Max-Weight Scheduling in Queueing Networks with. Heavy-Tailed Traffic

Transcription

1 Max-Weight Scheduling in Queueing Networks with Heavy-Tailed Traic arxiv: v1 [cs.ni] 1 Aug 011 Mihalis G. Markakis, Eytan H. Modiano, and John N. Tsitsiklis Abstract We consider the problem o packet scheduling in single-hop queueing networks, and analyze the impact o heavy-tailed traic on the perormance o Max-Weight scheduling. As a perormance metric we use the delay stability o traic lows: a traic low is delay stable i its expected steady-state delay is inite, and delay unstable otherwise. First, we show that a heavy-tailed traic low is delay unstable under any scheduling policy. Then, we ocus on the celebrated Max-Weight scheduling policy, and show that a light-tailed low that conlicts with a heavytailed low is also delay unstable. This is true irrespective o the rate or the tail distribution o the light-tailed low, or other scheduling constraints in the network. Surprisingly, we show that a light-tailed low can be delay unstable, even when it does not conlict with heavy-tailed traic. Furthermore, delay stability in this case may depend on the rate o the light-tailed low. Finally, we turn our attention to the class o Max-Weight-α scheduling policies; we show that i the α-parameters are chosen suitably, then the sum o the α-moments o the steady-state queue lengths is inite. We provide an explicit upper bound or the latter quantity, rom which we derive results related to the delay stability o traic lows, and the scaling o moments o steady-state queue lengths with traic intensity. This work was supported by NSF Grants CNS and CCF , and ARO MURI Grant W911NF The authors are with the Laboratory or Inormation and Decision Systems, at the Massachusetts Institute o Technology, Cambridge, MA, USA. 1

2 1 Introduction We study the impact o heavy-tailed traic on the perormance o scheduling policies in singlehop queueing networks. Single-hop network models have been used extensively to capture the dynamics and scheduling decisions in real-world communication networks, such as wireless uplinks and downlinks, switches, wireless ad hoc networks, sensor networks, and call centers. In all these systems, one cannot serve all queues simultaneously, e.g., due to wireless intererence constraints, giving rise to a scheduling problem. Clearly, the overall perormance o the network depends critically on the scheduling policy applied. The ocus o this paper is on a well-studied class o scheduling policies, commonly reered to as Max-Weight policies. This class o policies was introduced in the seminal work o Tassiulas and Ephremides [4], and since then numerous studies have analyzed the perormance o such policies in dierent settings, e.g., see [1, 9], and the reerences therein. A remarkable property o Max- Weight policies is their throughput optimality, i.e., their ability to stabilize a queueing network whenever this is possible, without any inormation on the arriving traic. Moreover, it has been shown that policies rom this class achieve low, or even optimal, average delay or speciic network topologies, when the arriving traic is light-tailed [8, 16, 0, 3, 5]. 1 However, the perormance o Max-Weight scheduling in the presence o heavy-tailed traic is not well understood. We are motivated to study networks with heavy-tailed traic by signiicant evidence that traic in real-world communication networks exhibits strong correlations and statistical similarity over dierent time scales. This observation was irst made by Leland et al. [13] through analysis o Ethernet traic traces. Subsequent empirical studies have documented this phenomenon in other networks, while accompanying theoretical studies have associated it with arrival processes that have heavy tails; see [17] or an overview. The impact o heavy tails has been analyzed extensively in the context o single or multi-server queues; see the survey papers [, 4], and the reerences therein. However, the related work is rather limited in the context o queueing networks, e.g., see the paper by Borst et al. [3], which studies the Generalized Processor Sharing policy. 1 On the other hand, when Max-Weight scheduling is combined with Back-Pressure routing in the context o multi-hop networks, there is evidence that delay perormance can be poor, e.g., see the discussion in [5].

3 This paper aims to ill a gap in the literature, by analyzing the impact o heavy-tailed traic on the perormance o Max-Weight scheduling in single-hop queueing networks. In particular, we study the delay stability o traic lows: a traic low is delay stable i its expected steady-state delay is inite, and delay unstable otherwise. Our previous work [15] gives some preliminary results in this direction, in a simple system with two parallel queues and a single server. The main contributions o this paper include: i) in a single-hop queueing network under the Max-Weight scheduling policy, we show that any light-tailed low that conlicts with a heavy-tailed low is delay unstable; ii) surprisingly, we also show that or certain admissible arrival rates, a light-tailed low can be delay unstable even i it does not conlict with heavy-tailed traic; iii) we analyze the Max- Weight-α scheduling policy, and show that i the α-parameters are chosen suitably, then the sum o the α-moments o the steady-state queue lengths is inite. We use this result to prove that by proper choice o the α-parameters, all light-tailed lows are delay stable. Moreover, we show that Max-Weight-α achieves the optimal scaling o higher moments o steady-state queue lengths with traic intensity. The rest o the paper is organized as ollows. Section contains a detailed presentation o the model that we analyze, namely, a single-hop queueing network. It also deines ormally the notions o heavy-tailed and light-tailed traic, and o delay stability. In Section 3 we motivate the subsequent development by presenting, inormally and through simple examples, the main results o the paper. In Section 4 we analyze the perormance o the celebrated Max-Weight scheduling policy. Our general results are accompanied by examples, which illustrate their implications in practical network settings. Section 5 contains the analysis o the parameterized Max-Weight-α scheduling policy, and the perormance that it achieves in terms o delay stability. This section also includes results about the scaling o moments o steady-state queue lengths with the traic intensity and the size o the network, accompanied by several examples. We conclude with a discussion o our indings and uture research directions in Section 6. The appendices contain some background material and most o the proos o our results. 3

