1 Deay Anaysis o Maximum Weight Scheduing in Wireess Ad Hoc Networks ong Bao e, Krishna Jagannathan, and Eytan Modiano Abstract This paper studies deay properties o the weknown maximum weight scheduing agorithm in wireess ad hoc networks. We consider wireess networks with either onehop or muti-hop ows. Speciicay, this paper shows that the maximum weight scheduing agorithm achieves order optima deay or wireess ad hoc networks with singe-hop traic ows i the number o activated inks in one typica schedue has the same order with the number o inks in the network. This condition woud be satisied or most practica wireess networks. This resut hods or both i.i.d and Markov moduated arriva processes with two states. For the muti-hop ow case, we aso derive tight backog bounds in the order sense. I. INTRODUCTION Wireess scheduing has been known to be a key probem or throughput/capacity optimization in wireess networks. The we-known maximum weight scheduing agorithm has been proposed by Tassiuas in his semina paper 1] where he proved its throughput optimaity. atter deveopments in this area incude extension o this maximum weight scheduing agorithm to wireess networks with rate/power contro ], 3], network contro when oered traic is outside the capacity region 4], and other scheduing poicies with owercompexity 5]-8]. Whie most existing works in the area o stochastic network contro ocused on throughput perormance o optima and suboptima scheduing poicies, deay properties o most scheduing poicies proposed or wireess ad hoc networks remain unknown. In this paper, we study backog/deay properties o the maximum weight scheduing agorithm in wireess ad hoc networks. There are some recent works which investigated backog/deay bounds or the supoptima maxima scheduing agorithm in wireess ad hoc networks and maximum weight scheduing agorithm in the downink/upink o ceuar networks. Speciicay, in 13] Neey showed that maxima scheduing achieves deay scaing o O (1/(1 ρ)) or traic inside the reduced stabiity region derived in 8]. This reduced stabiity region can be as sma as 1/I o the capacity region, where I is the maximum number o inks in any ink intererence set which do not interere with one another. In 14], 15], Neey aso proved the order optima deay or the maximum weight scheduing agorithm in the wireess ceuar upink/downink with ON/OFF wireess inks. Note that the capacity region in the ceuar setting can be expicity described which signiicanty eases the backog/deay anaysis. Average backog bounds were derived or maximum weight This work was supported by NSERC Postdoctora Feowship and by ARO Muri grant number W911NF-08-1-038. The authors are with Communications and Networking Research Group, Massachusetts Institute o Technoogy, Cambridge, MA, USA. Emais: ongbe, krishnaj, modiano}@mit.edu. scheduing in severa works ], 4], 10], 9]. These backog bounds were obtained by bounding the maximum transmission rates and the number o arrivas in each time sot which are, thereore, not tight in genera. There are some other works which investigate exponents or the tais o queue backogs in the wireess ceuar setting 11], 1]. In this paper, we consider a wireess ad hoc network with either one-hop and muti-hop traic ows. We show that average deay or the case o one-hop traic ows scaes as O (1/(1 ρ)) i we can construct a set o distinct schedues to cover the network where the number o activated inks in each o these schedues has the same order with the number o network inks. This condition woud be satisied or most practica arge-scae wireess networks. This deay scaing hods or both i.i.d and Markov moduated traic arriva processes with at most two states. These resuts are stated in Propositions 4 and 5 o the paper. To the best o our knowedge, these are the irst deay optima resuts or the maximum weight scheduing agorithm in wireess ad hoc networks. For wireess ad hoc networks with mutihop traic ows, we aso derive a tight backog bound which scaes as O (N/(1 ρ)) where N is the number o wireess nodes. The remaining o this paper is organized as oows. Deay anaysis or singe-hop traic ows is presented in section II. In section III, we derive backog bounds or wireess networks