Throughput-Optimal LIFO Policy for Bounded Delay in the Presence of Heavy-Tailed Traffic

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1 Throughput-Optimal Policy or Bounded Delay in the Presence o Heavy-Tailed Traic Shih-Chun Lin, Pu Wang, Ian F. Akyildiz, and Min Luo BWN Lab, School o Electrical and Computer Engineering, Georgia Institute o Technology, Atlanta, GA Department o Electrical Engineering and Computer Science, Wichita State University, Wichita, KS Shannon Lab, Huawei Technologies Co., Ltd. Santa Clara slin88@ece.gatech.edu; pu.wang@wichita.edu; ian@ece.gatech.edu; min.ch.luo@huawei.com Abstract Scheduling is one o the most important resource allocation or networked systems. Conventional scheduling policies are primarily developed under light-tailed (LT) traic assumptions. However, recent empirical studies show that heavytailed (HT) traic lows have emerged in a variety o networked systems, such as cellular networks, the Internet, and data centers. The highly bursty nature o HT traic undamentally challenges the applicability o the conventional scheduling policies. This paper aims to develop novel throughput-optimal scheduling algorithms under hybrid HT and LT traic lows, where classic optimal policies (e.g., maximum-weight/backpressure schemes), developed under LT assumption, are not throughput-optimal anymore. To counter this problem, a delay-based maximumweight scheduling policy with the last-in irst-out () service discipline, namely -DMWS, is proposed with the proved throughput optimality under hybrid HT and LT traic. The throughput optimality o -DMWS gives that a networked system can support the largest set o incoming traic lows, while guaranteeing bounded queueing delay to each queue, no matter the queue has HT or LT traic arrival. Speciically, by exploiting asymptotic queueing analysis, -DMWS is proved to achieve throughout optimality without requiring any knowledge o traic statistic inormation (e.g., the tailness or burstiness o traic lows). Simulation results validate the derived theories and conirm that -DMWS achieves bounded delay or all lows under challenging HT environments. I. INTRODUCTION Because o the emerging multimedia, data center, and the Internet applications over mobile devices, network traic in both radio access networks and wired core networks has tremendously grown in past ew years while the network capacity is rather limited. To address such a challenge, throughputoptimal scheduling is highly demanding, which determines the optimal transmission time or traic lows, supports the largest set o traic rates, and maintains the desired network stability. As being an important class o throughput-optimal scheduling, the maximum-weight scheduling (MWS) policy and many o its variants [1] are o great interests. On one hand, MWS policy can achieve throughput optimality without requiring any knowledge o the statistic inormation o arrival traic lows and time-varying channel conditions. On the other hand, MWS policy has shown its throughput optimality in a variety o network settings, such as the Internet, cellular networks, WiFi networks, satellite networks, ad-hoc and sensor networks, and high-perormance computing clusters. Generally developed under the assumption o light-tailed (LT) traic lows (e.g., Markovian or Poisson traic), the celebrated MWS policies achieve strong stability or LT lows, which guarantees that each low has bounded average queueing delay [2] whenever the incoming traic rates are within the network stability region. However, this LT assumption has large discrepancy rom recent large-scale empirical studies, which show the emergence o heavy-tailed (HT) traic in a variety o networked systems, such as WLAN [3], mobile ad-hoc networks [4], cellular networks [5], and data center networks [6]. Such HT traic is mainly caused by the inherent heavy-tailed distribution in traic sources (e.g., the ile size on the Internet servers, the message size on cellular base stations, the low length o data centers, and the rame length o VBR video streams [7]). Dierent rom conventional LT traic, HT traic exhibits high burstiness or dependence over a long range o time scale. Such a highly bursty nature can induce signiicant degradation in network stability [8], thus having a destructive impact on the throughput optimality o scheduling policies. In particular, it is proved that the celebrated MWS policies are not throughputoptimal