector ( θ c ) ∈ Λ with θ c ≥ a c ∀ c ∈ F , there exists a stationary randomizedalgorithm STAT that stabilizes the network with input ratevector ( µ ST ATs ( c ) b ( c ) ( t )) and scheduling parameters ( µ c,ST ATmn ( t )) independent of queue backlogs, such that the expected admit-ted rates are: E { µ c,ST ATs ( c ) b ( c ) ( t ) } = θ c , ∀ t, ∀ c ∈ F . In addition, ∀ t , ∀ n ∈ N , ∀ c , the flow constraint is satisfied: E { X i :( n,i ) ∈L µ c,ST ATni ( t ) − X j :( j,n ) ∈L c µ c,ST ATjn ( t ) } = 0 . Note that it is not necessary for the randomized algorithmSTAT to satisfy the average end-to-end delay constraints.Similar formulations of STAT and their proofs have been givenin [5] and [6], so we omit the proof of Lemma 2 for brevity.
Remark 4 : According to the STAT algorithm in Lemma2, we assign the input rates of the virtual queues at trans-port layer as R ST ATc ( t ) = µ c,ST ATs ( c ) b ( c ) ( t ) . Thus, we also have E { R ST ATc ( t ) } = θ c . According to the update equation (6),it is easy to show that the virtual queues under STAT arebounded above by µ M and the time-average of R ST ATc ( t ) satisfies: r ST ATc = θ c . Note that ( θ c ) can take values as ( r ∗ ǫ,c ) or ( r ∗ ǫ,c + ǫ ) or ( r ∗ ǫ,c − ǫ ′ ) , where we recall ( r ∗ ǫ,c + ǫ ) ∈ Λ and r ∗ ǫ,c ≥ a c + ǫ ′ ∀ c ∈ F .To prove Theorem 1, we first let Q ( t ) =(( U cn ( t )) , ( U cs ( c ) ( t )) , ( X c ( t )) , ( Z c ( t ))) and define theLyapunov function L ( Q ( t )) as follows: L ( Q ( t )) = 12 { X c ∈F q M − µ M q M U cs ( c ) ( t ) + X c ∈F X c ( t ) + X c ∈F Z c ( t ) + X c ∈F X n ∈N q M U cn ( t ) U cs ( c ) ( t ) } . (18) Note that the random access works cited above either do not provide delayguarantees or are not readily extended to multi-hop settings.
It is obvious that L ( Q (0)) = 0 . We denote the Lyapunov driftby ∆( t ) = E { L ( Q ( t + 1)) − L ( Q ( t )) | Q ( t ) } . (19)From the queue dynamics (3)(6), we have: X c ∈F X n ∈N q M U cn ( t + 1) U cs ( c ) ( t + 1) ≤ X c ∈F q M ( R c ( t ) + U cs ( c ) ( t )) X n ∈N U cn ( t + 1) ≤ µ M q M N K + X c ∈F q M U cs ( c ) ( t ) X n ∈N { U cn ( t ) + ( X i :( n,i ) ∈L µ cni ( t )) + ( X j :( j,n ) ∈L c µ cjn ( t )) − U cn ( t )( X i µ cni ( t ) − X j µ cjn ( t )) } , (20)where we recall that R c ( t ) ≤ µ M and we employ Lemma 1to deduce the second inequality.From (20), we have