with mutihop traic ows. II. ANAYSIS OF SINGE-HOP FOW CASE A. System Modes and Assumptions We mode a wireess ad hoc network as directed graph G = (V, E) where V is the set o wireess nodes and E is the set o wireess inks. Suppose the cardinaities o V and E are N and, respectivey. We consider singe-hop traic ows in this section. Data rom a ows traversing a particuar ink is buered at the corresponding transmitter o the ink. Assume time is sotted with ixed-size sot intervas. For now, traic arriving to source nodes o singe-hop ows is assumed to be independent and identicay distributed (i.i.d) over time. Assume that packets arriving during time sot t can ony be transmitted rom time sot t + 1 at the eariest. et denote by A (t) the number o packets arriving at ink in time sot t and µ (t) the number o packet transmitted on ink in time sot t. For simpicity, assume that µ (t) = 1 i ink is schedued in time sot t, otherwise µ (t) = 0. In the remaining o this paper, we wi use r to describe a coumn vector with eements r denoting quantities such as queue ength, schedued inks, etc. The queue evoution or the ow at ink can be written
as oows: (t + 1) = (t) µ (t) + A (t). (1) Assume that ony backogged inks are schedued, it can be veriied that this queue evoution equation hods or arbitrary (t) and µ (t). Regarding the scheduing, we consider the we-known maximum weight scheduing agorithm which is known to achieve the capacity region 1]. The maximum weight scheduing agorithm determines the optima schedue µ (t) based on the ink queue backogs as oows: µ (t) = argmax µ(t) S (t)µ (t) () where S denotes the set o a possibe easibe schedues according some intererence constraints. In this section, we are going to derive the deay bound or this scheduing poicy assuming that arriva traic is stricty within the capacity region so that the maximum weight scheduing agorithm wi stabiize the network 1]-3]. B. Backog/Deay Anaysis or i.i.d Arriva Traic In this subsection, we obtain a deay bound or the aorementioned scheduing scheme using the yapunov drit technique 1]-3]. Traic arriving to a transmitting buer o wireess ink is assumed to be i.i.d over time with average arriva rate λ. In the oowing, we use a resut which was stated in 13]. emma 1: (Theorem 1 rom 13]) et (t) be the queue backog vector in time sot t and ( (t)) be a yapunov unction. Aso, deine a one-step yapunov drit as oows: (t) = E ( (t + 1)) ( (t)) } (3) where the expectation E(.) is taken over the randomness o queue backogs (t) and system dynamics given the queue backogs (t). I the yapunov drit satisies then we have 1 t 1 im sup t t τ=0 (t) E g(t)} E (t)} (4) E (t)} im sup t 1 t 1 t τ=0 E g(t)}. (5) Now, consider the oowing quadratic yapunov unction ( (t)) = (t). (6) We have the oowing resut or the yapunov drit. Proposition 1: The yapunov drit satisies the oowing reation or any time sot t: where (t) = E B(t)} + E (t) (A (t) µ (t))} (7) B(t) = A (t) µ (t)] = A (t) + A (t)µ (t) + µ (t). (8) The proo o this proposition oows directy by appying the queue evoution reation in (1), and is omitted or brevity. As shown in 1], the capacity region coincides with the } convex hu o a possibe easibe schedues. et S = Ri be the set o a possibe schedues where one particuar schedue R i is a coumn vector o dimension with the -th eement equa 1 i ink is schedued and equa 0 otherwise. For any arriva rate vector λ stricty inside the capacity region, we have λ β iri (9) where denotes the cardinaity o set S, β i < 1 and <, denotes both a reguar inequaity and an eementwise inequaity. We have the oowing reation N λ = tr λ β i tr Ri tr µ β i < tr µ (10) where.] tr denotes the vector transposition and µ is the optima schedued vector given backog vector. It can be veriied that these resuts hod by using the reations in (9) and (). Now, we state a bound on the tota queue backogs or i.i.d. arriva traic in the oowing proposition. Proposition : Assume that the arriva rate vector λ is stricty inside the capacity region so that there exists a vector ɛ such that λ+ ɛ is inside the capacity region where ɛ is a vector with a eements equa to ɛ. Aso, assume that a arriva processes on a wireess inks have bounded second moments. Then, the network is stabe and the tota average queue backog can be bounded as + E A (t) } λ (11) ɛ where = λ is the tota ink arriva rates. Proo: Using (10) or backog vector (t), we have ] tr ] ] tr (t) λ + ɛ (t) µ (t). (1) Hence, ] tr ] tr ] tr (t) λ (t) µ (t) (t) ɛ. (13) Note that the second term o (7) can be written as E (t) (A (t) µ (t))} = E (t) (λ µ (t))} = E (t) ] tr ( λ µ(t) ) }. (14) Using (13) and (14) in (7) with µ(t) representing an optima schedued vector, we have (t) E B(t)} (t) ] tr ɛ