4 Model and Problem Formulation We start with a detailed presentation o the queueing model considered in this paper, together with some necessary deinitions and notation. We denote by R +, Z +, and N the sets o nonnegative reals, nonnegative integers, and positive integers, respectively. The cartesian products o M copies o R + and Z + are denoted by R M + and Z M +, respectively. We assume that time is slotted and that arrivals occur at the end o each time slot. The topology o the network is captured by a directed graph G = (N, E), where N is the set o nodes and E is the set o (directed) edges. Our model involves single-hop traic lows: data arrives at the source node o an edge, or transmission to the node at the other end o the edge, where it exits the network. More ormally, let F N be the number o traic lows o the network. A traic low {1,..., F } consists o a discrete time stochastic arrival process {A (t); t Z + }, a source node s(), and a destination node d(), with s(), d() N, and (s(), d()) E. We assume that each arrival process {A (t); t Z + } takes values in Z +, and is independent and identically distributed (IID) over time. Furthermore, the arrival processes associated with dierent traic lows are mutually independent. We denote by λ = E[A (0)] > 0 the rate o traic low, and by λ = (λ ; = 1,..., F ) the vector o the rates o all traic lows. Deinition 1: (Heavy Tails) A traic low is heavy-tailed i E[A (0)] =, and light-tailed otherwise. The traic o low is buered in a dedicated queue at node s() (queue, henceorth.) Our modeling assumptions imply that the set o traic lows can be identiied with the set o edges and the set o queues o the network. The service discipline within each queue is assumed to be First Come, First Served. The stochastic process {Q (t); t Z + } captures the evolution o the length o queue. Since our motivation comes rom communication networks, A (t) will be interpreted as the number o packets that queue receives at the end o time slot t, and Q (t) as the total number o packets in queue at the beginning o time slot t. The arrivals and the lengths o the various queues at time slot t are captured by the vectors A(t) = (A (t); = 1,..., F ) and 4

5 Q(t) = (Q (t); = 1,..., F ), respectively. In the context o a communication network, a batch o packets arriving to a queue at any given time slot can be viewed as a single entity, e.g., as a ile that needs to be transmitted. We deine the end-to-end delay o a ile o low to be the number o time slots that the ile spends in the network, starting rom the time slot right ater it arrives at s(), until the time slot that its last packet reaches d(). For k N, we denote by D (k) the end-to-end delay o the k th ile o queue. The vector D(k) = (D (k); = 1,..., F ) captures the end-to-end delay o the k th iles o the dierent traic lows. In general, not all edges can be activated simultaneously, e.g., due to intererence in wireless networks, or matching constraints in a switch. Consequently, not all traic lows can be served simultaneously. A set o traic lows that can be served simultaneously is called a easible schedule. We denote by S the set o all easible schedules, which is assumed to be an arbitrary subset o the powerset o {1,..., F }. For simplicity, we assume that all attempted transmissions o data are successul, that all packets have the same size, and that the transmission rate along any edge is equal to one packet per time slot. We denote by S (t) {0, 1} the number o packets that are scheduled or transmission rom queue at time slot t. Note that this is not necessarily equal to the number o packets that are transmitted because the queue may be empty. Let us now deine ormally the notion o a scheduling policy. The past history and present state o the system at time slot t N is captured by the vector H(t) = (Q(0), A(0),..., Q(t 1), A(t 1), Q(t)). At time slot 0, we have H(0) = (Q(0)). A (causal) scheduling policy is a sequence π = (µ 0, µ 1,...) o unctions µ t : H(t) S, t Z +, used to determine scheduling decisions, according to S(t) = µ t (H(t)). Using the notation above, the dynamics o queue take the orm: Q (t + 1) = Q (t) + A (t) S (t) 1 {Q (t)>0}, 5

6 or all t Z +, where 1 {Q (t)>0} denotes the indicator unction o the event {Q (t) > 0}. The vector o initial queue lengths Q(0) is assumed to be an arbitrary element o Z F +. We restrict our attention to scheduling policies that are regenerative, i.e., policies under which the network starts aresh probabilistically in certain time slots. More precisely, under a regenerative policy there exists a sequence o stopping times {τ n ; n Z + } with the olowing properties. i) The sequence {τ n+1 τ n ; n Z + } is IID. ii) Let X(t) = (Q(t), A(t), S(t)), and consider the processes that describe the cycles o the network, namely, C 0 = {X(t); 0 t < τ 0 }, and C n = {X(τ n 1 + t); 0 t < τ n τ n 1 }, n N; then, {C n ; n N} is an IID sequence, independent o C 0. iii) The (lattice) distribution o the cycle lengths, τ n+1 τ n, has span equal to one and inite expectation. Properties (i) and (ii) imply that the queueing network evolves like a (possibly delayed) regenerative process. Property (iii) states that this process is aperiodic and positive recurrent, which will be crucial or the stability o the network. The ollowing deinition gives the precise notion o stability that we use in this paper. Deinition : (Stability) The single-hop queueing network described above is stable under a speciic scheduling policy, i the vector-valued sequences {Q(t); t Z + } and {D(k); k N} converge in distribution, and their limiting distributions do not depend on the initial queue lengths Q(0). Notice that our deinition o stability is slightly dierent than the commonly used deinition (positive recurrence o the Markov chain o queue lengths), since it includes the convergence o the sequence o ile delays {D(k); k N}. The reason is that in this paper we study properties o the limiting distribution o {D(k); k N} and, naturally, we need to ensure that this limiting distribution exists. Under a stabilizing scheduling policy, we denote by Q = (Q ; = 1,..., F ) and D = (D ; = 1,..., F ) the limiting distributions o {Q(t); t Z + } and {D(k); k N}, respectively. The dependence o these limiting distributions on the scheduling policy has been suppressed rom the notation, but will be clear rom the context. We reer to Q as the steady-state length o queue. Similarly, we reer to D as the steady-state delay o a ile o traic low. We note that 6

