Joint Rate Control and Scheduling for Real-Time Wireless Networks

Transcription

1 Joit Rate Cotrol ad Schedulig for Real-Time Wireless Networks Shuai Zuo, I-Hog Hou, Tie Liu, Aathram Swami, ad Prithwish Basu Abstract This paper studies wireless etworks with multiple real-time flows that have striget reuiremets o both per-packet delay ad log-term average delivery ratio. Each flow dyamically adjusts its traffic load based o its observatio of etwork status. Whe the reuiremets of per-packet delay ad delivery ratio are satisfied, each flow obtais some utility based o its traffic load. We aim to desig joit rate cotrol ad schedulig policies that maximize the total utility i the system. We first show that the problem of maximizig total utility ca be formulated as a submodular optimizatio problem with expoetially may costraits. We the propose two simple distributed policies that reuire almost o coordiatio betwee differet etities i the etwork. The total utilities uder these two policies ca be made arbitrarily close to the theoretical upper-boud. Extesive simulatios also show that they achieve much better performace tha state-ofthe-art policies. I. INTRODUCTION The demads for real-time applicatios such as video streamig ad olie gamig i wireless etwork have bee icreasig drastically over the decade. Accordig to a recet Cisco report [], mobile video traffic will icrease eleve-fold betwee 5 ad, ad accout for three uarters of the overall mobile traffic by. These applicatios ca lead to severe etwork cogestio i wireless etworks. Further, real-time applicatios have some uiue features, such as hard per-packet delay reuiremet ad delivery ratio reuiremet, that separate them from traditioal o-real-time applicatios. Directly employig stadard cogestio cotrol ad resource allocatio policies without explicitly addressig these features ca thus lead to poor etwork performace. I this paper, we study wireless systems that cosist of multiple real-time flows where each flow ca dyamically adjust its traffic load. Each flow has a hard per-packet delay reuiremet to meet real-time costrait, as well as a packet delivery ratio reuiremet to esure data itegrity. Whe both delay ad delivery ratio reuiremets are satisfied, each flow obtais some utility based o its traffic load. We ca cosider the applicatio of video call, This material is based upo work supported by, or i part by, the U. S. Army Research Laboratory ad the U. S. Army Research Office uder cotract/grat umber W9NF Shuai Zuo, I-Hog Hou ad Tie Liu are with the Departmet of ECE, Texas A&M Uiversity. s: {zuosh9, ihou, tieliu}@tamu.edu Aathram Swami is with Army Research Labs. aathram.swami.civ@mail.mil Prithwish Basu is with Raytheo BBN Techologies. pbasu@bb.com such as Skype ad Facetime, as a motivatig example for our problem. I video call, the applicatio ca chage its traffic load by chagig its video ecodig scheme, where better video uality leads to larger traffic load. Packets of video call eed to be delivered withi a certai delay boud, typically aroud ms, or ed users may experiece severe coversatio iterruptios due to delay. A high ratio of packets, typically aroud 9% - 95%, eed to be successfully delivered to esure video frames ca be decoded ad played smoothly. We aim to desig joit rate cotrol ad schedulig policies that achieve the maximum system-wide total utility. We propose a aalytical model that icorporates the aforemetioed features of real-time applicatios ad the stochastic ature of wireless trasmissios. The problem of maximizig total utility ca the be modeled as a submodular optimizatio problem with expoetially may costraits. While there exists polyomialtime algorithms that solve the submodular optimizatio problem, they still icur high computatioal complexity ad rely o cetralized algorithms. We the propose two distributed policies where traffic load is determied by each flow, packet schedulig is determied by the access poit (AP), ad there is almost o coordiatio betwee flows ad the AP. These two policies have very low complexity ad ca be easily implemeted. We prove that both distributed policies satisfy all real-time applicatio reuiremets. Moreover, both policies achieve a total utility that ca be made arbitrarily close to the theoretical upper-boud. Our proposed policies are further evaluated by s- simulatios. We compare our policies agaist three other policies based o state-of-the-art mechaisms. Simulatio results demostrate that our policies achieve much better performace tha other policies uder a wide rage of differet scearios. The rest of the paper is orgaized as follows: Sectio II summarizes some existig studies. Sectio III itroduces our aalytical model. Sectio IV shows that the problem ca be modeled as a submodular optimizatio problem, ad itroduces a cetralized algorithm to solve it. Sectio V describes our two proposed distributed policies. Sectio VI aalyzes the performace of our policies ad proves that they achieve ear-optimal total utility. Sectio VII demostrates our simulatio results. Fially, Sectio VIII cocludes the paper.