anymore in the presence o HT traic lows because MWS can lead to unbounded average queueing delay even i the arrival traic rates are within network capacity region [8]. Such surprising phenomenon attributes to the act that the queues with HT traic arrivals (HT queues) inherently experience heavy-tail distributed queueing delay, which implies that HT queues have much higher chance to experience very large queueing delay, compared with the queues with LT arrivals (LT queues). As a result, based on MWS policies [8], [9], HT queues will receive much more service opportunities, while LT queues are starved or scheduling service and their queueing delay can be o unbounded mean. To counter such challenges, maximum power weight scheduling policies (MPWS) are investigated recently [10], [11], which make scheduling decisions based on queue backlog raised up to the α-th power, where α is determined by the burstiness or heavy tailness o traic lows. Intuitively, by properly selecting α to allocate more service opportunities to LT queues, MPWS can guarantee that all LT queues experience bounded average queueing delay, completely shielding those LT queues rom the destructive impact o HT traic. Despite such promising eature, MPWS cannot ensure the delay boundness o the HT queues and thus is still not a throughput-optimal scheduling policy. Moreover, /16/$ IEEE

2 MPWS policy requires the statistical inormation (i.e., tailness or burstiness o arrival lows), which is diicult to estimate. To counter above challenges, in this paper, we propose a delay-based maximum-weight scheduling policy with service discipline (-DMWS) and prove its throughput optimality in the presence o HT traic. Speciically, rather than adopting queue backlog as link weight, we ocus on delaybased scheduling, which exploits the head-o-line (HoL) delay metric in inter-queue scheduling decisions (i.e., determining the serving order or the packets rom dierent queues). Moreover, instead o using the classic FIFO service discipline, we exploit service discipline or intra-queue scheduling (i.e., determining the serving order or the packets within each queue). Furthermore, by exploiting asymptotic queueing delay analysis along with moment theory, we prove that - DMWS is throughput-optimal with respect to strong stability in the presence o heavy tails. That is, we show that with -DMWS, no matter the incoming traic lows are HT or LT, all queues will experience bounded average queueing delay as long as the incoming traic rates are within the network capacity region. Such a throughput optimality eature is o great importance, since it prevents the QoS perormance o LT traic rom being signiicantly degraded by the bursty HT traic. Simulation results conirm the throughput optimality o -DMWS and show that -DMWS brings considerable delay reduction as compared to classic maximum-weight scheduling policies [12]. To the best o our knowledge, this work is the irst throughput-optimal scheduling or bounded delay with emerging HT traic. The rest o the paper is organized as ollows. Section II introduces the system model and preliminaries. Section III proposes the -DMWS policy or system stabilization with hybrid HT and LT traic. Section IV presents the perormance evaluation and Section V concludes the paper. A. System Model II. SYSTEM MODEL AND PRELIMINARIES We consider a multi-queue single-server queueing system, where F queues share a single server. Such a queueing system can be used to model downlink/uplink scheduling in cellular or WiFi networks, where multiple users compete a single wireless channel or a set o wireless channels. We consider a time-slotted system. A traic low {1,..., F} is a discrete-time stochastic arrival process {A (t);t Z + }, which represents the total number o packets that arrive at a queue at the beginning o the time slot t and is independent and identically distributed rom time slot to time slot. The arrival processes or traic lows are mutually independent. Let λ = E[A (0)] > 0 be the rate o traic low and λ = (λ 1,...,λ F ) be the rate vector. Let stochastic processes {Q (t);t Z + } and {D (t);t Z + } be the number o packets and the queueing delay, respectively, in queue at the beginning o time slot t. Q(t) = (Q 1 (t),...,q F (t)) captures the queue backlog at time slot t, and its initial state Q(0) can be an arbitrary element o Z F +. Moreover, not all traic lows can be served simultaneously