Scheduling with Soft Deadlines for Input Queued Switches

Transcription

1 Scheduling with Soft Deadlines for Input Queued Switches Aditya Dua and Nicholas Bambos Abstract We study the problem of deadline aware scheduling for input queued (IQ) switches. While most research on scheduler design for IQ switches has focused on maximizing throughput and optimizing switch performance for non-realtime traffic, packet deadlines are a key consideration in the context of real-time applications like multimedia streaming and video telephony. In this paper, we introduce the notion of soft deadlines and study the scheduling problem within a dynamic programming (DP) framework. We demonstrate that by partitioning the set of switch configurations into orthogonal and complete subsets, the scheduling problem for an IQ switch can be transformed into one of scheduling parallel queues on a single server. We establish key structural properties of the optimal policy for the latter problem. We show that operating the switch using a single configuration subset is good enough to support any uniform admissible load. Further, we show that a scheduling policy which combines subset based operation with a randomized subset selection rule can support any admissible load. The randomized policy requires a Birkhoffvon Neumann (BV) decomposition of the input load vector. To eliminate this dependence, we propose two classes of heuristic scheduling policies, namely p(τ )- and p(τ)- MaxProj, which select the operational subset based on periodic maximum weight matching (p). We demonstrate the efficacy of the proposed policies via simulations. The proposed policies have a computational complexity of only O(N 2 ) per time-slot, and outperform under many scenarios. I. INTRODUCTION Real-time applications like multimedia streaming, video on demand (VoD), video telephony etc. continue to gain popularity amongst internet users. These applications have stringent quality-of-service (QoS) requirements, especially with regard to packet deadlines. Scheduling algorithms employed at packet switches in the network play a key role in QoS provisioning for such applications. While the initial emphasis of the switching community was on studying Output-Queued (OQ) switches [1], [2], Input-Queued (IQ) switches have received considerable attention owing to their scalable architecture. However, sophisticated scheduling algorithms are typically needed to efficiently control the switching fabric of an IQ switch. Most research on scheduling for IQ switches has focused on throughput maximization and stability. Numerous scheduling algorithms based on maximum weight matching () and its approximations ( [3]- [5] etc.) have been proposed, which provably achieve 1% throughput. The average delay performance of such scheduling algorithms has been studied both analytically and via simulations ( [5], [6] etc.). This vast A. Dua is with the Department of Electrical Engineering, Stanford University, Stanford, CA 9435, USA dua@stanford.edu N. Bambos is with the Faculty of Electrical Engineering and Faculty of Management Science and Engineering, Stanford University, Stanford, CA 9435, USA bambos@stanford.edu body of literature, while important in its own right, does not address the question of QoS provisioning for deadline constrained traffic. Liu et. al. [7] studied scheduling of multiclass periodic traffic flows through IQ switches. They conjectured that it is possible to schedule periodic traffic through an IQ switch so that each packet from a flow leaves the switch before the next packet of the flow arrives, provided the line utilization does not exceed unity. They proposed heuristic slot assignment rules based on the earliest deadline first (EDF) and minimum laxity first (MLF) policies. Giles et. al. [8] (nested period scheduling) and Rai et. al. [9] (uniform weight round robin) also proposed heuristic scheduling policies for scheduling periodic flows. Li et. al. [1] presented a frame based scheduler with guaranteed delay and jitter bounds for leakybucket constrained traffic. Chang et. al. proposed schemes for providing delay guarantees in IQ switches based on the Birkhoff von Neumann (BV) decomposition in [11], and based on EDF for load balanced switches (see [12]) in [13]. Our focus in this work is on developing switch scheduling algorithms for traffic streams which are associated with service profiles. A service profile associated with a traffic stream reflects the relative or inter-packet deadlines (IPD) between successive packets in the traffic stream. The service profile determines the ideal inter-departure times of packets in the stream from the switch. A deviation from this service profile results in missed packet deadlines, and hence QoS degradation at the receiver. If there were no congestion in the network, packets in a stream would depart the switch in accordance with their associated service profile. However, contention for shared network resources leads to deviations from the service profile. The goal of the scheduler is to minimize these deviations, or equivalently, adhere to the service profile for each stream as faithfully as possible. We do not make any assumptions on the periodicity or any other statistics of traffic streams arriving to the switch. Internet traffic can be bursty due to several reasons bursty nature of sources (for example, variable bit rate (VBR) video), multiplexing of streams, jitter induced by upstream switches etc. Consequently, traffic streams arriving to the switch may not be periodic even over small time-scales. It is therefore crucial to design schedulers agnostic to specifics of the input traffic streams. The scheduling problem with strict/hard inter-packet deadline constraints is NP-complete. To surmount the computational complexity associated with the problem, we introduce the notion of soft deadlines. Packets are allowed to violate these soft deadlines, while incurring a penalty for doing so. Packets also incur a penalty for being ahead of their