7 under a regenerative policy (i one exists), the queueing network is guaranteed to be stable. This is because the sequences o queue lengths and ile delays are (possibly delayed) aperiodic and positive recurrent regenerative processes, and, hence, converge in distribution; see []. The stability o the queueing network depends on the rates o the various traic lows relative to the transmission rates o the edges and the scheduling constraints. This relation is captured by the stability region o the network. Deinition 3: (Stability Region) [4] The stability region o the single-hop queueing network described above, denoted by Λ, is the set o rate vectors: { λ R F + ζ s R +, s S : λ ζ s s, } ζ s < 1. s S s S In other words, a rate vector λ belongs to Λ i there exists a convex combination o easible schedules that covers the rates o all traic lows. I a rate vector is in the stability region o the network, then the traic corresponding to this vector is called admissible, and there exists a scheduling policy under which the network is stable. Deinition 4: (Traic Intensity) The traic intensity o a rate vector λ Λ is a real number in [0,1) deined as: { } ρ(λ) = in ζ s λ ζ s s, ζ s R +, s S. s S s S Clearly, arriving traic with rate vector λ is admissible i and only i ρ(λ) < 1. Throughout this paper we assume that the traic is admissible. Let us now deine the property that we use to evaluate the perormance o scheduling policies, namely, the delay stability o a traic low. Deinition 5: (Delay Stability) A traic low is delay stable under a speciic scheduling policy i the queueing network is stable under that policy and E[D ] < ; otherwise, the traic low is delay unstable. 7

8 The ollowing lemma relates the steady-state quantities E[Q ] and E[D ], and will help us prove delay stability results. Lemma 1: Consider the single-hop queueing network described above under a regenerative scheduling policy. Then, E[Q ] < E[D ] <, {1,..., F }. Proo. see Appendix 1.1. Theorem 1: (Delay Instability o Heavy Tails) Consider the single-hop queueing network described above under a regenerative scheduling policy. Every heavy-tailed traic low is delay unstable. Proo. (Sketch) The result ollows easily rom the Pollaczek-Khinchine ormula or the expected delay in a M/G/1 queue, and a stochastic comparison argument. The main idea is that in a heavytailed traic low, the probability that a very big ile arrives to the respective queue is relatively high. Combined with the First Come, First Served discipline within the queue, this implies that a large number o iles, arriving ater the big one, experience very large delays. This is true even i the queue gets served whenever it is nonempty, namely, i the queue is given preemptive priority. Consequently, under any scheduling policy, there is relatively high probability that a large number o iles experiences very large delays. This then implies that a heavy-tailed traic low is delay unstable. For a ormal proo see Appendix. Since there is little we can do about the delay stability o heavy-tailed lows, we turn our attention to light-tailed traic. The Pollaczek-Khinchine ormula or the expected delay in a M/G/1 queue implies that the intrinsic burstiness o light-tailed traic is not suicient to cause delay instability. However, scheduling in a queueing network couples the statistics o dierent traic lows. We will see that this coupling can cause light-tailed lows to become delay unstable, giving rise to a orm o propagation o delay instability. 8

9 3 Overview o Main Results In this section we introduce, inormally and through simple examples, the main results o the paper and the basic intuition behind them. Let us start with the queueing system o Figure 1, which consists o two parallel queues and a single server. Traic low 1 is assumed to be heavy-tailed, whereas traic low is light-tailed. Service is allocated according to the Max-Weight scheduling policy, which is equivalent to Serve the Longest Queue in this simple setting. Theorem 1 implies that traic low 1 is delay unstable. Our indings imply that traic low is also delay unstable, even though it is light-tailed. The intuition behind this result is that queue 1 is occasionally very long (ininite, in steady-state expectation) because o its heavy-tailed arrivals. When this happens, and under the Max-Weight policy, queue has to build up to a similar length in order to receive service. A very long queue then implies very large delays or the iles o that queue under First Come, First Served, which leads to delay instability. Figure 1: Delay instability in parallel queues with heavy-tailed traic. Systems o parallel queues have been analyzed extensively in the literature. One o the main reasons is that their simple dynamics oten lead to elegant analysis and clean results. However, realworld communication networks are much more complex. In this paper we go beyond parallel queues and analyze queueing networks with more complicated structure. A simple example is the queueing network o Figure, where traic low 1 is assumed to be heavy-tailed, whereas traic lows and 3 are light-tailed. The server can serve either queue 1 alone, or queues and 3 simultaneously. This example could represent a wireless network with intererence constraints. In this setting the Max-Weight policy compares the length o queue 1 to the sum o the lengths o queues and 3, and serves the heavier schedule. 9

10 Figure : Propagation o delay instability: conlicting with heavy-tailed traic. The intuition rom the previous example suggests that at least one o the queues and 3 has to build up to the order o magnitude o queue 1, in order or these two queues to receive service. In other words, we expect that at least one o the traic lows and 3 will be delay unstable under Max-Weight. Our indings imply that, in act, both traic lows are delay unstable. The main idea behind this result is the ollowing: with positive probability, the arrival processes to queues and 3 exhibit their average behavior. In that case, the corresponding queues build up slowly and together, which implies that when they claim the server they have both built up to the order o magnitude o queue 1. The simple networks o Figures 1 and illustrate special cases o a general result: every lighttailed low that conlicts with a heavy-tailed low is delay unstable. For more details see Theorem in Section 4.1. Figure 3: Propagation o delay instability: concurring with heavy-tailed traic. Going one step urther, consider the queueing network o Figure 3. Traic low 1 is assumed to be heavy-tailed, whereas traic lows and 3 are light-tailed. The server can serve either queues 1 and simultaneously, or queue 3 alone. In this setting the Max-Weight policy compares the length 10