2 II. RELATED WORK Sice the pioeerig work of Kelly [], the study of cotrollig etworks usig a optimizatio approach has cotiued for over two decades. Li ad Shroff [4] proposed a joit rate cotrol ad schedulig algorithm for o real-time applicatios i multi-hop wireless etworks. Similarly, Eryilmaz ad Srikat [4] desiged ad aalyzed a joit algorithm with rate cotrol, routig ad schedulig for o real-time cliets i wireless etworks. Further, Lua et al. [5] studied the utility maximizatio problem ad proposed a joit cotrol algorithm for video streamig applicatios. However, these works do ot cosider the deadlie reuiremets i real-time applicatios. Withi this decade, there have bee a umber of works aimed at the cotrol of wireless applicatios with striget deadlie reuiremets. Hou [6] ad Hou et al. [7] proposed feasible schedulig algorithms for ureliable wireless etworks subject to deadlie costraits. Li ad Eryilmaz [], Xiog et al. [8] ad Mao [6] explored the joit cotrol problem for multi-hop wireless etworks where cliets have deadlie costraits. However, these studies assumed that the trasmissio rates are fixed, whereas the trasmissio rate for real-time applicatios may chage i respose to the etwork state ad i tur affect cotrol decisios. Che et al. [] studied the sigle user wireless system with real-time video trasmissio ad formulated this problem as a Markov Decisio Problem (MDP). However, such MDP approach suffers from the curse of dimesioality ad is thus ifeasible for systems with multiple users. Kag et al. [] also focused o a sigle user system ad their work does t provide a aalysis of optimality. Huag et al. [] ad Zhao ad Li [] studied video streamig wireless etworks with multiple deadlie costraits users. However, [] did t provide the optimality guaratees for their heuristic algorithm. The optimality of algorithms i [] is based o the assumptio that the system icludes a large umber of chaels ad users. III. SYSTEM MODEL AND PROBLEM FORMULATION We exted the model i [7] to icorporate dyamic rate cotrol policies for real-time applicatios. Cosider a wireless system with oe access poit (AP) servig M wireless cliets. Each cliet is associated with a dowlik real-time flow that ca adaptively adjust its data rate based o etwork cogestio. Time is slotted ad idexed as t =,,..., where the duratio of a time slot is chose to be the amout of time eeded to trasmit a data packet ad a ACK. Wireless trasmissios ca be ureliable, ad we say that each trasmissio for cliet m is successful with probability p m. By reuirig ACKs for all trasmissios, the AP has the feedback iformatio about whether a trasmissio is successful, ad may retrasmit the same packet if previous trasmissios fail. I the dowlik sceario, sources of real-time flows have reliable ad wired coectios to the AP, ad ca therefore obtai feedback iformatio, such as chael state ad whether a trasmissio is successful from the AP. Each real-time flow has some striget per-packet delay boud. Specifically, we say that the flow of cliet m geerates packets periodically with period T m, ad the delay boud of m is the same as its period. Therefore, if a period of m starts at time t, the all packets of m geerated at time t eed to be delivered o or before the ed of the period at time t + T m. Packets that are ot delivered withi their respective delay bouds are dropped from the system. Our theoretical aalysis focuses o the special case where all cliets have the same T m T, ad all their first periods start at time t =. We ca the group time slots ito itervals, where each iterval cosists of all time slots i (kt,(k +)T], for some iteger k. All flows geerate packets at the begiig of each iterval, ad all these packets are either delivered before the ed of the iterval, or dropped from the system at the ed of the iterval. I Sectio VII, we will demostrate that our proposed policies achieve good performace eve whe differet cliets have differet periods. We cosider real-time applicatios that ca adjust their data rates, such as video applicatios that ca choose amog differet resolutios. At the begiig of each iterval, the flow of cliet m geerates r m packets, where r m lies i [,R m ]. The uality of experiece of cliet m is the determied by r m, ad we say it receives utility U m (r m ), where U m ( ) is assumed to be a icreasig ad strictly cocave fuctio. O the other had, cliet m reuires that at least a portio m of its packets are delivered o time. Whe its flow geerates r m packets per iterval, cliet m reuires that its throughput be at least r m m packets per iterval. We ote that, i our model, deliverig more tha m of cliet m s packets does ot further improve its utility. This assumptio is based o existig studies o Skype video uality [], [9], where they show that the video uality has a sigificat drop whe packet loss rate exceeds 6 %, but stays rather stable as log as the packet loss rate is below 6%. We aim to fid policies that joitly cotrol the data rate of each flow ad the packet schedulig decisios of the AP so as to maximize the total utility, m U m(r m ), while esurig that the throughput of each cliet m is at least r m m. I order to simplify the formulatio of this problem, we first divide each cliet ito R m sub-cliets, where each sub-cliet ca geerate at most oe packet i each iterval. We set p = p m ad = m if is a sub-cliet of m. We say that the i-th sub-cliet of m geerates a packet if ad oly if r m i. Further, whe a sub-cliet who is the i-th sub-cliet of m geerates a packet, it receives a utility u i := U m (i) U m (i ). Let N be the set of all sub-cliets, we the have m U m(r m ) = N u (sub-cliet geerates a packet), where ( ) is the idicator fuctio. Our problem is to fid the optimal subset O of sub-cliets that maximizes O u, while esurig that the throughput of each sub-cliet O is at least.