12 ( X c ∈F X n ∈N q M ( U cn ( t + 1) U cs ( c ) ( t + 1) − U cn ( t ) U cs ( c ) ( t ))) ≤ X c ∈F (2 N − µ M ) U cs ( c ) ( t ) q M + 12 N Kq M µ M − X c ∈F X n ∈N U cn ( t ) U cs ( c ) ( t ) q M ( X j :( n,j ) ∈L µ cnj ( t ) − X i :( i,n ) ∈L c µ cin ( t )) , (21)where we employ the fact deduced from (4)(5) that P i µ cni ( t ) ≤ and P j µ cjn ( t ) ≤ when n = b ( c ) and P j µ cjn ( t ) ≤ µ M when n = b ( c ) . Note that we use thesummation index i and j interchangeably for convenience ofanalysis.From the queue length dynamics (6) and by employingLemma 1, we have: X c ∈F q M − µ M q M ( U cs ( c ) ( t + 1) − U cs ( c ) ( t ) ) ≤ X c ∈F q M − µ M q M ( µ cs ( c ) b ( c ) ( t ) + R c ( t ) − U cs ( c ) ( t )( µ cs ( c ) b ( c ) ( t ) − R c ( t ))) ≤ K q M − µ M q M µ M − q M − µ M q M X c ∈F U cs ( c ) ( t )( µ cs ( c ) b ( c ) ( t ) − R c ( t )) . (22) From the virtual queue dynamics (9), we have: X c ∈F ( X c ( t + 1) − X c ( t ) ) ≤ X c ∈F ( ρ c R c ( t ) + ( X n ∈N U cn ( t )) − X c ( t )( ρ c R c ( t ) − X n ∈N U cn ( t ))) ≤ µ M X c ∈F ρ c + 12 KN q M − X c ∈F X c ( t ) ρ c R c ( t ) + N q M X c ∈F X c ( t ) . (23)From the virtual queue dynamics (8), we have: X c ∈F ( Z c ( t + 1) − Z c ( t ) ) ≤ X c ∈F ( R c ( t ) + a c − Z c ( t )( R c ( t ) − a c )) ≤ Kµ M + 12 X c ∈F a c − X c ∈F Z c ( t ) R c ( t ) + X c ∈F a c Z c ( t ) . (24)Substituting (21)(22)(23)(24) into the Lyapunov drift (19) andsubtracting V P c E { R c ( t ) | Q ( t ) } from both sides, we thenhave: ∆( t ) − V X c ∈F E { R c ( t ) | Q ( t ) }≤ B + X c ∈F E { R c ( t )( ( q M − µ M ) U cs ( c ) ( t ) q M − X c ( t ) ρ c − Z c ( t ) − V ) | Q ( t ) } + N q M X c ∈F X c ( t ) + X c ∈F a c Z c ( t )+ 12 X c ∈F (2 N − µ M ) U cs ( c ) ( t ) q M − E { q M − µ M q M X c ∈F U cs ( c ) ( t ) µ cs ( c ) b ( c ) ( t )+ X c ∈F X n ∈N U cn ( t ) U cs ( c ) ( t ) q M ( X j :( n,j ) ∈L µ cnj ( t ) − X i :( i,n ) ∈L c µ cin ( t )) | Q ( t ) } . (25)We can rewrite the last term of RHS of (25) by simple algebraas − E { X c ∈F X ( m,n ) ∈L µ cmn ( t ) U cs ( c ) ( t ) q M ( U cm ( t ) − U cn ( t ))+ X c ∈F µ cs ( c ) b ( c ) ( t ) U cs ( c ) ( t ) q M ( q M − µ M − U cb ( c ) ( t )) | Q ( t ) } . (26)Then, the second term and the last term of the RHS of(25) are minimized by the congestion controller (11) and the scheduling policy (12), respectively, over a set of feasible al-gorithms including the stationary randomized algorithm STATintroduced in Lemma 2 and Remark 4. We can substituteinto the second term of RHS of (25) a stationary randomizedalgorithm with admitted arrival rate vector ( r ∗ ǫ,c ) and into thelast term with a stationary randomized algorithm with admittedarrival rate vector ( r ∗ ǫ,c + ǫ ) . Thus, we have: ∆( t ) − V X c ∈F E { R c ( t ) | Q ( t ) }≤ B − V X c ∈F r ∗ ǫ,c − X c ∈F U cs ( c ) ( t ) q M ( ǫ ( q M − µ M ) − N − µ M − X c ∈F ( r ∗ ǫ,c − a c ) Z c ( t ) − X c ∈F ( ρ c r ∗ ǫ,c − N q M ) X c ( t ) . (27)When (15) holds, we can find ǫ > such that ǫ ≤ ρ c r ∗ ǫ,c − N q M ∀ c ∈ F and ǫ ≤ ǫ ( q M − µ M ) − N − µ M q M . Recall that ǫ ′ is defined such that r ∗ ǫ,c ≥ a c + ǫ ′ ∀ c ∈ F . Thus, we have: ∆( t ) − V X c ∈F E { R c ( t ) | Q ( t ) }≤ B − δ X c ∈F ( X c ( t ) + U cs ( c ) ( t ) + Z c ( t )) − V X c ∈F r ∗ ǫ,c , (28)where δ , min { ǫ , ǫ ′ } .We take the expectation with respect to the distribution of Q on both sides of (28) and take the time average on τ =0 , ..., t − , which leads to t E { L ( Q ( t )) } − Vt t − X τ =0 X c ∈F E { R c ( τ ) }≤ B − V X c ∈F r ∗ ǫ,c − δt t − X τ =0 X c ∈F E { X c ( τ ) + U cs ( c ) ( τ ) + Z c ( τ ) } . (29)Since lim sup t →∞ t P t − τ =0 P c E { R c ( τ ) } is bounded above(say, by a constant B R with B R ≤ Kµ M ) and E { L ( Q ( t )) } isnonnegative, by taking limsup of t on both sides of (29), wehave: lim sup t →∞ t t − X τ =0 X c ∈F E { X c ( τ ) + U cs ( c ) ( τ ) + Z c ( τ ) }≤ Bδ + Vδ [lim sup t →∞ t t − X τ =0 X c ∈F E { R c ( τ ) } − X c ∈F r ∗ ǫ,c ] ≤ B ′ δ , (30)which proves (17). By taking liminf of t on both sides of (29), we have lim inf t →∞ t t − X τ =0 X c ∈F E { R c ( τ ) }≥ δV lim inf t →∞ t t − X τ =0 X c ∈F E { X c ( τ ) + U cs ( c ) ( τ ) + Z c ( τ ) }− BV + X c ∈F r ∗ ǫ,c , (31)which proves (16) since the first term of the RHS of (31) isnonnegative. V. F URTHER D ISCUSSIONS
A. Suboptimal Algorithms
Solving MWM optimization problem can be NP-hard de-pending on the underlying interference model as indicatedin [33]. In this section, we introduce a group of suboptimalalgorithms that aim to achieve at least a γ fraction of theoptimal throughput. We denote the scheduling parameters ofsuboptimal algorithms by ( µ c,SUBmn ( t )) . For convenience, wealso denote the scheduling parameters of ALG by ( µ c,OP Tmn ( t )) .Algorithms are called suboptimal if the scheduling parameters ( µ c,SUBmn ( t )) satisfy the following: X m,n µ c ∗ mn ( t ) ,SUBmn ( t ) w mn ( t ) ≥ γ X m,n µ c ∗ mn ( t ) ,OP Tmn ( t ) w mn ( t ) , (32)where γ ∈ (0 , is constant and we recall that c ∗ mn ( t ) and w mn ( t ) are defined in Section IV.A. In addition, the congestioncontroller of suboptimal algorithms is the same as that of ALG (11).Following the same analysis of
ALG , Proposition 1 holds forsuboptimal algorithms, i.e., the queue backlogs are boundedabove by q M , and we derive the following theorem: Theorem 2:
Given that q M > N − µ M γǫ + µ M and ρ c > N q M γr ∗ ǫ,c ∀ c ∈ F , ∃ ǫ > s.t. γr ∗ ǫ,c ≥ a c + ǫ ∀ c ∈ F , (33)a suboptimal algorithm ensures that the virtual queues have atime-averaged bound: lim sup t →∞ t t − X τ =0 X c ∈F E { U cs ( c ) ( τ ) + X c ( τ ) + Z c ( τ ) } ≤ ¯ Bδ , (34)where ¯ B , B + γV B R . In addition, a suboptimal algorithmcan achieve a throughput lim inf t →∞ t t − X τ =0 X c ∈F E { R c ( τ ) } ≥ γ X c ∈F r ∗ ǫ,c − BV . (35)
Proof:
The proof is provided in Appendix A.