2 Q 11 Q 12 Q 21 Q Input Ports Output Ports (a) (b) (c) Fig. 1. (a) A 2 2 IQ switch with VOQs to prevent HOL blocking; Q ij contains packets destined from input port i to output port j. (b) and (c) Two possible configurations/matchings for a 2 2 switch. deadlines. This ensures fairness amongst various streams and prevents streams from receiving more service than they need to meet their deadline constraints. Our goal is to design low-complexity schedulers which minimize aggregate soft deadline violation over all streams traversing the switch. Soft deadlines can be thought of as a modeling tool used to develop approximations to the scheduling problem with hard deadline constraints, which would arise in the context of applications like video telephony. On the other hand, soft deadlines are also representative of realistic scenarios like lossless multimedia streaming, where delayed packets are not dropped. Such applications abound on the internet today. As we mentioned earlier, most work on scheduling for IQ switches has focused on throughput and stability related issues. The typical metric for delay in such work is average queuing delay, a quantity measured on a macro time-scale. However, our work represents a paradigm shift in conventional scheduler design principles because the focus is now on micro time-scales (on the order of packet deadlines). The goal is no longer to minimize the average delay per packet, but to ensure that each packet meets its deadline requirement as closely as possible. The performance of scheduling algorithms in this setting is to be gauged by the fraction of packets which meet their deadlines, or the goodput, rather than throughput or average delay. A. Paper outline and contributions We introduce some terminology and mathematically formulate the problem of scheduling with soft deadlines for an IQ switch within a dynamic program (DP) framework [14] in Section II. In Section III, we propose a partitioning of the switch configurations into orthogonal and complete subsets. We argue that if the switch is operated using configurations from only one of these subsets, the switch scheduling problem is transformed into the problem of scheduling parallel queues on a single server. We analyze in detail the single server scheduling problem in Section IV and provide key structural properties of the optimal policy. We invoke a Lyapunov technique [15] to show that with single subset based operation, the switch can support any uniform and admissible load. In Section V, we propose a class of policies which combine the notion of subset based operation with. We show that a randomized subset selection policy in conjunction with computed over the chosen subset can support any admissible load. However, the randomized policy requires knowledge of traffic statistics. To eliminate this dependence, we propose two families of heuristic policies in Section VI, which compute the operational configuration subset periodically based on, rather than based on knowledge of traffic statistics. We demonstrate the efficacy of the proposed policies via simulation in Section VII. The proposed policies often outperform both in terms of goodput and average delay per late packet and have only O(N 2 ) complexity, compared to O(N 3 ) for. We conclude the paper in Section VIII. II. MODEL CONSTRUCTION AND PROBLEM FORMULATION A. Model construction Consider an IQ switch with virtual output queues (VOQs) at the input ports to prevent HOL blocking. There are a total of N 2 VOQs, each one corresponding to a pair of input and output ports. The ports are numbered 1,..., N. The i th VOQ contains packets destined from input port (i 1)/N +1 to output port (i 1) mod N +1. A 2 2 switch is depicted in Fig. 1. In every time-slot, the switch can transfer at most one packet from each input port to one of the output ports. At most one packet can be sent to any output port in a time-slot. It is the task of the scheduler to determine which input-output port pairs are connected in each time-slot. Each VOQ is associated with a traffic stream comprised of packets arriving to it. Traffic streams with deadline constrained packets can be described using a sequence x = (x 1, x 2,...), where x τ {, 1}. We call x the expected service sequence (ESS) for the traffic stream. Suppose that the k th 1 in the ESS occurs at location t and the (k + 1) st 1 occurs at location t + d k, that is, x t = 1, x t+1 =,...,x t+d k 1 =, x t+d k = 1. The interpretation is that if the stream is being serviced in accordance with its ESS, the inter-departure time between the k th and (k + 1) st packets is exactly d k time-slots. Equivalently, d k is the inter-packet deadline (IPD) between the k th and (k + 1) st packets. ESS formally embodies the notion of service profiles introduced in Section I. We define the cumulative expected service sequence (CESS) X = (X 1, X 2,...), where X t Throughout this paper, sequences and vectors are denoted in boldface.