11 o queue 3 to the sum o the lengths o queues 1 and, and serves the heavier schedule. The intuition rom the previous examples suggests that traic low 3 is delay unstable, but the real question is the delay stability o traic low. One would expect that this low is delay stable: it is light-tailed itsel, and is served together with a heavy-tailed low, which should result in more service opportunities under Max-Weight. Surprisingly though, we show that there exist arrival rates within the stability region o this network, such that traic low is delay unstable. The key observation here is that even though traic low does not conlict with heavy-tailed traic, it does conlict with traic low 3, which is delay unstable because it conlicts with heavy-tailed traic. For more details see Propositions 1, 3, and 4 in Sections 4. and 4.3. The examples above suggest that in queueing networks with heavy-tailed traic, delay instability not only appears but propagates through the network under the Max-Weight policy. Seeking a remedy to this situation, we turn to the more general Max-Weight-α scheduling policy. This policy assigns a positive α-parameter to each traic low, and instead o comparing the lengths o the queues/schedules, and serving the longest one, it compares the lengths o the queues to the respective α-powers. Our indings imply that in the network o Figure 1, we can guarantee that traic low is delay stable, provided the α-parameter or traic low 1 is suiciently small. In other words, we prevent the propagation o delay instability. This is a special case o a general result: i the α-parameters o the Max-Weight-α policy are chosen suitably, then the sum o the α-moments o the steady-state queue lengths is inite. For more details see Theorem 3 in Section Max-Weight Scheduling In this section we evaluate the perormance o the Max-Weight scheduling policy, with respect to the delay stability o traic lows. Inormally speaking, the weight o a easible schedule is the sum o the lengths o all queues included in it. As its name suggests, the Max-Weight policy activates a easible schedule with the maximum weight at any given time slot. More ormally, under 11

12 the Max-Weight policy, the scheduling vector S(t) belongs to the set: S(t) arg max (s ) S { F Q (t) s }. I this set includes multiple easible schedules, then one o them is chosen uniormly at random. The ollowing lemma states that the network is stable under the Max-Weight policy. Essentially, this result is well-known, e.g., or light-tailed traic, see [4]; or more general arrivals, see [3]. A subtle point is that in this paper we adopt a somewhat dierent deinition or stability. So, we have to ensure that, apart rom the sequences o queue lengths, the sequences o ile delays converge as well. Lemma : (Stability under Max-Weight) The single-hop queueing network described in =1 Section is stable under the Max-Weight scheduling policy. Proo. Consider the single-hop queueing network o Section under the Max-Weight scheduling policy. It can be veriied that the sequence {Q(t); t Z + } is a time-homogeneous, irreducible, and aperiodic Markov chain on the countable state-space Z F +. Proposition o [3] implies that this Markov chain is also positive recurrent. Hence, {Q(t); t Z + } converges in distribution, and its limiting distribution does not depend on Q(0). Based on this, it can be veriied that the sequence {D(k); k N} is a (possibly delayed) aperiodic and positive recurrent regenerative process. Thereore, it also converges in distribution, and its limiting distribution does not depend on Q(0); see []. 4.1 Conlicting with Heavy-Tailed Flows In this section we state one o the main results o the paper, which generalizes our observations rom the simple networks o Figures 1 and. Beore we give the result, though, let us deine precisely the notion o conlict between traic lows. Deinition 6: The traic low conlicts with, and vice versa, i there exists no easible schedule in S that includes both and. 1

13 Theorem : (Conlicting with Heavy Tails) Consider the single-hop queueing network described in Section under the Max-Weight scheduling policy. Every light-tailed low that conlicts with a heavy-tailed low is delay unstable. Proo. (Sketch) Let h and l be a heavy-tailed and a light-tailed traic low, respectively, and suppose that l conlicts with h. Queue h is occasionally very long (ininite, in steady-state expectation), due to the heavy-tailed nature o the traic that it receives. In order or queue l to get served, the weight o at least one easible schedule that includes l has to build up to the order o magnitude o queue h. However, with positive probability, the arrival processes o all easible schedules that include l exhibit their average behavior. In that case, queue l builds up at a roughly constant rate, or a time period o the order o magnitude o queue 1. Combined with Lemma 1, this implies that traic low l is delay unstable. For a ormal proo see Appendix 3. We emphasize the generality o this result. Namely, a light-tailed low that conlicts with heavytailed traic is delay unstable, irrespective o: i) its rate; ii) the tail asymptotics o its underlying distribution; iii) whether it is scheduled alone or with other traic lows. Hence, we view Theorem as capturing a universal phenomenon or the propagation o delay instability. 4. Concurring with Heavy-Tailed Flows So ar we have shown that: i) a heavy-tailed traic low is delay unstable under any regenerative scheduling policy; and ii) a light-tailed traic low that conlicts with a heavy-tailed low is delay unstable under the Max-Weight scheduling policy. It seems reasonable, however, that a light-tailed low that does not conlict with heavy-tailed traic should be delay stable. Unortunately, this is not always the case. We demonstrate this by means o simple examples. Let us come back to the queueing network o Figure 3. The easible schedules o this network are {1, } and {3}, and all queues are served at unit rate, whenever the respective schedules are activated. The rate vector λ = (λ 1, λ, λ 3 ) is assumed admissible. The ollowing proposition shows that traic low is delay unstable i its rate is suiciently high. 13