Remark 5 : From Theorem 2, given an arbitrarily small ǫ ,we show that a suboptimal algorithm can at least achievea throughput arbitrarily close to a fraction γ of the optimal results P c ∈F r ∗ ǫ,c . Suboptimal algorithms include the well-known Greedy Maximal Matching (GMM) algorithm [34]with γ = as well as the solutions to the maximumweighted independent set (MWIS) optimization problem suchas GWMAX and GWMIN proposed in [35] with γ = ,where ∆ is the maximum degree of the network topology G .The delay bound and throughput tradeoff in Theorem 1 stillhold in Theorem 2. B. Arbitrary Arrival Rates at Transport Layer
Note that in the previous model description, we assumedthat the flow sources are constantly backlogged, that is, thecongestion controller (11) can always guarantee R c ( t ) = µ M when q M − µ M q M U cs ( c ) ( t ) − X c ( t ) ρ c − Z c ( t ) − V ≤ . In thissubsection, we present an optimal algorithm when the flowshave arbitrary arrival rates at the transport layer.Let A c ( t ) denote the arrival rate of flow c packets atthe beginning of the time slot t at the transport layer. Weassume that A c ( t ) is i.i.d. with respect to t with mean λ c . Forsimplicity of analysis, we assume ( λ c ) to be in the exterior ofthe capacity region Λ so that a congestion controller is neededand we assume that A c ( t ) is bounded above by µ M ∀ c ∈ F . Let L c ( t ) denote the backlog of flow c data at the transportlayer which is updated as follows: L c ( t + 1) = min { [ L c ( t ) + A c ( t ) − µ cs ( c ) b ( c ) ( t )] + , L M } , (36)where L M ≥ is the buffer size for flow c at the transportlayer. Note that we have L M = 0 and L c ( t ) = 0 if there isno buffer for flow c at the transport layer.Following the idea introduced in [5], we construct a virtualqueue Y c ( t ) and an auxiliary variable v c ( t ) for each virtualinput rate R c ( t ) , with queue dynamics for Y c ( t ) as follows Y c ( t + 1) = [ Y c ( t ) − R c ( t )] + + v c ( t ) , (37)where initially we have Y c (0) = 0 . The intuition is that v c ( t ) serves as the function of R c ( t ) in congestion controller (11)and we note that when Y c ( t ) is stable, we have r c ≥ v c , where v c is the time average rate for v c ( t ) , recalling that r c is thetime average rate for R c ( t ) . Thus, when Y c ( t ) and U cs ( c ) ( t ) arestable, if we can ensure the value P c v c is arbitrarily closeto the optimal value P c r ∗ ǫ,c , then so is the throughput P c µ c since µ c ≥ r c ≥ v c .Now we provide the optimal algorithm for arbitrary arrivalrates at the transport layer:
1) Congestion Controller : min ≤ v c ( t ) ≤ µ M v c ( t )( ηY c ( t ) − V ) , (38) min R c ( t ) R c ( t )( q M − µ M q M U cs ( c ) ( t ) − ηY c ( t ) − X c ( t ) ρ c − Z c ( t )) (39)s.t. ≤ R c ( t ) ≤ min { L c ( t ) + A c ( t ) , µ M } where η > is a weight associated with the virtual queue Y c ( t ) . Note that (38) and (39) can be solved independently. Note that our analysis also works for the case when A c ( t ) is boundedabove by some constant A M ∀ c ∈ F , where A M ≥ µ M . Specifically, when ηY c ( t ) − V ≥ , v c ( t ) is set to zero;Otherwise, v c ( t ) = µ M . When q M − µ M q M U cs ( c ) ( t ) − ηY c ( t ) − X c ( t ) ρ c − Z c ( t ) ≥ , R c ( t ) is set to zero; Otherwise, R c ( t ) = min { L c ( t ) + A c ( t ) , µ M } .