3 PSfrag Lead Cost Lag Expected Service Lead Cost Received Service Lag Cost Time Deviation Fig. 2. The left side depicts the notion of expected and received service, and deviation from expected service. The right side depicts a typical deviation cost associated with a VOQ. t x τ. X t represents the total number of packets of the τ=1 stream which should ideally have departed the switch by the end of time-slot t. By definition, X t t. For large t, we can think of X t /t [, 1) as the average rate associated with the stream. For example, if the IPDs are constant (say d), X t = t/d and the average rate is 1/d. We assume that lim n Xt /t exists for every traffic stream, so that the notion of average rate is well defined. To capture deviations from expected service caused due to resource contention, we associate with each traffic stream the received service sequence (RSS) y = (y 1, y 2,...), where y τ {, 1}. Then, y τ = 1 if the switch forwards a packet from the traffic stream in the τ th time-slot, and y τ else. Similar to CESS, we define the cumulative received service t sequence (CRSS) Y = (Y 1, Y 2,...), where Y t y τ. τ=1 Y t represents the total amount of service the traffic stream has received till the end of time-slot t. Ideally, we would like Y t = X t t, or equivalently x τ = y τ τ. If Y t > X t, the stream has received more service than it requires. In this case, we say that the stream is leading. If Y t < X t, the stream has received less than its expected share of service. In this case, we say that the stream is lagging. Every traffic stream traversing the switch experiences fluctuations relative to its ESS, which can be quantified using the notion of deviation, d t Y t X t. The objective of the scheduler is to keep d t as close to as possible at all times, for every traffic stream. The idea is pictorially depicted in Fig 2 (left side). Since we allow for deviations from the expected service sequence, we can think of packets as having soft deadlines. Packets which miss their soft deadlines are not dropped (which would be the case for hard deadlines), but are penalized for doing so. Packets are also penalized for being ahead of their scheduled deadlines. A good scheduler ought to minimize both the number and magnitude of soft deadline violations. B. Problem formulation We define the state of the switch as the N 2 -tuple d t = (d t 1,...,d t N ), where d t 2 i denotes the deviation of the traffic stream associated with the i th VOQ in time-slot t. We denote x t = (x t 1,..., x t N ), where x t 2 i is the t th element of the ESS of the traffic stream associated with the i th VOQ. Similarly, X t = (X1, t..., XN t ), where X t 2 i is the t th element of the CESS of the traffic stream associated with the i th VOQ. With the i th VOQ we associate the cost function φ i (k) = f + i (k +) + f i (k ), which measures the cost associated with deviation k Z, where k + = max(k, ) and k = min(k, ). We assume that f + i is a convex, non-negative and non-decreasing function, and f i is a convex, non-negative and non-increasing function. Thus, f + i measures the cost of positive deviation (or lead) and f i measures the cost of negative deviation (or lag). A typical cost function is depicted in Fig. 2 (right side). Finally, we N 2 define the sum cost function Φ(d t ) φ i (d t i) as the sum of deviation cost of all VOQs. We consider a finite time-horizon of T time-slots. For ease of exposition, we assume that all VOQs are infinitely backlogged. In each time-slot, the switch can be configured in one of N! possible configurations. A configuration specifies the input-output port pairs which are connected, or equivalently, the set of VOQs which are served simultaneously. Each switch configuration can be represented by a configuration vector of length N 2. The i th entry of the j th configuration vector v j is 1 if the i th VOQ is served when the switch is set in configuration j, and else. We denote the set of all possible configuration vectors by V. An admissible policy Π T is a sequence of switch configurations (v 1,...,v T ) V T in time-slots t = 1,...,T. Given the initial state of the system d, the objective is to find the policy Π T which satisfies { T } Π T = argmin Φ(d t Π T ), (1) Π T t=1 where d t Π T denotes the state of the switch in time-slot t under scheduling policy Π T. For given vectors x 1,...,x t, Π T can be computed numerically via dynamic programming. For a 2 2 switch, we showed in [16] that the optimal policy is myopic, that is, it is greedy with respect to instantaneous costs. It follows