due to wireless intererence. Accordingly, a set o lows that can be served simultaneously is called a easible schedule. In our network model, each easible schedule only contains one low, because only one queue can be served during each time slot. Let S denote the set o all easible schedules. Then, or each easible schedule π S, let π (t) denote the number o packets transmitted rom queue under schedule π at time t. For simplicity, we assume that the service rate is one packet per time slot. Thus, the average service rate o queue under schedule π is E[π (t)] = π 1. Accordingly, the timevarying scheduling vector S(t) = (π 1 (t),...,π F (t)),π S is determined by the proposed scheduling policy. Hence, the set o processes {Q(t), D(t), S(t)} with Q(0) completely captures the dynamic o the entire stochastic queueing system. B. Mathematical Preliminaries Deinition 1 (Heavy Tail) A random variable (r.v.) X is heavy-tailed (HT), i or all θ > 0, lim x e θx Pr(X > x) =, or equivalently, E[e zx ] =, z > 0. A r.v. is light-tailed (LT), i it is not HT, or equivalently, i there exists z > 0 so that E[e zx ] <. A HT r.v. has tail distribution decreases slower than exponentially (e.g., Pareto and log-normal); a LT r.v. has tail distribution decreases exponentially or even aster (e.g., exponential and Gamma). From the existence o the moments, we deine the tail index o a nonnegative r.v. X as κ(x) := sup{k 0 : E[X k ] }, (1) which deines the maximum order o inite moments that X can have. Moreover, to show the suicient condition or inite tail indexes [13], we have the ollowing: A nonnegative r.v. X has κ(x) i and only i the tail distribution o X satisies logpr(x > t) lim = κ(x). (2) In the ollowing, we deine an important class o HT distributions (i.e., regularly varying distributions). Deinition 2 (Regularly Varying) A r.v. X is called regularly varying with tail index β > 0, denoted by X RV(β), i Pr(X > x) x β L(x), where or any two real unctions a(t) and b(t), a(t) b(t) denote lim t a(t)/b(t) = 1 and L(x) is a slowly varying unction. Regularly varying distributions are a generalization o Pareto/Zip/power-law distributions and can eectively, stochastically characterize a wide range o network attributes, including the rame length o VBR traic, the session duration o network users in WLANs, the iles sizes at the Internet severs, and the message sizes at cellular base stations. The smaller values o β imply heavier tail. In particular, i the arrival process A (t) ollows regular varying distribution, A (t) RV(β), then κ(a (t)) = β. This indicates that i 0 < β < 1, X has ininite mean and variance. I 1 < β < 2, X has inite mean and ininite variance.

3 C. System Stability and Throughput Optimality Deinition 3 (Steady-state Stability) Given the queueing system described in Section II-A, i there exists a scheduling policy under which the Markov chain o queue lengths is positive Harris recurrent (i.e., {Q(t);t Z + } converges in distribution), then the queueing network is steady-state stable. Steady-state stability only guarantees the convergence o the steady-state distribution or queue backlog. To characterize the delay perormance o queueing systems, strong stability needs to be adopted as ollows. Deinition 4 (Strong Stability) A queueing system is strongly stable, i all traic lows experience bounded average queueing delay (i.e., E[W ] <, F ). The throughput-optimality o scheduling algorithms characterizes their capability o achieving strong stability in the network capacity region, deined as ollows. Deinition 5 (Network Capacity Region [1]) The network capacity region Φ o the queueing system is the set o all traic rate vectors that are admissible by the system (i.e., λ can be covered by a convex combination o easible schedules). Mathematically, Φ := {λ R F + λ σ componentwise, or some σ Co(S)}, where Co(S) denotes the convex hull o all easible schedules. Deinition 6 (Throughput Optimality) A scheduling policy is throughput-optimal, i it can achieve strong stability or any admissible rate vector (i.e., any rates within the network capacity region). III. DELAY-BASED MAXWEIGHT SCHEDULING WITH SERVICE DISCIPLINE (-DMWS) MWS policies activate a easible scheduler with the maximum weight at any given time slot. The classic MWS policies adopt FIFO service discipline or intra-queue scheduling and utilize queue-length as the weight to regulate inter-queue scheduling. That is, at each time