3 Hou et al. [7] have derived a ecessary ad sufficiet coditio for checkig whether it is feasible to esure a throughput of for each O. Theorem : A throughput of for each O is feasible if ad oly if F(S), S O, () p S where F(S) := E[mi{T, S γ }], ad γ is a geometric radom variable with mea p. By defiitio, F(S) is icreasig i the sese that F(S ) F(S ) if S S. The coditio () is the euivalet to ( geerates a packet) F(S), S N. p S Let x := p ( geerates a packet). The problem of maximizig total utility ca the be formulated as Max u p x s.t. x F(S), S S x {, p }, We relax the last costrait to trasform this iteger programmig problem ito a liear programmig problem: Max u p x () s.t. x F(S), S () S x p, (4) I this paper, we aim to desig joit rate cotrol ad schedulig policies that solve problem ()-(4). IV. A CENTRALIZED GREEDY ALGORITHM I this sectio, we propose a cetralized greedy algorithm that solves problem ()-(4) optimally usig techiues developed for submodular optimizatio. Let Ω be a oempty fiite set ad R be the set of real umbers. A real-valued fuctio f : Ω R is called submodular over the subsets of Ω if, for X Y Ω, ad / Y, we have f(x {}) f(x) f(y {}) f(y) It has bee show that F(S) := E[mi{T, S γ }] is a submodular fuctio. Lemma (Theorem, [8]): F(S) is a submodular fuctio over all subsets of N. Sice F(S) is a submodular fuctio, the problem ()-(4) is called a submodular optimizatio problem with vector costraits [5], ad there exists a greedy algorithm that solves the problem. The greedy algorithm works as follows: it first sorts all sub-cliets i the order of u p up... It calculates x,x,...x N i this order oe by oe. The assigmet of x is accordig to x = ˆF({,,...,}) ˆF({,,..., }), where ˆF( ) is defied as ˆF(S) = mi {F(Z)+ }. Z S p S\Z The greedy algorithm is summarized i Alg.. It has bee show that the greedy algorithm fids the optimal solutio uder some mild coditios. Algorithm Greedy algorithm : x, : Sort all sub-cliets such that up up... : 4: x = ˆF({,,...,}) ˆF({,,..., }) 5: +, go to step 4 Theorem (Theorem., [5]): Let x,x,... be the optimal solutio to ()-(4). If N x = F(N), the the greedy algorithm produces the optimal solutio. Accordig to [9], give F(Z) for every subset Z S, ˆF(S) ca be computed i O(( N 5 EO+ N 6 )log N ) time, where EO is the time to evaluate fuctio F(Z). However, calculatig F(Z) is difficult sice it does ot have a simple closed-form expressio. Moreover, the greedy algorithm is a cetralized algorithm ad it does ot provide ay solutio to packet schedulig. I most real-time applicatios, rate cotrol occurs at the cliet side while the schedulig of the trasmissio of packets aturally occurs at the AP side. Therefore, distributed solutios that oly reuires limited coordiatio betwee the AP ad cliets are highly desirable. V. DISTRIBUTED POLICIES FOR JOINT RATE CONTROL AND SCHEDULING I this sectio, we propose two joit rate cotrol ad schedulig policies. Oe is a fully distributed policy that reuires almost o iformatio exchage betwee the AP ad cliets. The other policy is similar to the first oe, but employs a global variable to achieve a better covergece rate. Both of our policies have two compoets: First, each cliet m determies r m, which is the umber of packets it geerates i each iterval. This is euivalet to determiig the values of x i problem ()-(4). We call this compoet the rate cotrol problem. Secod, the AP determies its schedulig decisios so as to provide a throughput of m r m to each cliet m. We call the secod compoet the schedulig problem. We itroduce variable x (k) to capture the cliet s solutio to the rate cotrol problem, defied as follows: At the begiig of each iterval k, The cliet sets the value x (k) {, p } for its correspodig sub-cliet. If x (k) = p, sub-cliet geerates a real packet i