2) Scheduling Policy : The scheduling algorithm is the sameas that of
ALG provided in Section IV.B, except for the up-dated constraints: ≤ µ cs ( c ) b ( c ) ( t ) ≤ min { L c ( t )+ A c ( t ) , µ M } .Since the scheduling policy is not changed, Proposition 1still holds. And we present a theorem below for the perfor-mance of the algorithm: Theorem 3:
Given that q M > N − µ M ǫ + µ M and ρ c > N q M r ∗ ǫ,c ∀ c ∈ F , the algorithm ensures that the virtual queues have a time-averaged bound: lim sup t →∞ t t − X τ =0 X c ∈F E { U cs ( c ) ( τ )+ X c ( τ )+ Z c ( τ )+ Y c ( τ ) } ≤ B δ ′ , where B , B + Kηµ M + V B R and δ ′ is constant positivenumber. In addition, the algorithm can achieve a throughput lim inf t →∞ t t − X τ =0 X c ∈F E { v c ( τ ) } ≥ X c ∈F r ∗ ǫ,c − B V , where B , B + Kηµ M . Proof:
The proof is provided in Appendix B.Theorem 3 shows that optimality is preserved and O ( ǫ ) delay scaling is kept. C. Employing Delayed Queue Backlog Information
Recall that in
ALG , congestion controller (11) is performedat the transport layer and link weight assignment in (13) isperformed locally at each link. Thus, in order to account for thepropagation delay of queue information, we employ delayedqueue backlog of ( X c ( t )) in (11) and employ delayed queuebacklog of ( U cs ( c ) ( t )) for links in L in (13). Specifically, werewrite (11) in ALG as: min R c ( t )( ( q M − µ M ) U cs ( c ) ( t ) q M − X c ( t − T ) ρ c − Z c ( t ) − V ) , (40)where T is an integer number that is larger than the maximumpropagation delay from a source to a node, and we rewrite (13)as: w cmn ( t ) = U cs ( c ) ( t − T ) q M [ U cm ( t ) − U cn ( t )] , if ( m, n ) ∈ L ,U cs ( c ) ( t ) q M [ q M − µ M − U cb ( c ) ( t )] , if ( m, n ) = ( s ( c ) , b ( c )) , , otherwise. (41)Proposition 1 still holds and we present a theorem for thescheduling algorithm using delayed queue backlog informa-tion, which maintains the throughput optimality and O ( ǫ ) scaling in delay bound: Theorem 4:
Given that q M > N − µ M ǫ + µ M and ρ c > N q M r ∗ ǫ,c ∀ c ∈ F , the algorithm ensures that the virtual queues have a time-averaged bound: lim sup t →∞ t t − X τ =0 X c ∈F E { U cs ( c ) ( τ ) + X c ( τ ) + Z c ( τ ) } ≤ B δ , where B , B + V B R and B , B + KN µ M T + N q M T µ M ρ c + Kρ c µ M T . In addition, the algorithm canachieve a throughput lim inf t →∞ t t − X τ =0 X c ∈F E { R c ( τ ) } ≥ X c ∈F r ∗ ǫ,c − B V .
Proof:
The proof is provided in Appendix C.On employing delayed queue backlogs, we can extend thecentralized optimization problem (12) to distributed implemen-tations with methods introduced in Remark 3.