slot, the queue with the maximum queue length is served, i.e., Q FIFO (t) = max j F QFIFO j (t), F. (3) The ollowing theorem shows the instability o MWS algorithms in the presence o HT traic lows. Theorem 1 (Network Instability o MWS [10], [11]) Under MWS, i one o queues has a HT traic arrival with tail index smaller than 2 (i.e.,a RV(α) withα < 2), all queues have bounded average queueing delay (i.e., E[D ] =, F ). The above theorem indicates the classic MWS is not throughput-optimal anymore with HT traic. Such surprising phenomenon can be attributed to the act that HT queues inherently experience heavy-tail distributed queueing delay, which implies that compared with LT queues, HT queues have much higher chance to experience very large queueing delay, which can be o unbounded mean. As a result, based on MWS policies, HT queues will receive much more service opportunities, while LT queues are seriously starved and thus their average queueing delay can be unbounded. To counter such a problem, we propose a -DMWS policy. Dierent rom the classic MWS policies, the proposed -DMWS exploits service discipline or intra-queue scheduling and employs the HoL delay as the weight or inter-queue scheduling. In this case, the HoL packet delay (t)} is the queueing delay {D (t)} experienced so ar by the HoL packet in the queue. Speciically, at each time slot, the -DMWS policy serves queue with the maximum HoL delay, i.e., {W W (t) = max j F W j (t), F (4) where ties are broken randomly. Despite its simplicity, - DMWS is the irst throughput-optimal scheduling or bounded delay with heavy-tailed traic. Theorem 2 (Throughput Optimality o -DMWS) The -DMWS policy is throughput optimal by ensuring that all queues have bounded average queueing delay E[D ] <, F whenever (1) all arrival traic lows have bounded mean λ = E[A (t)], F. That is, all arrival traic lows have a tail index larger than 1, i.e., min F κ(a (t)) > 1; (2) the incoming traic rates are within the network capacity region. Intuitively speaking, the promising eature o -DMWS comes rom the optimal asymptotic delay perormance o the HT queues under service discipline, which can guarantees that HT queues experience suiciently reduced delay, which are o bounded mean. Then, by adopting HoL delay as the weight metric, we can ensure that the delay o LT queues is at least o the same order as the delay o HT queues. This implies the queueing delay o LT queues is also o the bounded mean. Accordingly, we can show that -DMWS is throughput-optimal. More importantly, -DMWS does not require any knowledge o the statistical inormation o traic arrivals. Despite the intuitive advantage o -DMWS, proving its throughput optimality is very challenging. To prove the throughput optimality o delay-based maximum-weight scheduling (DMWS) with FIFO, luid model-based schemes are generally adopted, which establish the linear relationship between queueing delay and queue length through luid model solutions. This implies the throughput optimality o DMWS, since queue-length based policies are throughputoptimal. However, under HT environment, luid model based schemes cannot be applied because Little s law does not hold or HT queues with discipline, which indicates that the linear relationship between queueing delay and queue length may not hold or HT queues in the luid domain. To counter this challenge, we adopt our developed asymptotic queueing analysis tools to prove the throughput optimality o -DMWS. In the ollowing subsections, we irst introduce the advantage o service discipline or HT traic by showing the asymptotic queueing delay o queues

4 under single-queue, single-server scenario in Section III-A. Then, we prove the throughput optimality in Section III-B by deriving the asymptotic queueing delay o each queue under the -DMWS policy. A. Asymptotic Queueing Delay o Discipline Inormally speaking, discipline allows the waiting time or queueing delay to be the same degree heavier as the service time in the case o HT arrival. Speciically, as shown in [14], the tail distribution D o the queueing delay o a single queue with FIFO discipline ollows Pr(D > 1 1 ρ E[B] x x FIFO) ρ 1 ρ Pr(Br > x) = ρ Pr(B > u)du, where (x) g(x) denotes lim x (x)/g(x) = 1, and B, B r and ρ denotes the service time, the residual service time and the traic load, respectively. I traic arrival A(t) is HT with tail index κ(a(t), i.e., A(t) RV(κ(A(t)), the service time or