4 4 the k-th iterval. If x (k) =, sub-cliet geerates a dummy packet. Therefore, the flow of cliet m geerates a total umber of ( pm m :sub-cliets of m x (k)) real packets, ad (R m pm m :sub-cliets of m x (k)) dummy packets. We reuire m to geerate dummy packets so that it always has R m packets for trasmissios K k= i each iterval. Defie x limif E[x(k)] K K as the log-term average of x (k). The log-term average umber of real packets geerated by sub-cliet is the p x, ad it reuires a throughput of p x. The AP employs a schedulig policy to determie the trasmissio of packets i each iterval. Let w (k) be the umber of time slots that the AP used for trasmittig the packet for sub-cliet i iterval k. Due to ureliable wireless trasmissios, the value of w (k) depeds o both the employed schedulig policy ad the radom packet loss evets i the iterval. Defie w K k= limif E[w(k)] K K. Hou et al. [7] have established the followig relatios betwee w ad throughput of. Lemma 4 (Lemma, [7]): The throughput of subcliet is at least p x if ad oly if w x. Give {x (k)} ad {w (k)}, we defie a deficiecy for each sub-cliet as follows: D (k +) = D (k)+x (k) w (k), (5) with D () =, for all. By Lemma 4, the throughput of is at least p x if limsup k D (k) k. A. Our first rate cotrol ad schedulig policy, amed as the fully distributed policy, employs the deficiecy give i (5) ad it works as follows. Rate Cotrol Policy of Cliets: Let V > be a cotrol parameter. At the begiig of each iterval k, the AP sets x (k) to p, ad geerates a real packet for sub-cliet,. It sets x (k) to ad geerates a dummy packet if u < D(k) p V. Schedulig Policy of the AP: At the begiig of each iterval, the AP sorts all sub-cliets accordig to their deficiecies such that D (k) D (k)... ad schedules packet trasmissios for the sub-cliets accordig to this order. I particular, i each time slot i the iterval, the AP schedules the sub-cliet with the largest D (k) amog those whose packets have ot bee delivered ad D (k) >. If the AP has delivered all packets for all sub-cliets with D (k) >, the the AP idles for the remaiig time slots i the iterval. May real-time applicatios geerate packets that have depedecies. For example, i scalable video codig, a if u D(k) p V We ote that our schedulig policy is ot work-coservig i the sese that it does ot trasmit ay packets with D (k), eve whe there are o packets with D (k) > to trasmit. This artificial costrait is eeded for the performace aalysis i Sectio VI. Oe ca obviously modify the policy by allowig it to schedule packets with D (k). Whether such a modified policy remais optimal, ad whether it improves covergece speed, is a iterestig uestio for future research. video frame is separated ito a base layer ad several ehacemet layers. Ehacemet layers caot be decoded without the base layer, while the base layer ca be decoded idepedetly. It is the reuired to trasmit the base layer before trasmittig ehacemet layers. For these applicatios, whe the AP schedules a sub-cliet of cliet m, it trasmits the first udelivered packet of m. Algorithm Fully distributed policy : D, : for each iterval do : x p {u D p }, V 4: w, 5: Sort all sub-cliets such that D D... 6: j 7: for each time slot i the iterval do 8: if D j > the 9: Trasmit for j : w j w j + : if the trasmissio is successful the : j j + : ed if 4: ed if 5: ed for 6: D D +x w, 7: ed for Our fully distributed policy is summarized i Alg.. I each iterval, the AP computes D (k) ad x (k) for all sub-cliets. Coseuetly, each flow geerates packets accordig to it s correspodig x s. After that, the AP oly eeds to sort packet trasmissios for all sub-cliets accordig to their deficiecies. The complexity o the AP s side is O( N log N ). I the ext sectio, we will demostrate that our policy coverges to a poit that ca be made arbitrarily close to the optimum solutio to ()- (4). B. We propose a secod policy, which is called the policy with a small overhead. I this policy, the AP computes Φ(k) D (k), ad broadcasts its value to all flows. Each flow the determies its rate based o both Φ(k) ad D (k). The policy is described as the followig: Rate Cotrol Policy of Cliets: At the begiig of each iterval k, the AP sets x (k) to p, ad geerates a real packet, if u Φ(k) p V D(k) p V. Schedulig Policy of the AP: The schedulig policy is exactly the same as the fully distributed policy. We ote that the policy with a small overhead is very similar to the fully distributed policy. The oly differece is that the rate cotrol policy i the policy with a small overhead ivolves both Φ(k) ad D (k). Ituitively, the value of Φ(k) reflects the cogestio of the whole etwork. By takig Φ(k) ito accout, the policy with a small overhead should achieve better covergece rate,