4 v 1 C(v 1 ) C 2 (v 1 ) v 2 C(v 2 ) C 2 (v 2 ) Fig. 3. subsets of orthogonal configurations for a 3 3 switch. The three leftmost configurations are generated by v 1 = [e T 1 et 2 et 3 ]T and the three rightmost configurations are generated by v 2 = [e T 1 et 3 et 2 ]T. that computing the myopic policy reduces to a maximum weight matching () problem on a bi-partite graph, where the edge weights are (functions of) deviations of the VOQs. While the optimal policy for bigger switches (N > 2) is not myopic, numerical results indicate that the myopic policy is near-optimal. Regardless, is computationally intensive with a complexity of O(N 3 ), and not well-suited to implementation in high-speed switches with a large number of ports. We argue that it is not essential to compute a maximum weight matching over all possible N! switch configurations in every time-slot. In fact, by appropriate partitioning of the configuration set V into subsets of size N and computing a maximum weight matching over only of these subsets in every time-slot, very good performance can be achieved with only O(N 2 ) complexity. Let us see how such subsets of configuration vectors can be constructed. III. CONFIGURATION SUBSETS AND META-QUEUES A. Construction of orthogonal, complete subsets Recall that in each-time slot the switch can be operated in one of N! possible configurations chosen from the set V. Denote by e i the standard i th unit vector in R N. Note that v = [e T 1 et 2...eT N ]T V is an admissible configuration vector which corresponds to connecting input port i to output port i for i = 1,..., N. For an arbitrary N 2 -length vector w = [w T 1 w T 2...w T N ]T where w 1,...,w N are N-length vectors, define the circular shift operator C as C(w) = [w T N w T 1 w T 2...w T N 1] T. (2) Now, recursively define C k (w) = C(C k 1 (w)) for k N. By convention, C (w) = w. Also note that C N (w) = w and C k (w) = C k mod N (w). Thus, starting with a configuration vector v V, we can generate N distinct configuration vectors (including v) by applying C to v N 1 times. It is easily verified that circular shifts of v are admissible configuration vectors. We say that the configuration vector v V generates the configuration subset S v = {v, C(v),..., C N 1 (v)} V, and refer to v as a generator vector. How many such subsets can be generated? To answer this question, let π be a permutation of {2,...,N}. Now consider the configuration vector v π = [e T 1 e T π(2)...et π(n) ]T. As before, we can use the circular shift operator C to generate x T denotes the transpose of vector x. subset S vπ = {v π, C(v π ),..., C N 1 (v π )} V. It is easy to check that S vπ S vπ = for π π. There are (N 1)! possible choices for the permutation π. Thus, we can partition the configuration set V into (N 1)! configuration subsets of size N each. We denote the i th generator vector by v i and the corresponding generated subset by S vi. The configuration subsets satisfy S vi = V, S vi S vj =, i j. (3) i The construction is pictorially depicted for N = 3 in Fig. 3. Let us further explore some interesting properties of these configuration subsets. Denote by x, y the inner-product or dot-product between two vectors x,y R n. Now, note that for any v V, v, C k (v) = for k = 1,..., N. This implies that all configuration vectors within subset S v are orthogonal to each other. Equivalently, each input-output port pair (or the corresponding VOQ) is served by only one configuration from subset S v. Also, note that by construction every input-output pair is served by some configuration vector in the subset. We say that the subset S v is complete. We capture these properties through the relation N 1 j= C j (v i ) = 1, i = 1,...,(N 1)! (4) where 1 denotes an N 2 -vector with all unit entries. B. Meta-queues and problem transformation Consider some generator vector v (for example, v = [e T 1 e T 2...eT N ]T ) and the generated subset S v. If the switch is set in the configuration specified by C k (v), VOQs k, N + k,..., N(N 1) + k get served. VOQs which are served concurrently by a configuration (N of them for each configuration) can be clumped into a single meta-queue. We associate one such meta-queue with each configuration vector in the subset S v, resulting in a total of N meta-queues. The first packet in a meta-queue is obtained by fusing the HOL packets of the N constituent VOQs. These N packets are served concurrently when the switch is set in the configuration associated with this meta-queue. Subsequent meta-packets in the meta-queue are constructed similarly. The configuration vectors in S v are orthogonal, implying that each VOQ is uniquely associated with a meta-queue. Since S v is complete by construction, every VOQ gets associated with a meta-queue.