14 Proposition 1: (Concurring with Heavy Tails) Consider the single-hop queueing network o Figure 3 under the Max-Weight scheduling policy. I the arriving traic is admissible and the rates satisy λ > (1 + λ 1 λ 3 )/, then traic low is delay unstable. Proo. (Sketch) Let us irst give the intuition or the special case, where λ 1 = λ 3. Consider sample paths or which a very large ile arrives to queue 1; this is a relatively likely event, since traic low 1 is heavy-tailed. Queue 3 will build up to the order o magnitude o the large ile in queue 1 in order to receive service. Starting rom the time slot that the weights o the two schedules become equal, the Max-Weight policy will be draining the weights o the two schedules at the same rate. The period o time until they empty is o the order o magnitude o the large ile in queue 1. Now assume that queue stays small throughout this period. I the traic lows 1 and 3 exhibit their average behavior, then each easible schedule will be activated once every two time slots, since λ 1 = λ 3. However, i λ > 1/, queue will build up to the order o magnitude o the large ile in queue 1, which is a contradiction. The intuition or the more general case is based on the ollowing luid argument : assume that the arrivals at each queue {1,, 3} are a luid with rate λ. The departures rom queue during periods when all queues are nonempty are also assumed to be a luid with rate µ. The Max-Weight policy has the property o draining the weights o the two easible schedules at the same rate. Hence, the departure rates are the solution to the ollowing system o linear equations: λ 1 + λ µ 1 µ = λ 3 µ 3 µ 1 + µ 3 = 1 µ 1 = µ. The last two equations ollow rom the acts that Max-Weight is a work-conserving policy, and that queues 1 and are served simultaneously. I the rate at which luid arrives to queue is greater than the rate at which it departs, i.e., λ > µ = 1 + λ 1 + λ λ 3, 3 14

15 or, equivalently, λ > 1 + λ 1 λ 3, then queue builds up over long periods o time, which, combined with Lemma 1, implies the delay instability o low. A ormal proo essentially shows that this luid model is a aithul approximation o the actual stochastic system (with nonvanishing probability), whenever queue 1 receives a large ile; see Appendix 4. Proposition 1, as well as Propositions 3 and 4 o the next section, capture a rate-dependent phenomenon or the propagation o delay instability. We conjecture that a converse to Proposition 1 also holds; namely, that queue is delay stable i the arriving traic is admissible and λ < (1 + λ 1 λ 3 )/. 4.3 Practical Examples and Implications We illustrate the implications o the results presented so ar in the context o speciic network topologies, oten used to model real-world communication networks. Example 1: (Parallel Queues) Consider the network o Figure 4, consisting o n parallel queues and a single server. Networks o parallel queues are oten used to model wireless uplinks, downlinks, and call centers. Traic low 1 is assumed to be heavy-tailed, whereas the other traic lows are light-tailed. The scheduling constraints o parallel queues require that no two queues can be served simultaneously. The server is allocated according to the Max-Weight scheduling policy, which in this setting is equivalent to Serve the Longest Queue. Proposition : Consider the system o parallel queues depicted in Figure 4, under the Max- Weight scheduling policy. I traic low 1 is heavy-tailed, then all traic lows are delay unstable. Proo. The result ollows easily rom Theorems 1 and. Example : (Input-Queued Switch) Consider the input-queued switch depicted in Figure 5. Input-queued switches are oten used to model internet routers. Traic low (1,1) is 15

16 Figure 4: Delay instability in parallel queues under Max-Weight scheduling: i traic low 1 is heavy tailed (black), then all traic lows are delay unstable (gray.) assumed to be heavy-tailed, whereas all other lows are light-tailed. The scheduling constraints o an input-queued switch require that every easible schedule has to be a matching between the sets o input and output ports. Thus, the easible schedules o the network are {(1, 1), (, )} and {(1, ), (, 1)}. In this setting the Max-Weight scheduling policy activates a matching with the maximum weight. Figure 5: Delay instability in a data switch under Max-Weight scheduling: i traic low (1,1) is heavy tailed (black), then traic lows (1,) and (,1) are delay unstable (gray.) Traic low (,) is also delay unstable, i its rate is suiciently high. Proposition 3: Consider the input-queued switch depicted in Figure 5, under the Max- Weight scheduling policy. I traic low (1,1) is heavy-tailed, then traic lows (1,1), (1,), and (,1) are all delay unstable. I, additionally, λ > ( + λ 11 λ 1 λ 1 )/3, then traic low (,) is also delay unstable. Proo. The irst part o the result ollows rom Theorems 1 and. Regarding the second part, we 16

17 provide the calculations or the associated luid model, which justiy the particular threshold or λ : assume that the arrivals at each queue {(1, 1), (1, ), (, 1), (, )} are a luid with rate λ. The departures rom queue during periods when all queues are nonempty are also assumed to be a luid with rate µ. The Max-Weight policy has the property o draining the weights o the two easible schedules at the same rate. Hence, the departure rates are the solution to the ollowing system o linear equations: λ 11 + λ µ 11 µ = λ 1 + λ 1 µ 1 µ 1 µ 11 + µ 1 = 1 µ 11 = µ µ 1 = µ 1. The second equation is a consequence o the work-conserving nature o the Max-Weight policy. The last two equations ollow rom the acts that queue (1,1) is served simultaneously with queue (,), and queue (1,) is served simultaneously with queue (,1). I the rate at which luid arrives to queue (,) is greater than the rate at which it departs, i.e., i λ > µ = + λ 11 + λ λ 1 λ 1, 4 or, equivalently, i λ > + λ 11 λ 1 λ 1, 3 then queue (,) builds up over long periods o time, which, combined with Lemma 1, implies the delay instability o low (,). The proo that the stochastic model ollows the luid model is similar to the proo o Proposition 1 and is omitted. Example 3: (Wireless Ring) Consider the wireless ring network o Figure 6. The network consists o 6 nodes, each o which receives traic that it transmits to its neighboring node in the clockwise direction. Traic low 1 is assumed to be heavy-tailed, whereas all other lows are lighttailed. Due to wireless intererence, i a link o the network is activated, then the links within 17