5 5 which we ideed demostrate by simulatios i Sectio VII. The detailed algorithm for the policy with a small overhead is described i Alg. ad it has a complexity of O( N log N ). Algorithm Policy with a small overhead : D, : for each iterval do : Φ D 4: x p {u Φ p D V p }, V 5: w, 6: Sort all sub-cliets such that D D... 7: j 8: for each time slot i the iterval do 9: if D j > the : Trasmit for j : w j w j + : if the trasmissio is successful the : j j + 4: ed if 5: ed if 6: ed for 7: D D +x w, 8: ed for VI. PERFORMANCE ANALYSIS I this sectio, we show that our two proposed policies are both asymptotically optimal ad feasible, as formally stated i Theorem 5 ad Theorem 6, respectively. Theorem 5: K Let x be the value of k= limif E[x(k)] K K uder the fully distributed policy. For ay cotrolled parameter V >, the logterm average total utility achieved uder this policy satisfies: u p x u p x B V, where {x } is the optimal solutio to problem ()-(4), ad B is a bouded costat. Further, the fully distributed policy satisfies all throughput reuiremets by providig a throughput of at least p x to each sub-cliet. Proof: Let D(k) be the vector cotaiig all D (k), ad defie a Lyapuov fuctio with respect to {D (k)} as follows: ( ) L(D(k)) D (k). We the have, uder ay policy ad costat V, E[L(D(k +)) L(D(k)) D(k)] = E[(D (k)+(x (k) w (k))) D (k) D(k)] = E[(x (k) w (k)) +D (k)(x (k) w (k)) D(k)] B +E[ D (k)(x (k) w (k)) D(k)], where B is a bouded costat sice x (k) ad w (k) are both bouded for all. Therefore, E[L(D(k +)) L(D(k)) V u p x (k) D(k)] E[ D (k)(x (k) w (k)) V u p x (k) D(k)]+B = E[ E[ E[ E[ (D (k) V u p )x (k) D(k)] D (k)w (k) D(k)]+B (D (k) + V u p )x (k) D(k)] D (k) + w (k) D(k)]+B, where D (k) + := max{,d (k)}, ad the last ieuality follows because the fully distributed policy does ot trasmit for sub-cliets with egative D (k). Recall that x (k) p, ad the fully distributed policy sets x (k) = p if ad oly if D (k) V up. Therefore, the fully distributed policy miimizes E[ (D (k) + V up )x (k) D(k)]. Moreover, Theorem i [] has show that the schedulig policy employed by the fully distributed policy maximizes E[ D (k) + w (k) D(k)]. Thus, the fully distributed policy miimizes E[ (D (k) + V up )x (k) D(k)] E[ D (k) + w (k) D(k)]. Next, cosider a policy that sets x (k) = x, ad employs a statioary radomized schedulig policy that provides a throughput of p x. Such a schedulig policy must exist sice {x } satisfies all costraits i problem ()-(4). Let w be the expected umber of trasmissios that the policy schedules i a iterval. By Lemma 4, w x. We ow have, uder the fully distributed policy, E[L(D(k +)) L(D(k)) V u p x (k) D(k)] E[ (D (k) + V u p )x (k) D(k)] E[ B D (k) + w (k) D(k)]+B, (D (k) + V u p )x V u p x. D (k) + w +B Summig the above ieuality over k {,,...,K } ad dividig by K yields: E[L(D(K))] E[L(D())] V K K B V K k= E[ u p x (k)] u p x. (6)

6 6 Sice L(D()) = ad L(D(k)), we further have lim if K = K K k= u p x E[ u p x (k)] u p x B V. Next, we show that the fully distributed policy provides a throughput of at least p x to each sub-cliet. Sice x (k) p for all ad k, from (6), we get ad hece (B + V K (B +V E[D (K)] = E[L(D(K))] K k= E[ D (K) ] u )K, E[ u p x (k)])k (B +V Dividig by K ad lettig K yields u )K. (B +V u )K E[ D (K) ] lim lim =, K K K K ad hece the throughput of each sub-cliet is at least p x. Theorem 6: K Let x be the value of k= limif E[x(k)] K K uder the policy with a small overhead. For ay cotrol parameter V >, u p x u p x B V, where B is a bouded costat. Further, the fully distributed policy satisfies all throughput reuiremets. Proof: The proof cosists of two parts: we first prove the optimality of the policy, ad the prove that it provides a throughput of at least p x to each sub-cliet. For each subset S N of sub-cliets, defie J S (k) S D (k). J S (k) ca be expressed recursively as follows, J S (k +) = J S (k)+ S x (k) Sw (k), with J S () =. Let D(k) be the vector cotaiig all D (k). Defie a Lyapuov fuctio as: L(D(k)) J S(k). N S We the have, E[L(D(k +)) L(D(k)) D(k)] E[ N J S(k)( x (k) w (k)) D(k)]+B S S S = E[ (x N (k) w (k))( J S (k)) D(k)]+B S: S = E[ (x N (k) w (k))( D l (k)) D(k)]+B, S: S l S (7) where B is a bouded costat. For each, there exists N subsets of N that cotais. Also, for ay two differet sub-cliets ad l, there exists N subsets of N that cotais both ad l. Therefore, S: S l S D l(k) = N (D (k) + Φ(k)), where Φ(k) = l D l(k) as defied by the policy with a small overhead. Therefore, E[L(D(k +)) L(D(k)) V u p x (k) D(k)] E[ E[ E[ E[ (Φ(k)+D (k) V u p )x (k) D(k)] (Φ(k)+D (k))w (k) D(k)]+B ([Φ(k)+D (k)] + V u p )x (k) D(k)] (Φ(k)+D (k)) + w (k) D(k)]+B. By its desig, the policy with a small overhead miimizes E[ E[ ([Φ(k)+D (k)] + V u p )x (k) D(k)] (Φ(k)+D (k)) + w (k) D(k)]. Now, cosider a policy that sets x (k) = x for all k, ad employs a statioary radomized schedulig policy so that the average umber of trasmissios scheduled for sub-cliet i each iterval is w x. Therefore, for ay give D(k), we have, uder the policy with a small overhead, E[L(D(k +)) L(D(k)) V u p x (k) D(k)] B V ad therefore, ([Φ(k)+D (k)] + V u p )x (Φ(k)+D (k)) + w +B u p x, E[L(D(K))] E[L(D())] V K K B V K k= E[ u p x (k)] u p x. (8)

7 7 Sice L(D()) = ad L(D(k)), we thus have lim if K K K k= E[ u p x (k)] = u p x O the other had, otice that x (K) p for all ad K. Therefore, from (8), we get N E[JS (K)] = E[L(D(K))] S (B + V K (B +V K k= u )K, E[ u p x (k)])k u p x B 45 V. Total utility Roud Robi Optimal (a) Fixed chael reliability ad hece E[ J S (K) ] lim lim K K K N (B +V u )K K =, for ay subset S. This also implies that E[ D lim (K) ] K K =, ad the throughput of subcliet is at least p x. VII. SIMULATION RESULTS We have implemeted the fully distributed policy ad the policy with a small overhead i s-, ad we demostrate our simulatio results i this sectio. We cosider both cases where all cliets have the same period, T m T, m, ad where differet cliets have differet periods. We compare our policies with three competig policies. The first policy applies a Additive Icrease Multiplicative Decrease () rate cotrol scheme. I particular, cliet m icreases r m by if all its real packets i the previous iterval are delivered o time, ad sets r m to r m /, otherwise. The secod policy applies a Additive Icrease Additive Decrease () rate cotrol scheme, where cliet m icreases r m by if all its real packets i the previous iterval are delivered o time, ad decreases r m by, otherwise. Both policies employ the schedulig policy of our fully distributed policy for packet schedulig. The third policy employs the rate cotrol policy of the fully distributed policy, but uses roud-robi for packet schedulig. We evaluate these policies through three differet metrics: The first is the total utility, m U m(r m ), where r m is the average umber of packets that cliet m geerates per iterval. The secod is the total deficiecy, defied as D(k)+. The AP delivers the reuired throughput, k D(k)+ m r m, to each cliet m if ad oly if k coverges to. Fially, we also evaluate the total variace of the umber of packets that cliet m geerates i each iterval. May existig studies have show that a high variace i data rate ca result i poor uality of experiece (QoE). A desirable policy should therefore achieve low total variace. Total utility 5 5 Roud Robi Optimal A. Homogeeous Period (b) Fixed QoS reuiremets Fig. : Total utility We cosider a WiFi system where the AP serves 4 cliets, each of which has R m = 6. Therefore, there are 4 sub-cliets i the system. The iterval legth is set to be ms, ad each iterval is composed of 5 cosecutive time slots whe the AP trasmits at Mb/s. We cosider two differet scearios. I the first sceario, cliets have differet service reuiremets, but the same chael reliability. Accordig to the study i [7], we choose the utility fuctio of cliet m as 5log(r m +.)+(5+m)r m. I additio, we set m = (8+m)% ad p m =.65. I the secod sceario, cliets have differet wireless chael coditios, but the same service reuiremets. We set p =.65, p =.55, p =.6 ad p 4 =.5. The utility fuctio ofmis 5(log(r m +.)+5, ad we choose m =.9 for all cliets. Simulatio results o the three performace metrics are show i Figs., ad. I both scearios, our policies achieve higher total utility tha the other three policies, ad also satisfy the throughput reuiremets of all flows. Further, our policies result i performace with smaller variaces tha other policies. It is also observed i the simulatio that the policy with a small overhead seems to have better trasitioal performace tha the