A(t) ollows HT distribution which asymptotically behaves as x κ(a(t)), i.e., logpr(b > t FIFO) lim = κ(a(t)). (5) As a result, the queueing delay tail distribution behaves as x 1 κ(a(t)), i.e., logpr(d > t FIFO) lim = κ(a(t))+1, (6) which is one degree heavier than the service time tail distribution in (5). This implies by Deinition 2 that i traic arrival process has unbounded variance, i.e., κ(a(t)) < 2, then the queueing delay has unbounded mean. On the other hand, the queueing delay tail distribution with discipline becomes Pr(D > x ) 1 1 ρ Pr(B > x(1 ρ)). This means that i traic arrival process is HT with tail indexκ(a(t)), the service time or A(t) ollows HT distribution which asymptotically behaves as logpr(b > t ) lim = κ(a(t)). (7) As a result, the queueing delay tail distribution behaves as logpr(d > t ) lim = κ(a(t)). (8) It means that the queueing delay is as heavy as that o the service time which has the same tail index o arrival process. This implies by Deinition 2 that the average queueing delay is bounded as long as service time is o bounded mean or has a tail index larger than one, i.e., κ(a(t) > 1. B. Throughput Optimality Analysis o -DMWS In this section, we irst investigate the asymptotic queueing delay perormance under general work-conserving scheduling policies in Lemma 3, which gives the upper bound o queueing delay tail index under -DMWS. In Lemma 2, we derive the asymptotic delay perormance under -DMWS, which yields the lower bound o queueing delay tail index. Then, we prove the throughput optimality o -DMWS by showing that the upper and lower bounds coincide and are larger than one as long as the arrival traic has tail index larger than one. This indicates that all queues are o bounded average queueing delay under -DMWS. Beore going to the main theorems, we irst introduce Lemma 1 that is used in the later proos. Lemma 1: For any work-conserving scheduling with singlehop hybrid traic, we have the ollowing: Under FIFO, under, κ( F Q ) = [min F κ(a (t))] 1; (9) κ(max F D ) min F κ(a (t)). (10) Proo: The result or FIFO service discipline is obtained by the analysis o the ictitious queue. Speciically, consider a ictitious q v, which has the arrival process A v (t) = F A (t) under the original single-hop queueing system. As we adopt regularly varying distributions or HT lows, i.e., low HT with A (t) RV(β ) and κ(a (t)) = β, it implies by regular variation that the arrival A v (t) RV(min F β ). Let D v, Q v, and Q v (t) denote the steady-state queueing delay, the steady-state queue backlog, and the queue backlog at time t o q v, respectively. Regarding FIFO service discipline, it ollows by Eq. (6) that κ(d v ) = (min F β ) 1. Under any work-conserving scheduling policy in the original queueing system, we have Q v (t) = F Q (t), which implies that Q v = F Q. By distributional Little s law, we have Pr(Q v > t) Pr(E[A v (t)]d v > t), which indicates that κ( F Q ) = (min F β ) 1, and prove the result or FIFO discipline. The result or service discipline is obtained by union bound and regular variation. Speciically, we have that Pr(max F D > t) = Pr(D 1 > t D F > t) Pr(D 1 > t) + + Pr(D F > t). This, combing with Eq. (2) and Deinition 2, implies that κ(max F D ) min F κ(a (t)); thus, we complete the proo. Lemma 2: Consider any work-conserving scheduling with arrival traic satisying min F κ(a (t)) > 1. Under FIFO discipline, the steady-state queueing delay D o low with κ(a (t)) ollows κ(a (t)) 1 κ(d ) [min F κ(a (t))] 1; (11) under discipline, the steady-state delay D ollows κ(a (t)) κ(d ) min F κ(a (t)) (12) whenever incoming traic rates are within the network capacity region. Proo: By Deinition 5 and the work in [1], the prerequisite that the incoming rates are within the convex hull o the set o all easible schedules ensures the steady-state stability o queueing systems and thus the existence o steadystate queueing distributions. This implies that the steady-state queue backlog Q and delay D exist or all F, and the asymptotic queueing delay analysis can be readily applied. We prove the asymptotic results in Eq. (11) with queueq as ollows. Speciically, it is evident that the queue backlog Q