3 = E B(t)} ɛ E (t)}. (15) Using the resut in emma 1 in (15), we have im sup t 1 t 1 t τ=0 E (τ)} B ɛ (16) where B = 1 t 1 im sup t t τ=0 E B(τ)}. From (8), using the act that µ (t) 1 and arriva processes have bounded second moments, we have B <. Thereore, the queueing network is strongy stabe. Because it evoves according to an ergodic Markov chain with countabe state space, the imiting time averages o queue backogs equa to the corresponding steady state averages. To cacuate B, we note that under the stabiity condition we have 1 t 1 im sup t τ=0 µ (τ) = λ. Aso, note that µ (t) = t µ (t) because µ (t) = 0, 1} depending on whether ink is schedued in time sot t or not. As a consequence, B can be written as B = = + E A (t) } λ + λ E A (t) } λ. (17) Because time average imits o queue backogs are equa to their steady state averages, using (17) the inequaity (16) can be rewritten as + E A (t) } λ. (18) ɛ Hence, the proposition is proved. 1) Deay Bound: Appying itte s aw to (11), we can obtain a deay bound as oows: E A (t) } λ ] W 1 + 1. (19) ɛ Now, in order to understand the scaing o this deay bound, we need to determine the reationship between the traic oading actor ρ and the parameter ɛ. et us denote by Λ the capacity region. Assume that the arriva rate vector λ = (λ1, λ,, λ ) tr is stricty inside the capacity region Λ, then there exists a oading actor ρ < 1 such that λ ρλ. (0) In the oowing, we state a deay bound by choosing a straightorward oading actor ρ as a unction o ɛ. Proposition 3: I arriva rate is in ρ-scaed capacity region as described in (0), the average tota deay can be bounded as 1 + 1 E A (t) } ] ] λ W. (1) (1 ρ) In the specia case where the arriva process on each wireess ink is Poisson, we have W (1 1 λ ). () 1 ρ Proo: The proo oows by using the act that we can choose ɛ = (1 ρ) 1 where 1 is an a-one vector with dimension such that λ+ ɛ Λ or any λ ρλ. Speciicay, by pugging ɛ = (1 ρ) into the deay bound in (19), we can obtain (1). Now, we show that λ+ ɛ Λ or ɛ = (1 ρ). Note that or any λ ρλ, we can write λ = ρ β ir i where β i < 1. Deine e i be a vector o dimension with a zeros except a one at the i-th position. It can be easiy seen that any e i (i = 1,,, ) represents a easibe schedue (with ink i being activated). Aso, note that e i = 1. Hence, we have the oowing resut 1 ρ 1 +ρ β i Ri = 1 ρ e i +ρ β iri Λ. (3) When the arriva processes are Poisson, we have E A (t) } = λ + λ. Using this reationship in the deay bound (1), we obtain (). Note that the term 1 E E A (t) } is typicay O(1) or any traic satisying A (t) A max. In act, in such cases 1 we have E E A (t) } A max. Hence, the deay bound stated in Proposition 3 is typicay O(/(1 ρ)). C. Tighter Deay Bound In the oowing, we state a tighter deay bound under speciic assumptions which can be achieved by expoiting underying intererence constraints and network topoogy. Proposition 4: Assume that the arriva rate is in the ρ- scaed capacity region as described in (0). Aso, assume that we can ind a set o easibe schedues, namey Ψ = s i, i = 1,,, T }, satisying the oowing assumptions For any schedue s i Ψ, i ink is activated in s i then ink is not activated in any other s j Ψ or j i (i.e., any ink shoud beong to one and ony one schedue in the set Ψ). et E be the set o inks activated by a schedues in Ψ, then E = E where reca that E is the set o a network inks (i.e., the union o activated inks by a schedues covers the whoe network). et K i denote the number o activated inks in schedue s i and K min = min i K i. Then, we have the oowing deay bound E A (t) } λ ] W 1 + 1 K min (1 ρ) /. (4) Beore proving this proposition, we note that or wireess networks such that K min = O(), proposition 4 impies that the network deay typicay scaes as O(1/(1 ρ)). This condition woud hod i the network topoogy is suicienty sparse and uniorm so that the most baanced set o schedues Ψ (i.e., amost a schedues in Ψ have the same number o activated inks in the order sense) satisies K min = O(). Note that this condition woud be satisied or most practica wireess networks because a typica schedue woud activate most inks in the network. We wi provide one such network exampe ater the proo. Proo: The proo or this proposition oows the same ine as that or proposition 3. However, a tighter deay bound