18 two-hop distance must be inactive; this is the so-called two-hop intererence model. Thus, the easible schedules o the network are {1, 4}, {, 5}, and {3, 6}. Figure 6: Delay instability in a wireless ring network under Max-Weight scheduling: i traic low 1 is heavy tailed (black), then traic lows, 3, 5, and 6 are delay unstable (gray.) Traic low 4 is also delay unstable, i its rate is suiciently high. Proposition 4: Consider the wireless ring network depicted in Figure 6, under the Max-Weight scheduling policy. I traic low 1 is heavy-tailed, then traic lows 1,, 3, 5, and 6 are all delay unstable. I, additionally, λ 4 > ( + λ 1 λ λ 3 λ 5 λ 6 )/4, then traic low 4 is also delay unstable. Proo. The irst part o the result ollows rom Theorems 1 and. Regarding the second part, we provide the analysis o the associated luid model: assume that the arrivals at each queue {1,, 3, 4, 5, 6} are a luid with rate λ. The departures rom queue during periods when all queues are nonempty are also assumed to be a luid with rate µ. The Max-Weight policy has the property o draining the weights o the three easible schedules at the same rate. Hence, the departure rates are the solution to the ollowing system o linear equations: λ 1 + λ 4 µ 1 µ 4 = λ + λ 5 µ µ 5 λ 1 + λ 4 µ 1 µ 4 = λ 3 + λ 6 µ 3 µ 6 µ 1 + µ + µ 3 = 1 µ 1 = µ 4 µ = µ 5 µ 3 = µ 6. 18

19 The third equation is a consequence o the work-conserving nature o the Max-Weight policy. The last three equations ollow rom the acts that queue 1 is served simultaneously with queue 4, and similarly or queues and 5, and queues 3 and 6. I the rate at which luid arrives to queue 4 is greater than the rate at which it departs, i.e., i λ 4 > µ 4 = + λ 1 + λ 4 λ λ 3 λ 5 λ 6, 6 or, equivalently, i λ 4 > + λ 1 λ λ 3 λ 5 λ 6, 4 then queue 4 builds up over long periods o time, which, combined with Lemma 1, implies the delay instability o low 4. A detailed proo is omitted or brevity. 5 Max-Weight-α Scheduling The results o the previous section suggest that Max-Weight scheduling perorms poorly in the presence o heavy-tailed traic. The reason is that by treating heavy-tailed and light-tailed lows equally, there are very long stretches o time during which heavy-tailed traic dominates the service. This leads some light-tailed lows to experience very large delays and, eventually, to become delay unstable. Intuitively, by discriminating against heavy-tailed lows one should be able to improve the overall perormance o the network, namely to mitigate the propagation o delay instability. One way to do this is by giving preemptive priority to the light-tailed lows. However, priority-based scheduling policies are undesirable because o airness considerations, and also because they can be unstable in many network settings, e.g., see [1, 18]. Instead, we ocus on the Max-Weight-α scheduling policy: given constants α > 0, or all {1,..., F }, the scheduling vector S(t) belongs to the set: S(t) arg max (s ) S { F =1 } Q α (t) s. 19

20 I this set includes multiple easible schedules, one o them is chosen uniormly at random. By choosing smaller values o the α-parameters or heavy-tailed lows and larger values or light-tailed lows, we give a orm o partial priority to light-tailed traic. 5.1 The Main Result Let us start with a preview o the main result o this section: i the α-parameters o the Max- Weight-α policy are chosen such that E[A α +1 (0)] <, or all {1,..., F }, then the network is stable and the steady-state queue lengths satisy: E[Q α ] <, {1,..., F }. An earlier work by Eryilmaz et al. has given a similar result or the case o parallel queues with a single server; see Theorem 1 o [6]. In this paper we extend their result to a general single-hop network setting. Moreover, we provide an explicit upper bound to the sum o the α-moments o the steady-state queue lengths. Beore we do that we need the ollowing deinition. Deinition 7: (Covering Number o Feasible Schedules) The covering number k o the set o easible schedules is deined as the smallest number k or which there exist s 1,..., s k S with k i=1 si = {1,..., F }. Notice that the quantity k is a structural property o the queueing network, and is not related to the scheduling policy or the statistics o the arriving traic: it is the minimum number o time slots required to serve at least one packet rom each low. Theorem 3: (Max-Weight-α Scheduling) Consider the single-hop queueing network described in Section under the Max-Weight-α scheduling policy. Let the intensity o the arriving traic be ρ < 1. I E[A α +1 (0)] <, or all {1,..., F }, then the queueing network is stable and the steady-state queue lengths satisy: F =1 E[Q α ] F =1 ( ) H ρ, k, α, E[A α +1 (0)], 0