D. Arbitrary Link Capacities and Arbitrary Interference Mod-els with Fading Channels
Recall that in the model description in Section III, the linkcapacity is assumed constant (one packet per slot) and node-exclusive model is employed. In this subsection, we extend themodel to arbitrary link capacities and arbitrary interferencemodels with fading channels of finite channel states. Thus,instead of (4), we have ( µ cmn ( t )) ( m,n ) ∈L ∈ I ( t ) , where I ( t ) is the feasible activation set for time slot t determined by theunderlying interference model and current channel states, withlink capacity constraints P c ∈F µ cmn ( t ) ≤ l mn , where l mn isthe arbitrarily chosen link capacity for a link ( m, n ) ∈ L .We define l n , max ( µ cmn ( t )) ∈ I ( t ) P c ∈F P m :( m,n ) ∈L µ cmn ( t ) .Note that it is clear that l n ≤ P m :( m,n ) ∈L l mn . Then wecan update the optimization (12) and weight assignment (13),respectively, as follows: max ( µ cmn ( t )) X m,n µ c ∗ mn ( t ) mn ( t ) w mn ( t ) s.t. ( µ cmn ( t )) ( m,n ) ∈L ∈ I ( t ) and µ s ( c ) b ( c ) ( t ) ≤ µ M ∀ c ∈ L .w cmn ( t ) = U cs ( c ) ( t ) q M [ U cm ( t ) − U cn ( t ) − l n ] , if ( m, n ) ∈ L ,U cs ( c ) ( t ) q M [ q M − µ M − U cb ( c ) ( t )] , if ( m, n ) = ( s ( c ) , b ( c )) , , otherwise.It is not difficult to check that Proposition 1 still holdswith q M ≥ max { max n ∈N l n , µ M } and Theorem 1 holdswith a different definition of constant B . The above modifiedalgorithm can be further extended to solve power allocationproblems, where we refer interested readers to our recent work[41]. GC FB HED A
Fig. 1. Network topology for simulations
VI. N
UMERICAL R ESULTS
In this section, we present the simulation results for theproposed optimal algorithm
ALG . Simulations are run inMatlab 2009A with results averaged over time slots. Inthe network topology illustrated in Figure 1, there are threesource-destination pairs ( A, G ) , ( D, E ) and ( F, H ) with samePoisson arrival rates and µ M = 2 . The required minimum datarate for the three flows are all set to . . We denote by BP theback-pressure scheduling algorithm in [1] with a congestioncontroller in [5], and denote by Finite Buffer the cross-layeralgorithm developed in [31] with buffer size equal to the queuelength limit q M . Note that it is shown in simulation results in[31] that Finite Buffer algorithm ensures much smaller internalqueue length (of nodes excluding the source node) than BPalgorithm (and queue length is related to delay performance).We set the control parameter V = 1000 , where in simulationswe find that a higher V cannot further improve the throughput.We first illustrate in Table I the throughput optimality of ALG when the sources are constantly backlogged. We loosenthe delay constraint as ρ c = 30 q M . As we increase the controlparameter q M , the ALG achieves a throughput approaching thethroughput of BP algorithm which is known to be optimal. Wealso note that this approximation in throughput results in worseaverage end-to-end delay performance, which complies withRemark 1.We then illustrate the throughput and delay tradeoff for boththe
ALG and its corresponding suboptimal GMM algorithm inFigure 2 for the case of arbitrary arrival rates at transport layerwith L M = 0 , where we set q M = 5 and ρ c = 50 for each flow c . Note that this pair of q M and ρ c shows that the bound in(15) is actually quite loose, and thus our algorithm can achievebetter delay performance than stated in (15). Figure 2 showsthat the average end-to-end delay under ALG is well belowthe constraint ( ρ c = 50 ) and lower than that under BP andFinite Buffer algorithms. The throughput of ALG is close to(although lower than) that of the optimal BP algorithm whenarrival rates are small ( ≤ . ). Specifically, when the arrivalrate is . , ALG achieves a throughput more than theGMM algorithm and . less than BP algorithm, with anaverage end-to-end delay . less than the BP algorithm.In the large-input-rate-region ( > . ), we also observe that thedelay in both the BP and Finite Buffer algorithm violates thedelay constraints. In addition, in the above illustrated scenarioswith backlogged and arbitrary arrival rates, the minimum TABLE IT
HROUGHPUT PERFORMANCE OF
ALG
WHEN SOURCES ARE BACKLOGGED AT THE TRANSPORT LAYER
ALG ( ρ c = 150 ) ALG ( ρ c = 300 ) ALG ( ρ c = 3000 ) ALG ( ρ c = 30000 ) BPThroughput (sum for three flows) . . . . . End-to-end delay (averaged over three flows) .
76 131 .