4 is achieved in this proposition by constructing ɛ rom the set o schedues Ψ each o which has at east K min activated inks. Now, consider the oowing inear combination o easibe schedues whose outcome ies inside the capacity region (1 ρ) T K i T j K s i + ρ β iri Λ. (5) j Thereore, the resut stated in proposition 4 oows by pugging ɛ = (1 ρ) K min into the deay bound (19). In the oowing, we provide a simpe exampe where the assumptions o the proposition hod. Exampe: Consider a grid network and one-hop (primary) intererence mode or the sake o simpicity as being shown in Fig. 1. In this igure, we aso show how to construct a set o easibe schedues Ψ that covers the whoe network graph (again, each schedue has the same ink pattern). To anayze its deay bound, assume that the size in one dimension o the grid network is H inks, then it can be veriied that = H(H+1). From the constructed set o schedues Ψ as shown in this igure, we have K min = (H + 1) H/. Thereore, using the resut in proposition 4, the deay can be bounded as H(H + 1) 1 + 1 E A (t) } ] ] λ W (H + 1) H/ (1 ρ) 1 + 1 E A (t) } ] ] λ (1 ρ) which scaes typicay as O(1/(1 ρ)). a time t. In order to obtain deay bound or this case, we wi use one resut proved in 13] which is stated in the oowing emma. emma : (rom section V.A o 13]) Deine C = E A (t 1)A (t)} or E 1 and C = 0 or E. For a ink, we have E (t)a (t)} E (t)} λ + C δ + σ. Now, we state deay bounds or the case o time-correated arrivas in the oowing proposition. Proposition 5: I the arriva traic is within the ρ-scaed capacity region and the assumptions in proposition 4 are satisied, then the network is stabe and the average deay can be bounded as where W B = 1 + 1 E B + C K min (1 ρ)/. (6) E A (t) }, C = 1 E 1 C σ + δ. (7) The proo oows by using resuts in emma and Proposition 1 so it is omitted or brevity. The term 1 E E A (t) } is typicay O(1) or any traic satisying A (t) A max. In act, in such cases we have B 1 + A max. It is not very diicut to see that or E 1, we have C λ λ max where λ max < 1 is the maximum conditiona rate over a inks and states. Hence, we can obtain the oowing deay bound W 1 + A max + max E1 λ max /(σ + δ )} K min (1 ρ)/ which scaes as O(1/(1 ρ)) or K min = O(). (8) Fig. 1. Grid networks with one-hop (primary) intererence mode. D. Anaysis or Time-Correated Arrivas with Two States Here, assume that arriva process A (t) or inks is either i.i.d. or moduated by a discrete time stationary and ergodic Markov chain Z (t) having two states (i.e., states 1 and ). et σ and δ be transition probabiities rom state 1 to state and rom state to state 1, respectivey. For each ink, deine the conditiona average arriva rates λ (m) as oows: λ (m) = E A (t) Z (t) = m}. Now, et denote by E 1 E as the set o inks with timecorreated arrivas where λ (1) λ (). Aso, assume that arriva traic to any other inks in E = E E 1 is either i.i.d or time-correated with two states satisying λ (1) = λ (). Assume that the moduating Markov chains o a arriva processes are stationary so that or a inks we have E A (t)} = λ or III. ANAYSIS OF MUTIHOP FOW CASE A. System Modes and Assumptions We consider the same network mode as section II. We assume that there is set o mutihop ows F where ow F has a ixed route rom a source node s() to a destination node d(). We denote the set o inks and nodes on the route o ow as () and R(), respectivey. For simpicity, we assume that packet arrivas to source nodes o a ows are i.i.d stochastic processes. We denote the queue ength o ow at node n at the beginning o time sot t as n(t) and the number o packets arriving at the source node o ow as A s() (t). Note that data packets o any ow are deivered to the higher ayer upon reaching the destination node, so d() (t)=0. In addition, et µ n(t) be the number o packets o ow transmitted rom node n aong ink (n, m) o its route which is buered at node m i m d(). Again, we assume that µ n(t) = 1 i we activate ink (n, m) on the route o ow and µ n(t) = 0, otherwise. Given the routes or a ows, the maximum weight scheduing agorithm is used or data deivery 1]. Speciicay, the scheduing is perormed in every time sot as oows:
5 Each ink (n, m) inds the maximum dierentia backogs as oows: w nm (t) = max n (t) m(t) }. (9) :(n,m) () Based on cacuated ink weights, a maximum weight schedue is ound as µ (t) = argmax w nm (t)µ nm (t). (30) µ(t) S (n,m) For any schedued ink, one packet is transmitted rom the buer o the ow achieving the maximum dierentia backog. The queue evoutions can be written as n(t + 1) = n(t) µ n(t) + π n(t) (31) where this equation hods because µ n(t) = 1 ony i n(t) 1 (i.e., we do not schedue inks with empty queues). Aso, πn(t) is the number o packets arriving to queue n(t) in time sot t which can be written as A πn(t) = n (t), i n = s() µ n 1 (t), otherwise. (3) B. Backog Bound The queue backog bound is stated in the oowing proposition. The tighter backog bound wi be deveoped ater that. Proposition 6: Assume that arriva processes or a ows are i.i.d over sots and have bounded second moments. Aso, assume that arriva rate ies within the ρ-scaed capacity region. Then, 1) The tota average queue backogs can be bounded as =1 n R() n(t) θ max max (N + D). (1 ρ) where ρ is a oading actor, max = max F } (), θ max is the maximum number o ows traversing any ink in the network, N is tota number o network nodes, and D = ( ) } E A s() (t) λ + λ. (33) =1 ) For Poisson arriva process, the backog can be bounded as θ max max (N + ) F λ n(t) (1 ρ) =1 n R() where = F λ. Proo: Consider the oowing yapunov unction ( ) (t) = =1 n R() ( n (t) ) (34) where again the expectation E(.) is taken over the randomness o queue backogs (t) and system dynamics given the queue backogs (t). The yapunov drit can be written as oows: ( (t) = E n (t + 1) ) ( n (t) ) ] =1 n R() = E π n (t) µ n(t) ] (35) =1 n R() +E n(t) πn(t) µ n(t) ].(36) =1 n R() Now, consider (35) we have E π n (t) µ n(t) ] N + B 1(t) (37) =1 n R() ] where B 1 (t) = =1 E A s() (t) µ s() }. (t) This inequaity hods because we have F,s() n π n (t) µ n(t) ] 1. Now, using (3), we can manipuate (36) as oows: E n(t) πn(t) µ n(t) ] =1 n R() = µ nm(t)wnm + s() (t)λ (38) =1 where µ nm(t) corresponds to the maximum weight schedue with queue backogs (t) and w nm = max :(n,m) () n (t) m(t) ] +. Note that we have written down n(t) m(t) ] + instead o n (t) m(t) ] because inks with negative weight wi not be schedued by the maximum weight scheduing agorithm. Note that we can rewrite =1 s() (t)λ as oows: s() (t)λ = =1 =1 (n,m) () λ n (t) m(t) ] :(n,m) () λ max n (t) m(t) ] + (39). :(n,m) () Suppose that λ = (λ ) F is stricty inside the capacity region. Then, there exists a vector ɛ such that λ+ ɛ is sti inside the capacity region. This impies that there exists a vector o ink rates (µ nm ) Co(S) such that 1] (λ + ɛ) = λ + θ nm ɛ µ nm (40) :(n,m) () :(n,m) () where Co(S) represents the convex hu o a easibe ink schedues and θ nm denotes the number o ows traversing ink (n, m). Using (38), (39), (40), we can rewrite (36) as (36) µ nm(t) + λ wnm ɛ :(n,m) () ( µ nm(t) + µ nm θ nm ɛ) w nm θ nm w nm. (41)
6 Now, we have the oowing n(t) () (n,m) () () (n,m) () (n,m) () n (t) m(t) ] + w nm. (4) Note that a simiar bounding technique or (n,m) () n(t) used in the above inequaity has been empoyed in 16] to prove the network stabiity. We instead aim at obtaining a tight upper-bound o queue backogs as a unction o the oading actor ρ, so the yapunov drit anaysis in this paper is very dierent rom that in 16]. Using (4), we have =1 (n,m) () max =1 (n,m) () n(t) () =1 w nm = max (n,m) () w nm θ n,m w nm. (43) where max = max F } (). Using (37), (41) and (43), the yapunov drit can be bounded as (t) N + B 1 (t) ɛ max =1 n R() n(t). (44) I the arriva processes