8 8 Total deficiecy.5.5 Roud Robi Variace Roud Robi (a) Fixed chael reliability (a) Fixed chael reliability Total deficiecy.5.5 Roud Robi Variace Roud Robi (b) Fixed QoS reuiremets Fig. : Total deficiecy (b) Fixed QoS reuiremets Fig. : Variace of r fully distributed policy i that its total utility coverges to the optimal value faster, ad its total deficiecy also coverges to faster. B. Heterogeeous Periods Next, we cosider systems where differet flows have differet periods T m. We cosider a WiFi system where the AP serves 4 cliets. Cliets ad have R m = ad T m = ms, while cliets ad 4 have R m = 7 ad T m = 5ms. The utility fuctios are U (r ) := 5log(r +.)+5r, U (r ) := 5log(r +.)+7r, U (r ) := 5log(r +.)+r ad U 4 (r 4 ) := 5log(r 4 +.)+ 5r 4. We set m = (8+m)% ad p m =.65. The simulatio results are show i Figs.4, 5, ad 6. Both of our policies still perform better tha the other three policies i both total utility ad total variace. This result suggests that our proposed policies ca still achieve better performace whe cliets have differet periods. VIII. CONCLUSION We ivestigated the utility maximizatio problem for real-time applicatios with striget deadlies. We formulate this problem as a submodular optimizatio problem. Total utility Roud Robi Fig. 4: Total utility i heterogeeous case Sice there is o stadard techiue that ca solve it efficietly, we propose two asymptotically optimal policies, both of which have very low complexity. The performace of our policies is further evaluated via s- simulatio. All simulatio results show that our policies achieve better performace tha three other competig policies.

9 9 Total deficiecy Variace Roud Robi Fig. 5: Total deficiecy i heterogeeous case Roud Robi [9] IWATA, S., AND ORLIN, J. A simple combiatorial algorithm for submodular fuctio miimizatio. I Proc. of the Twetieth Aual ACM-SIAM Symposium o Discrete Algorithms (9). [] JI, X., HUANG, J., CHIANG, M., LAFRUIT, G., AND CATTHOOR, F. Schedulig ad resource allocatio for svc streamig over ofdm dowlik systems. IEEE Tras. Circuits Syst. Video Tech 9, (9), [] KANG, S. H., AND ZAKHOR, A. Packet schedulig algorithm for wireless video streamig. I Packet Video (Apr ). [] KELLY, F. Chargig ad rate cotrol for elastic traffic, 997. [] LI, R., AND ERYILMAZ, A. Schedulig for ed-to-ed deadliecostraied traffic with reliability reuiremets i multi-hop etworks. I Proc. of IEEE INFOCOM (Apr. ), pp [4] LIN, X., AND SHROFF, N. Joit rate cotrol ad schedulig i multihop wireless etworks. I i Proceedigs of IEEE Coferece o Decisio ad Cotrol (Dec. 4). [5] LUNA, C. E., KONDI, L. P., AND KATSAGGELOS, A. K. Maximizig user utility i video streamig applicatios. IEEE Tras. Circuits Syst. Video Tech., (), [6] MAO, Z., KOKSAL, C. E., AND SHROFF, N. B. Olie packet schedulig with hard deadlies i multihop commuicatio etworks. I Proc. of IEEE INFOCOM (). [7] SCHROEDER, D., ESSAILI, A. E., STEINBACH, E., STAEHLE, D., AND SHEHADA, M. Low-complexity o-referece psr estimatio for h.64/avc ecoded video. I Packet Video Workshop (PV), th Iteratioal (Dec ). [8] XIONG, H., LI, R., AND ERYILMAZ, A. Delay-aware cross-layer desig for etwork utility maximizatio i multi-hop etworks. IEEE Joural o Selected Areas i Commuicatios 9, 5 (May ), [9] ZHANG, X., XU, Y., HU, H., LIU, Y., GUO, Z., AND WANG, Y. Profilig skype video calls: Rate cotrol ad video uality. I INFOCOM, Proceedigs IEEE (), IEEE, pp [] ZHAO, S., AND LIN, X. Rate-cotrol ad multi-chael schedulig for wireless live streamig with striget deadlies. I Proc. of IEEE INFOCOM (4). [] ZUO, S., AND HOU, I.