5 Consider a meta-queue which is obtained by clumping together VOQs indexed {i 1,...,i N }. Suppose that the metaqueue has been served Y t times by time-slot t. Since all constituent VOQs are served concurrently, each of them is served Y t times by time-slot t. Let Xi t j denote the t th entry in the CESS of the i th j VOQ. The deviation of the i th j VOQ in time-slot t is given by d t i j = Y t Xi t j. The CESS associated with the meta-queue in the single server model is computed as X t = max j=1,...,n {Xt i j }. By virtue of this definition, if the meta-queue is served in accordance with its CESS upto timeslot t, the constituent VOQ which requires the most service upto time-slot t is satisfied (and so are the others VOQs which require lesser service). Thus, the deviation associated with the meta-queue is d t = min j=1,...,n {dt i j }, the minimum of deviation of the constituent meta-queues. If the switch is operated using a single configuration subset, the problem of scheduling with soft deadlines in an N N switch reduces to the problem of scheduling N parallel meta-queues with soft deadlines on a single server. While the former problem has an N 2 -dimensional state-space and an N!-size action-space, the latter problem has an N-dimensional state-space and an N-size actionspace. This is a tremendous reduction in complexity! Two important questions arise at this point: (a) What is the loss in performance incurred (if any) by operating the switch using a single configuration subset? and (b) Can the performance loss be compensated for (if needed), without sacrificing the advantage of low computational complexity? We will attempt to answer these questions in the remainder of this paper. Before that, we turn our attention to the problem of scheduling parallel queues on a single server. IV. SCHEDULING PARALLEL QUEUES ON A SINGLE A. Problem Formulation SERVER WITH SOFT DEADLINES Consider a system comprised of N + 1 parallel queues being served by a single time-multiplexed server. The i th queue is denoted by Q i, for i =,...,N. In each timeslot, the server selects one of the queues according to some scheduling policy and serves the HOL packet of the chosen queue. While Q 1,..., Q N correspond to physical queues in the system competing for service, Q is a dummy queue, scheduling which is equivalent to idling the server. Each queue is associated with a deadline constrained traffic stream, which can be described using an expected service sequence, as in Section II. The ESS associated with dummy queue Q has all-zero entries. We will recycle much of our notation from Section II. Let d t i denote the deviation of the traffic stream associated with Q i in time-slot t. We will call d t = (d t,...,dt N ) the deviation vector in time-slot t. We denote x t = (x t,..., x t N ), where xt i {, 1} is the tth element of the ESS of the traffic stream associated with Q i. Similarly, we denote X t = (X, t...,xn t ), where Xt i is the tth element of the CESS of the traffic stream associated with Q i. With Q i, we associate the cost function φ i (k) = f i + (k) f i (k), which measures the cost associated with deviation k Z. The functions f + i and f i have the same properties as stated in Section II (non-negativity, monotonicity and convexity). We set φ (k) = k Z. We define the sum cost function Φ(d t ) i= φ i (d t i ) as the sum of deviation costs of all queues. Consider a finite time-horizon of T time-slots. As before, we assume that all queues are infinitely backlogged. At the beginning of each time-slot, the server selects one of the N + 1 queues for service. The configuration vector corresponding to scheduling Q i is e i (the i th unit vector in R N ), for i = 1,...,N. The configuration vector corresponding to scheduling Q is, the zero-vector. For notational ease, we set e =. We define an admissible policy as a sequence of scheduling decisions (i 1, i 2,..., i T ) {,..., N} T, corresponding to scheduling Q it in time-slot t for t = 1,...,T. We denote the set of admissible policies by Π (note Π = N T ). If Π T Π selects Q i in time-slot t, when the deviation vector is d t Π T, the new deviation vector in time-slot t + 1 is d t+1 Π T = d t Π T +e i x t. Given the initial deviation vector d, our objective is to find the policy Π T Π which minimizes the total T -period operating cost, for given x 1,...,x T. We will compute Π T via dynamic programming. B. Structural properties of the optimal policy Let the state of the