or a ows have bounded second moments then B 1 (t) is bounded. Under this condition, the yapunov drit wi be negative when the tota backog becomes arge enough. Hence, the network is stabe and the time average imits o queue backogs are equa to their steadystate averages. Aso, it is not very diicut to see that the time average imit o B 1 (t) is equa to D. Thereore, by using emma 1 and substituting time average imits o queue backogs by their steady-state averages, we have =1 n R() n max(n + D). (45) ɛ Now, to understand the scaing o this backog bound, we need to ind ɛ as a unction o the oading actor ρ as beore. Suppose arriva rate vector λ is inside the ρ-scaed capacity region, and et λ be a vector with (n, m)-th eements equa λ nm = :(n,m) () λ. Then, we have λ = ρ β iri (46) or some non-negative β i such that β i < 1. et θ max = max } θ nm where reca that θ nm is the number o ows traversing ink (n, m). We wi show that λ (1) = λ + ɛ Λ or ɛ = (1 ρ)/(θ max ). Now, et us construct a vector λ () with its (n, m)-th eements equa to () λ nm = :(n,m) () λ(1). Then, we have ( ) λ () = λ 1 ρ + θ nm θ max where ρ β iri + 1 ρ ( 1 ρ θ max θ nm ) e nm Λ (47) denotes a vector with (n, m)-th eements equa to the quantity inside the bracket. Substitute ɛ = (1 ρ)/(θ max ) into (45), we can obtain part 1) o the proposition. To prove part ) o the proposition, we need some manipuation o D or the Poisson arriva process. Speciicay, or Poisson process we have E ( A s() (t) ) } = λ + λ. Substitute this into (33), we have D = =1 λ. Pug this into the bound in part 1), we obtain part ) o the proposition. It can be shown that the average backogs derived in this proposition scae as O(N/(1 ρ)) or max = O(1). In act, using the simiar idea as that o proposition 4, a tighter backog bound can be obtained. Speciicay, i the arriva processes satisy assumptions in proposition 6 and the assumptions o proposition 4 hod, then the tota queue backogs can be bounded as =1 n R() n θ max max (N + D) K min (1 ρ)/. (48) This backog bound typicay scaes as O(N/(1 ρ)) or max = O(1) and K min = O(). REFERENCES 1]. Tassiuas and A. Ephremides, Stabiity properties o constrained queueing systems and scheduing poicies or maximum throughput in mutihop radio networks, IEEE Trans. Automatic Contro, vo. 37, no. 1, pp. 1936-1948, Dec. 199. ] M. Neey, E. Modiano, and C. Rohrs, Power aocation and routing in mutibeam sateites with time-varying channes, IEEE/ACM Trans. Networking, vo. 11, no. 1, pp. 138-15, Feb. 003. 3] M. Neey, E. Modiano, and C. Rohrs, Dynamic power aocation and routing or time varying wireess networks, IEEE INFOCOM 003. 4] M. Neey, E. Modiano, and C. i, Fairness and optima stochastic contro or heterogeneous networks, IEEE INFOCOM 005. 5] E. Modiano, D. Shah, and G. Zussman, Maximizing throughput in wireess networks via gossiping, ACM SIGMETRICS 006. 6] S. Sanghavi,. Bui, and R. Srikant, Distributed ink scheduing with constant overhead, ACM SIGMETRICS 007, June 007. 7] X. in and N. Shro, The impact o imperect scheduing on cross-ayer rate contro in wireess networks, IEEE INFOCOM 005. 8] P. Charporkar, K. Kar, and S. Sarkar, Throughput guarantees through maxima scheduing in wireess networks, Aerton 005, Sept. 005. 9] Y. Yi, A. Proutiere, and M. Chiang, Compexity o wireess scheduing: Impact and tradeos, ACM Mobihoc, May 008. 10] G. Gupta and N. B. Shro, Deay anaysis o scheduing poicies in wireess networks, Asiomar Conerence on Signas, Systems, and Computers, Oct. 008. 11] A. Stoyar, arge deviations o queues under os scheduing agorithms, Aerton 006, Sept. 006. 1] V. J. Venkataramanan and X. in, Structura properties o DP or queue-ength based wireess scheduing agorithms, Aerton 007, Sept. 007. 13] M. Neey, Deay anaysis or maxima scheduing in wireess networks with bursty traic, IEEE INFOCOM 008. 14] M. Neey, Order optima deay or opportunistic scheduing in mutiuser wireess upinks and downinks, Aerton 006, Sept. 006. 15] M. Neey, Deay anaysis or max weight opportunistic scheduing in wireess systems, Aerton 008, Sept. 008. 16]. Bui, R. Srikant, and A. Stoyar, Nove architecture and agorithms or deay reduction in back-pressure scheduing and routing, IEEE INFOCOM 009.