21 where ( ) H ρ, k, α, E[A α +1 (0)] = ( ) and K = α 1 α E[A α +1 (0)] + 1. k 1 ρ ( k 1 ρ ( ) E[A α +1 (0)] + 1, α 1, ) α K α + k 1 ρ K, α > 1, Proo. (Sketch) Consider the single-hop queueing network o Section under the Max-Weight-α scheduling policy. It can be veriied that the sequence {Q(t); t Z + } is a time-homogeneous, irreducible, and aperiodic Markov chain on the countable state-space Z F +. The act that this Markov chain is also positive recurrent, and the related moment bound, are based on drit analysis o the Lyapunov unction F 1 V (Q(t)) = α + 1 Qα +1 (t), =1 and use o the Foster-Lyapunov stability criterion. This implies that {Q(t); t Z + } converges in distribution, and its limiting distribution does not depend on Q(0). Based on this, it can be veriied that the sequence {D(k); k N} is a (possibly delayed) aperiodic and positive recurrent regenerative process. Hence, it also converges in distribution, and its limiting distribution does not depend on Q(0). For a ormal proo see Appendix Traic Burstiness and Delay Stability A irst corollary o Theorem 3 relates to the delay stability o light-tailed lows. Corollary 1: (Delay Stability under Max-Weight-α) Consider the single-hop queueing network described in Section under the Max-Weight-α scheduling policy. I the α-parameters o all light-tailed lows are equal to 1, and the α-parameters o heavy-tailed lows are suiciently small, then all light-tailed lows are delay stable. Proo. With the particular choice o α-parameters, Theorem 3 guarantees that the expected steadystate queue length o all light-tailed lows is inite. Lemma 1 relates this result to delay stability. 1

22 Combining this with Theorem 1, we conclude that when its α-parameters are chosen suitably, the Max-Weight-α policy delay-stabilizes a traic low, whenever this is possible. Max-Weight-α turns out to perorm well in terms o another criterion too. Theorem 3 implies that by choosing the α-parameters such that E[A α +1 (0)] <, or all {1,..., F }, the steadystate queue length moment E[Q α ] is inite, or all {1,..., F }. The ollowing proposition suggests that this is the best we can do under any regenerative scheduling policy. Proposition 5: Consider the single-hop queueing network described in Section under a regenerative scheduling policy. Then, E[A c+1 (0)] = = E[Q c ] =, {1,..., F }. Proo. This result is well-known in the context o a M/G/1 queue, e.g., see Section 3. o [4]. It can be proved similarly to Theorem 1. Thus, when its α-parameters are chosen suitably, the Max-Weight-α policy guarantees the initeness o the highest possible moments o steady-state queue lengths. 5.3 Scaling Results under Light-Tailed Traic Although this paper ocuses on heavy-tailed traic and its consequences, some implications o Theorem 3 are o general interest. In this section we assume that all traic lows in the network are light-tailed, and analyze how the sum o the α-moments o steady-state queue lengths scales with traic intensity and the size o the network. Corollary : (Scaling with Traic Intensity) Let us ix a single-hop queueing network and constants α 1 and B > 0. The Max-Weight-α scheduling policy is applied with α = α, or all {1,..., F }. Assume that the traic arriving to the network is admissible, and that the (α + 1)-moments o all traic lows are bounded rom above by B. Then, F =1 E[Q α ] M(k, α, B) (1 ρ) α,

23 where M(k, α, B) is a constant that depends only on k, α, and B. Moreover, under any stabilizing scheduling policy F E[Q α ] =1 M (α) (1 ρ) α, where M (α) is a constant that depends only on α. Proo. I α = α 1, or all {1,..., F }, then Theorem 3 implies that: F =1 E[Q α ] M(k, α, B) (1 ρ) α, where M(k, α, B) is a constant that depends only on k, α, and B. On the other hand, Theorem.1 o [1] implies that under any stabilizing scheduling policy there exists an absolute constant M, such that Utilizing Jensen s inequality, we have: F E[Q ] =1 M (1 ρ). F F E[Q α ] (E[Q ]) α =1 =1 1 ( F α. F α E[Q ]) =1 Consequently, there exists a constant M (α) that depends only on α, such that under any stabilizing scheduling policy. F E[Q α ] =1 M (α) (1 ρ) α, Similar scaling results appear in queueing theory, mostly in the context o single-server queues, e.g., see Chapter 3 o [11]. More recently, results o this lavor have been shown or particular 3

24 queueing networks, such as input-queued switches [19, 1]. All the related work, though, concerns the scaling o irst moments. Corollary gives the precise scaling o higher order steady-state queue length moments with traic intensity, and shows that Max-Weight-α achieves the optimal scaling. We now turn our attention to the perormance o the Max-Weight scheduling policy under Bernoulli traic, i.e., when each o the arrival processes {A (t); t Z + } is an independent Bernoulli process with parameter λ > 0. serve. We denote by S max the maximum number o traic lows that any easible schedule s S can Corollary 3: (Scaling under Bernoulli Traic) Consider the single-hop queueing network described in Section under the Max-Weight scheduling policy. Assume that the traic arriving to the network is Bernoulli, with traic intensity ρ < 1. Then, F E[Q ] k S max =1 ( 1 + ρ ). 1 ρ Proo. I all traic lows are light-tailed and all the α-parameters are equal to one, a more careul accounting in the proo o Theorem 3 provides the ollowing tighter upper bound: F E[Q ] =1 k ( 1 ρ S max + F E[A ). (0)] =1 I the traic arriving to the network is Bernoulli, then E[A (0)] = λ, or all {1,..., F }. Moreover, the act that the arriving traic has intensity ρ, implies the existence o nonnegative real numbers ζ s, or s S, such that: λ s S ζ s s, {1,..., F }, and F ζ s = ρ. =1 4