47 1 . × . × . × arrival rates and average end-to-end delay requirements aresatisfied for individual flows under ALG . As a side note, theaverage end-to-end delay in all four algorithms in Figure 2first decreases, which can be explained by the intuition that allthe algorithms are based on back-pressure of links (i.e., queuebacklog difference of links) and the queue backlog differencetends to be larger for each hop with a larger arrival rate. Whenarrival rate further increases, congestion level becomes highersince more packets are admitted into the network. arrival rate t h r o u g hpu t arrival rate a v e r ag ee nd - t o - e ndd e l a y ALGBPGMMFinite Buffer
ALGBPGMMFinite BufferDelay constraint
Fig. 2. Throughput and delay tradeoff under Alg. with performancescompared to Finite Buffer algorithm and BP algorithm, with varying arrivalrates at the transport layer.
VII. C
ONCLUSIONS AND F UTURE WORKS
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ACMSIGMETRICS’09 ∼ xued/distributed.pdf[41] D. Xue and E. Ekici, “Optimal Power Allocation in Multi-Hop WirelessNetworks with Finite Buffers”, in Proc. of IEEE International Conferenceon Communications (ICC 2011) , June 2011. A PPENDIX AP ROOF OF T HEOREM Proof:
Let ∆ SUB ( t ) denote the corresponding Lyapunovdrift of a suboptimal algorithm which takes the same form as(19). By analyzing (25)(26) which also hold for suboptimalalgorithms, we note that the second term of RHS of (25) isalways non-positive ensured by the congestion controller (11).Employing (32) to (25)(26), we derive the following ∆ SUB ( t ) − V X c ∈F E { R c ( t ) | Q ( t ) }≤ B + γ X c ∈F E { R c ( t )( ( q M − µ M ) U cs ( c ) ( t ) q M − X c ( t ) ρ c − Z c ( t ) − V ) | Q ( t ) } + N q M X c ∈F X c ( t ) + X c ∈F a c Z c ( t )+ 12 X c ∈F (2 N − µ M ) U cs ( c ) ( t ) q M − γ E { q M − µ M q M X c ∈F U cs ( c ) ( t ) µ c,SUBs ( c ) b ( c ) ( t )+ X c ∈F X n ∈N U cn ( t ) U cs ( c ) ( t ) q M ( X j µ c,SUBnj ( t ) − X i µ c,SUBin ( t )) | Q ( t ) } , (42)Following the steps in proving (27), we have from (42) ∆( t ) SUB − V X c ∈F E { R c ( t ) | Q ( t ) }≤ B − V γ X c ∈F r ∗ ǫ,c − X c ∈F U cs ( c ) ( t ) q M ( γǫ ( q M − µ M ) − N − µ M − X c ∈F ( γr ∗ ǫ,c − a c ) Z c ( t ) − X c ∈F ( γρ c r ∗ ǫ,c − N q M ) X c ( t ) . (43)Employing the conditions (33) and following the steps inproving (30) and (31), we can prove Theorem 2.A PPENDIX BP ROOF OF T HEOREM ( θ c ) introduced in Lemma 2 and givenflow c at node n , recall that ( A c ( t )) is i.i.d. with mean ( λ c ) and ( λ c ) > ( θ c ) element-wise. The flow control forSTAT can be given as: Admit µ c,ST ATs ( c ) b ( c ) ( t ) = A c ( t ) w.p. θ c λ c ;otherwise, µ c,ST ATs ( c ) b ( c ) ( t ) = 0 . Then E { µ c,ST ATs ( c ) b ( c ) ( t ) } = θ c , ∀ t .Now take v ST ATc ( t ) = R ST ATc ( t ) = µ c,ST ATs ( c ) b ( c ) ( t ) ∀ c ∈ F .Then we also have E { v ST ATc ( t ) } = E { R ST ATc ( t ) } = θ c .Note that R ST ATc ( t ) ≤ A c ( t ) ≤ min { L c ( t ) + A c ( t ) , µ M } and v ST ATc ( t ) ≤ µ M .Now we present the proof. Proof:
We define the Lyapunov function as L ( Q ′ ( t )) = L ( Q ( t )) + η P c ∈F Y c ( t ) and the Lyapunov drift as ∆ ′ ( t ) = E { L ( Q ′ ( t + 1)) − L ( Q ′ ( t )) | Q ′ ( t ) } , where Q ′ ( t ) =( Q ( t ) , ( Y c ( t ))) . From the virtual queue dynamics (37) and Lemma 1, we have η X c ∈F ( Y c ( t + 1) − Y c ( t ) ) ≤ η X c ∈F ( R c ( t ) + v c ( t ) − Y c ( t )( R c ( t ) − v c ( t ))) ≤ Kηµ M − X c ∈F ηY c ( t )( R c ( t ) − v c ( t )) . (44)Following the steps in deriving (25)(26), we have ∆ ′ ( t ) − V X c ∈F E { v c ( t ) | Q ′ ( t ) }≤ B + X c ∈F E { v c ( t )( ηY c ( t ) − V ) | Q ′ ( t ) } + X c ∈F E { R c ( t )( ( q M − µ M ) U cs ( c ) ( t ) q M − ηY c ( t ) − X c ( t ) ρ c − Z c ( t )) | Q ′ ( t ) } + N q M X c ∈F X c ( t ) + X c ∈F a c Z c ( t )+ 12 X c ∈F (2 N − µ M ) U cs ( c ) ( t ) q M − E { X c ∈F X ( m,n ) ∈L µ cmn ( t ) U cs ( c ) ( t ) q M ( U cm ( t ) − U cn ( t ))+ X c ∈F µ cs ( c ) b ( c ) ( t ) U cs ( c ) ( t ) q M ( q M − µ M − U cb ( c ) ( t )) | Q ′ ( t ) } , (45)The second term, third term and the last term of the RHSof (45) are minimized by the congestion controller (38),(39) and the scheduling policy (12), respectively, over a setof feasible algorithms including the stationary randomizedalgorithm STAT. Substitute into the second term of RHS of(45) a stationary randomized algorithm with admitted arrivalrate vector ( r ∗ ǫ,c − ǫ ′ ) , the third term a stationary randomizedalgorithm with admitted arrival rate vector ( r ∗ ǫ,c ) and the lastterm a stationary randomized algorithm with admitted arrivalrate vector ( r ∗ ǫ,c + ǫ ) . Then, following the steps in provingTheorem 1, we can prove Theorem 3.A PPENDIX CP ROOF OF T HEOREM Proof:
According to queue dynamics (6)(9), we obtain U cs ( c ) ( t ) − µ M T ≤ U cs ( c ) ( t − T ) ≤ U cs ( c ) ( t ) + µ M T,X c ( t ) − N q M T ≤ X c ( t − T ) ≤ X c ( t ) + ρ c µ M T. (46) Employing the above inequalities to (25)(26), we have ∆( t ) − V X c ∈F E { R c ( t ) | Q ( t ) }≤ B + X c ∈F E { R c ( t )( ( q M − µ M ) U cs ( c ) ( t ) q M − X c ( t − T ) ρ c − Z c ( t ) − V ) | Q ( t ) } + N q M X c ∈F X c ( t ) + X c ∈F a c Z c ( t ) + Kρ c µ M T + 12 KN µ M T + 12 X c ∈F (2 N − µ M ) U cs ( c ) ( t ) q M − E { X c ∈F X ( m,n ) ∈L µ cmn ( t ) U cs ( c ) ( t − T ) q M ( U cm ( t ) − U cn ( t ))+ X c ∈F µ cs ( c ) b ( c ) ( t ) U cs ( c ) ( t ) q M ( q M − µ M − U cb ( c ) ( t )) | Q ( t ) } . The second term and the last term of the RHS of the aboveinequality are minimized by the congestion controller (40)and the scheduling policy (12) with weight assignment (41),respectively, over a set of feasible algorithms including thestationary randomized algorithm STAT. Substitute into thesecond term of RHS a stationary randomized algorithm withadmitted arrival rate vector ( r ∗ ǫ,c ) and the last term a station-ary randomized algorithm with admitted arrival rate vector ( r ∗ ǫ,c + ǫ ))