5 is stochastically dominated by the composite queue backlog F Q, which by Lemma 1 and distributional Little s law that Pr(D > t) Pr(Q /λ > t), proves the lower bound o Eq. (11), i.e., κ(d ) [min F κ(a (t))] 1. As to the upper bound, we consider the best scheduling policy or q, which allows q to receive the service whenever q is not empty. This scheduling policy yields the best asymptotic results or the queue q, since q does not have to compete with other queues or the service and thus behaves like a single low queue with FIFO discipline. Invoking Eq. (6), the upper bound o Eq. (11) hods, i.e., κ(d ) κ(a (t)) 1, and thus proves Eq. (11) or the FIFO discipline. We prove Eq. (12) or service discipline as ollows. The lower bound is simply obtained rom the result in Lemma 1. Speciically, it is obvious that the queueing delay D is stochastically dominated by the maximum delay max F D, i.e., Pr(D > t) Pr(max F D > t) or all t 0. This, combing with Eq. (10), implies that κ(d ) κ(max F D ) min F κ(a (t)). The upper bound is obtained by using the best scheduling policy and Eq. (8). Speciically, consider the best scheduling or q, which implies that q has an exclusive access to the scheduler, like a single queue with discipline. Invoking Eq. (8), the upper bound o Eq. (12) hods, i.e., κ(d ) κ(a (t)); thus, we complete the entire theorem proo. Now, we derive the upper bound o the tail index or queueing delay. Lemma 3: Under -DMWS, the tail indexκ(d ) o the steady-state queueing delay D is upper bounded by κ(d ) min F κ(a (t)). (13) Proo: We exploit our asymptotic queueing analysis tools [10] to study queueing delay or -DMWS. Speciically, we construct a ictitious queueing system with F low queues { q } F, where each queue q has the same input process as q. Consider a particular queue q, and let all queues { q j } j except q have the exclusive access to their own scheduler without competing with each other (i.e., these queues operate as single low queues with discipline). The queue q receives service i and only i W = max W j F j. In such a system, it is easy to prove that the ictitious queue q has less queueing delay than the queue q in the original system, i.e., D (t) D (t). We assume that the ictitious system is in the steady state. Let Φ j denote the event where q j is not empty and all the other queues excluding q are empty, i.e., Φ j := { D j 0 D k = 0} (14) k,j and Pr(Φ j ) := Pr( D j > 0) k,j (1 Pr( D k > 0)). Thus, we have the lower bound o the d th moment o D as E[D d ] j Pr(Φ j )E[ D d Φ j ]. (15) In the rest o the proo, we will derive the lower bound o the conditional moments E[ D d Φ j]. Assume that the event Φ j occurs at time t. Let Bj r(t) and Be j (t) denote the residual and the expanded service time o the message currently in service or the queue q j. By renew theory and Eq. (8) under discipline, we have κ(b r j(t)) = κ(b e j(t)) = κ(a j (t)). (16) I the event Φ j occurs, then three possible events, namely (i) Υ 1, (ii) Υ 2, and (iii) Υ 3, occur to W (t). Speciically, we deine Υ 1 := { W (t) W j (t)+bj r(t)}, Υ 2 := { W j (t)+bj r (t) > W (t) W j (t)}, and Υ 3 := { (t) > (t)}. I (i) Υ 1 occurs, we have W j W I (ii) Υ 2 occurs, we have D (t) D j (t)+b r j(t). (17) D (t) D j (t). (18) In the case o (iii)υ 3, letτ denote the last time beoret that q receives service. This means that W (τ) > W j (τ). Combing with the act that W j (t) > W (t) > W (τ), it indicates that the burst being served at time t did not begin to receive service at time τ, i.e., t τ > Bj e(t). This implies that D (t) W (t) = W (τ)+(t τ) > Bj(t).(19) e Combining Eq. (15) and Eqs. (17)-(19), we obtain E[D] { d Pr(Φ j ) Pr(Υ 1 )E[ ( Dj (t)+bj(t) r ) d ] j +Pr(Υ 2 )E[ D j (t) d ]+Pr(Υ 3 )E[B e j(t) d ] } j Pr(Φ j ) { Pr(Υ 1 )(E[ D j (t) d ]+E[B r j (t)d ]) +Pr(Υ 2 )E[ D j (t) d ]+Pr(Υ 3 )E[B e j (t)d ] }. (20) This, combing with Eq. (8) and Eq. (16), implies that i the order o the moments d min j κ(a j (t)), then at least one o the terms on the right-hand side o Eq. (20) is ininite, which implies κ(d ) min j κ(a j(t)). (21) Moreover, since under any work-conserving scheduling policy, D is lowered bounded by that o a single exclusive queue with discipline. This implies that κ(d ) κ(a (t)), (22) which, combing with Eq. (21) completes the proo. Upon this stage, the throughput optimality o -DMWS is presented in the ollowing Theorem 2. Proo o Theorem 2: Given that incoming traic rates are within the network capacity region, this implies that the network is steady-state stable. Hence, we are ready to apply the asymptotic queueing analysis or -DMWS as ollows. First, the upper bound o κ(d ) under -DMWS is given by Lemma 3. Since discipline is work-conserving, by Theorem 2, the lower bound o κ(d ) under -DMWS