25 Consequently, F F E[A (0)] = =1 =1 λ F ζ s s =1 s S = F ζ s s s S =1 ζ s S max s S = ρ S max, and the result ollows. Example 4: (n Parallel Queues) Consider a single-server system with n parallel queues. The arriving traic is assumed to be Bernoulli, with traic intensity ρ < 1. In this case k = n and S max = 1. Corollary 3 implies that under the Max-Weight scheduling policy, the sum o the steady-state queue lengths is bounded rom above by: n E[Q i ] i=1 4n 1 ρ. The total queue length o a system o parallel queues under a work-conserving scheduling policy evolves like a Geo [B] /D/1 queue, rom which we iner that ) n i=1 E[Q i] = Θ. So, in the ( 1 1 ρ context o parallel queues, the scaling provided by Corollary 3 is tight with respect to the traic intensity, but not necessarily tight with respect to the size o the network. Example 5: (n n Input-Queued Switch) Consider a n n input-queued switch. The arriving traic is assumed to be Bernoulli, with traic intensity ρ < 1. In this case k = n and S max = n. Corollary 3 implies that under the Max-Weight scheduling policy, the sum o the 5

26 steady-state queue lengths is bounded rom above by: n n i=1 j=1 E[Q ij ] 4n 1 ρ. In the context o input-queued switches, the joint scaling provided by Corollary 3, in terms o both the traic intensity and the size o the network, is the tightest currently known. However, it should be noted that the correct scaling as n and ρ 1 is an open problem; see [19]. Example 6: (n n Grid) Consider a single-hop queueing network in a n n grid topology, under the one-hop intererence model. The arriving traic is assumed to be Bernoulli, with traic intensity ρ < 1. In this case k 4 and S max n /. Corollary 3 implies that under the Max-Weight scheduling policy, the sum o the steady-state queue lengths is bounded rom above by: n n i=1 j=1 E[Q ij ] 8n 1 ρ. 6 Discussion The main conclusion o this paper is that the celebrated Max-Weight scheduling policy perorms poorly in the presence o heavy-tailed traic. More speciically, our indings show that the phenomenon o delay instability not only arises, but can propagate to a signiicant part o the network. This is somewhat surprising, since Max-Weight is known to perorm very well in the presence o light-tailed traic, at least in single-hop queueing networks. Another important conclusion is that the Max-Weight-α scheduling policy can be used to alleviate the eects o heavy-tailed traic, and is even order optimal, i its α-parameters are chosen suitably. However, or Max-Weight-α to perorm well, accurate knowledge o the tail coeicients o all traic lows is required. I the α-parameters are not chosen appropriately, then in light o Proposition 5, this policy may also perorm poorly. O particular interest is the study o networks with time-varying channel state. In this class o models there exists an underlying state o the network which evolves in time, and the transmission 6

27 rates o the links are given by a unction o the state. Under certain conditions on the channel state evolution, it can be veriied that Theorems 1-3 carry over with minimal changes to this more general setting. An important direction or uture research is to consider queueing networks with correlated traic. The IID assumption that we made here acilitates the analysis and oers valuable insights, but is clearly restrictive. As alluded to earlier, evidence suggests that traic in real-world networks exhibits strong correlations, and phenomena such as sel-similarity and long-range dependence arise. Concrete results in this direction would be o great theoretical and practical interest. Reerences [1] M. Andrews, K. Kumaran, K. Ramanan, A. Stolyar, R. Vijayakumar, P. Whiting (004). Scheduling in a queueing system with asynchronously varying service rates. Probability in the Engineering and Inormational Sciences, 18, [] S. Borst, O. Boxma, R. Nunez-Queija, B. Zwart (003). The impact o the service discipline on delay asymptotics. Perormance Evaluation, 54, [3] S. Borst, M. Mandjes, M. van Uitert (003). Generalized processor sharing with light-tailed and heavy-tailed input. IEEE/ACM Transactions on Networking, 11, [4] O. Boxma, B. Zwart (007). Tails in scheduling. Perormance Evaluation Review, 34, [5] L. Bui, R. Srikant, A. Stolyar (009). Novel architectures and algorithms or delay reduction in back-pressure scheduling and routing. In: Proc. Inocom 009. [6] A. Eryilmaz, R. Srikant, J. Perkins (005). Stable scheduling policies or ading wireless channels. IEEE/ACM Transactions on Networking, 13, [7] R. Gallager (1996). Discrete stochastic processes. Kluwer Academic. 7

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29 [19] D. Shah, J. Tsitsiklis, Y. Zhong (011). Optimal scaling o average queue sizes in an inputqueued switch: an open problem. To appear in Queueing Systems. [0] D. Shah, D. Wischik (006). Optimal scheduling algorithms or input-queued switches. In: Proc. Inocom 006. [1] D. Shah, D. Wischik (008). Lower bound and optimality in switched networks. In: Proc. Allerton 008. [] K. Sigman, R. Wol (1993). A review o regenerative processes. SIAM Review, 35, [3] A. Stolyar (004). Maxweight scheduling in a generalized switch: state space collapse and workload minimization in heavy traic. The Annals o Applied Probability, 14, [4] L. Tassiulas, A. Ephremides (199). Stability properties o constrained queueing systems and scheduling policies or maximum throughput in multihop radio networks. IEEE Transactions on Automatic Control, 37, [5] L. Tassiulas, A. Ephremides (1993). Dynamic server allocation to parallel queues with randomly varying connectivity. IEEE Transactions on Inormation Theory. 39, [6] D. Williams (1991). Probability with Martingales. Cambridge University Press. Appendix 1 - Background Material 1.1 BASTA, Little s Law, and Delay Stability In this section we give the steady-state versions o two important results in queueing theory, the Bernoulli Arrivals See Time Averages property and Little s Law, which we later use to prove Lemma 1. Consider the single-hop queueing network described in Section. Let τ,k be the random time slot o the arrival o the k th ile to queue, k N, {1,..., F }. We assign two marks to this 9