-H. Olie schedulig for eergy efficiecy i real-time wireless etworks. I Proc. of Allerto (4) Fig. 6: Variace i heterogeeous case Shuai Zuo is curretly pursuig the Ph.D. degree with Texas A&M Uiversity, College Statio, TX, USA. His research iterests iclude wireless etworkig, real-time systems ad optimizatio. REFERENCES [] Cisco visual etworkig idex: Global mobile data traffic forecast update, 5c white paper. I Cisco (Feb 6). [] ASIRI, A., AND SUN, L. Performace aalysis of video calls usig skype. Advaces i Commuicatios, Computig, Networks ad Security Volume (), 55. [] CHEN, C., HEATH, R. W., BOVIK, A. C., AND DE VECIANA, G. Adaptive policies for real-time video trasmissio: A markov decisio process framework. I IEEE iteratioal Coferece o Image Processig (Sep ). [4] ERYILMAZ, A., AND SRIKANT, R. Joit cogestio cotrol, routig ad mac for stability ad fairess i wireless etworks. IEEE Joural o Selected Areas i Commuicatios 4 (6), [5] FUJISHIGE, S. Submodular fuctios ad optimizatio. Aals of Discrete Mathematics, (Ja 5), 89. [6] HOU, I.-H. Schedulig heterogeeous real-time traffic over fadig wireless chaels. IEEE/ACM Tras. o Networkig, 5 (Oct. 4), [7] HOU, I.-H., BORKAR, V., AND KUMAR, P. A theory of QoS for wireless. I Proc. of IEEE INFOCOM (9), pp [8] I-H. HOU, A. TRUONG, S. C., AND KUMAR, P. Optimality of periodwise static priority policies i real-time commuicatios. I Proc. of CDC (ivited) (Orlado, FL, Dec ). I-Hog Hou (S-M) received the B.S. i Electrical Egieerig from Natioal Taiwa Uiversity i 4, ad his M.S. ad Ph.D. i Computer Sciece from Uiversity of Illiois, Urbaa-Champaig i 8 ad, respectively. I, he joied the departmet of Electrical ad Computer Egieerig at the Texas A&M Uiversity, where he is curretly a assistat professor. His research iterests iclude wireless etworks, wireless sesor etworks, real-time systems, distributed systems, ad vehicular ad hoc etworks. Dr. Hou received the C.W. Gear Outstadig Graduate Studet Award from the Uiversity of Illiois at Urbaa-Champaig, ad the Silver Prize i the Asia Pacific Mathematics Olympiad.

10 Tie Liu (S99 M6 SM5) was bor i Jili, Chia i 976. He received his B.S. (998) ad M.S. () degrees, both i Electrical Egieerig, from Tsighua Uiversity, Beijig, Chia ad a secod M.S. degree i Mathematics (4) ad a Ph.D. degree i Electrical ad Computer Egieerig (6) from the Uiversity of Illiois at Urbaa-Champaig. Sice August 6 he has bee with Texas A&M Uiversity, where he is curretly a Associate Professor i the Departmet of Electrical ad Computer Egieerig. His primary research iterest is i the area of iformatio theory ad statistical iformatio processig. Dr. Liu received a M. E. Va Valkeburg Graduate Research Award (6) from the Uiversity of Illiois at Urbaa-Champaig ad a CAREER Award (9) from the Natioal Sciece Foudatio. He was a Techical Program Committee Co-Chair for the 8 IEEE GLOBECOM, a Geeral Co-Chair for the IEEE North America School of Iformatio Theory, ad a Associate Editor for Shao Theory for the IEEE Trasactios o Iformatio Theory durig 4-6. Aathram Swami is with the US Army Research Laboratory as the Army s Seior Research Scietist (ST) for Network Sciece. Prior to joiig ARL, he held positios with Uocal Corporatio, USC, CS- ad Malgudi Systems. He was a Statistical Cosultat to the Califoria Lottery, developed a Matlab-based toolbox for o-gaussia sigal processig. He has held visitig faculty positios at INP, Toulouse., ad curretly at Imperial College. He received the B.Tech. degree from IIT-Bombay; the M.S. degree from Rice Uiversity, ad the Ph.D. degree from the Uiversity of Souther Califoria (USC), all i Electrical Egieerig. Swami s work is i the broad area of etwork sciece, with emphasis o wireless commuicatio etworks. He is a ARL Fellow ad a Fellow of the IEEE. Prithwish Basu (Sr. Member 9) received a B.Tech. degree i Computer Sciece ad Egieerig from Idia Istitute of Techology, Delhi i 996, ad his M.S. ad Ph.D. i Computer Egieerig from Bosto Uiversity i 999 ad, respectively. He is curretly a Lead Scietist at Raytheo BBN Techologies i the Networkig ad Commuicatios Techologies busiess uit. His research iterests iclude etwork sciece, wireless ad hoc ad sesor etworks, ad mobile computig. He has co-authored over papers i etwork-related cofereces ad jourals ad has received best paper awards at IEEE NetSciComm ad PAKDD cofereces. He is a Associate Editor of the IEEE Trasactios of Mobile Computig. Dr. Basu received the MIT Techology Review s TR5 award i 6 (give to 5 iovators uder the age of 5), ad was raked 7th i the Idia Natioal Mathematics Olympiad i 99.