6 is obtained, i.e., κ(d ) min F κ(a (t)). Thereore, it ollows that the upper and lower bounds o κ(d ) coincide under -DMWS, i.e., κ(d ) = min F κ(a (t)). This, combing with the given condition min F κ(a (t)) > 1 and Eq. (1), completes the proo. IV. PERFORMANCE EVALUATION In this section, we validate our asymptotic queueing analysis. We choose Pareto and Poisson distributions to depict HT and LT distributions, respectively. Speciically, a r.v. X PAR(β,x m ), i it ollows Pareto distribution with a shape parameter β > 0 and a scale parameter x m > 0, i.e., P(X > x) = (x m /x) β. A r.v. X Poiss(λ), i it ollows Poisson distribution with mean λ > 0, i.e., P(X > x) = 1 e λ x i=0 λi /i!, where x is the loor unction. In the ollowing, we validate the throughput optimality o - DMWS with single-hop hybrid traic. All simulation results are evaluated over 10 5 time slots. A. Throughput-optimal -DMWS We consider a scenario where a HT low and a LT low sharing a single channel, i.e., A h (t) PAR(1.5,1) and A l (t) Poiss(3). All the ollowing tail distribution results are plotted on log-log coordinates, by which a HT distribution with tail index κ maniests itsel as a straight line with the slope equal to κ. We irst examine the perormance o hybrid traic under the classic MWS policy. To enable a air comparison, we also adopt delay as the weight metric or MWS instead o queue length and denote it by (FIFO-)DMWS. Fig. 1a shows that during 10 5 time slots, only around 10% o packets (as compared to HT traic) rom LT traic leave the scheduler and contribute to the cumulative packet delay, given the same packer arrival rate or both LT and HT lows. The reason is that most o packets rom LT traic are stuck in the queue due to the competitions with the HT lows. Moreover, Fig. 1b shows that under the DMWS policy, the queueing delay o LT low ollows heavy tailed distribution with a tail index smaller than one, as its tail distribution decays slower than the reerence Pareto distribution with tail index one. This means that the LT traic also has unbounded average delay as that o HT traic. Hence, under the DMWS policy with hybrid traic, the LT low necessarily has ininite average queueing delay, and DMWS is not throughput optimal. We next show that the bounded delay and thus throughput optimality can be achieved or hybrid traic by applying the -DMWS policy. Fig. 2a shows that LT traic can receive suicient service opportunities to sent a comparable number o packets as HT traic lows. Moreover, as shown in Fig. 2b that under the -DMWS policy, the queueing delay o LT low has a tail index greater than one, as their tail distributions decay aster than the reerence Pareto distribution with tail index one. This means the LT traic has bounded queueing delay. Furthermore, while the queueing delay o HT lows are also o bounded mean. Finally, a more complicated scenario under -DMWS is studied with three hybrid lows in Fig. 3, i.e., A h (t) PAR(1.5,1), A l1 (t) Poiss(3), Cumulative Packet Delay 1-F(x) Most packets o LT low are stuck in scheduling queue Number o Packets (a) Dynamics o cumulative packet delay. Reerence (Tail Index = 0.5) Packet Delay: D h Packet Delay: D l Reerence (Tail Index = 1) x (b) Tail distribution o packet delay. Fig. 1: Queueing delay o a HT low, i.e., A h (t) PAR(1.5,1), and a LT low, i.e., A l (t) Poiss(3), under (FIFO-)DMWS [12]. and A l2 (t) Poiss(2), and the same conclusion is reached accordingly. In particular, Fig. 3a shows that most o packets rom LT lows can exit the system. Fig. 3b urther indicates that or the tail distributions o queueing delay o all LT and HT lows have a slope or decaying rate greater than one, indicating the average bounded delay or all traic lows. Above results are consistent with Theorem 2, which implies that under hybrid HT and LT traic, -DMWS is throughput-optimal by achieving strong stability. V. CONCLUSION -DMWS is proposed to achieve throughput-optimality with heavy-tailed traic. In particular, -DMWS guarantees bounded average delay or hybrid HT and LT lows under any admissible traic arrivals without any knowledge o statistical inormation o the arrivals. Perormance evaluations validate our theoretical indings. The uture research will be the extension o -DMWS to multi-hop traic lows with possibly eedback loops. D h D l

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