system be the N-tuple n t = (n t 1,..., n t N ), where nt i {,..., t} denotes the number of times Q i has been served within the first t time-slots. Note that unlike Section II, we are not using the deviation vector as the state of the system. Since the server is allowed to idle, n t i t, t = 1,...,T. Observe that if Q i is served n i times up to time-slot t, its deviation at the end of time-slot t is given by d t i = d i + n i Xi t. Thus, given x1,...,x T, there is a unique correspondence between the system state vector and deviation vector in every time-slot. Without loss of generality, we assume d =. Let V t (n) denote the cost-to-go at the beginning of time-slot t. By definition, V t (n) is the total cost incurred by the optimal policy Π T starting from state n in time-slot t, until the end of the horizon in time-slot T. By the principle of optimality, the cost-to-go function is described by the following recursive DP equations for t = 1,...,T : V t (n) = min i=,...,n V t+1 (n + e }{{} i ) + Φ(n + e i X }{{} t, New state Deviation d t+1 ) (5) and V T+1 (n) = n. For notational convenience, we define Ω t (n) V t (n) + Φ(n X t ). With this definition, the DP equations can be re-written as V t (n) = min i=,...,n { Ω t+1 (n + e i ) }. (6) Finally, define the pairwise decision functions γ t ij(n) Ω t+1 (n + e i ) Ω t+1 (n + e j ), i j. (7)

6 N=3, T=4, t=3, n 3 =8, p=.3 15 N=3, T=4, t=3, n 3 =8, p=.1 n 2 n n n 1 Fig. 4. A typical partition of the (n 1, n 2 ) plane for fixed n 3 = 8, and p =.3. The states in which it is optimal to schedule Q 1, Q 2 and Q 3 are depicted by,, and respectively. Fig. 5. A typical partition of the (n 1, n 2 ) plane for fixed n 3 = 8, and p =.1. The states in which it is optimal to schedule Q 1, Q 2, and Q are depicted by, and respectively. It follows that the optimal policy favors Q i over Q j in state n in time-slot t if γij t (n), and favors Q j else. The pairwise decision functions γij t (n) satisfy the following important property: Lemma 1: The function γij t (n) is a non-decreasing function of n i, and a non-decreasing function of n j for i, j {,...,N}, i j, and t = 1,...,T. Equivalently, γij t (n + e i ) γij t (n), and γt ij (n) γt ij (n + e j). Lemma 1 implies that a two-dimensional projection of the N-dimensional state-space on the (n i, n j ) plane is partitioned into N + 1 distinct connected decision regions by Π T, corresponding to scheduling Q,..., Q N respectively. The result is established via a contradiction argument. For every t, as n t i increases for fixed nt j, j i, the optimal policy switches over from Q i to Q k for some k i. Thereafter, the optimal policy never switches back to Q i. For illustration, two typical partitions of the (n 1, n 2 ) plane for n 3 = 8, N = 3 are depicted in Fig. 4 and 5. For Fig. 4, the IPDs of all three traffic streams were generated using a geometric distribution with parameter p =.3, while for Fig. 5 they were generated using a geometric distribution with parameter p =.1. Since the ESSs are more sparse (more zeros) in the latter case, there are several states in which it is optimal to idle the server. C. The myopic policy While the switch-over property of the optimal policy is intuitively pleasing, it does not provide us the means to reduce computational complexity. The complexity of computing Π T increases exponentially, with both T and N. However, for a myopic policy, the complexity is O(N). The myopic policy chooses Q i (n,t) in state n in time-slot t such that i (n, t) = arg min i=,...,n { Φ(n + ei X t ) }. (8) By definition, the optimal policy is myopic for T = 1. It is therefore natural to ask: Is the optimal policy myopic for T > 1? It can be shown that the optimal policy is myopic for N = 2 T. However, the assertion does not hold in general for N > 2. Nevertheless, numerical results indicate that a myopic policy is near-optimal. Also, over a time-horizon T in which there are no pending deadlines for any of the queues (all ESSs have zero entries), the optimal policy is myopic. Theorem 1: The optimal scheduling policy for N = 2 is myopic, T N, x 1,...,x T. Theorem 2: The optimal scheduling policy is myopic (for any N, T ) if x t = t = 1,...,T. The foregoing discussion provides sufficient motivation for further studying the properties and performance of the myopic policy. We next consider the myopic policy for the special case of quadratic cost functions. D. The Largest Lag First () policy We now focus our attention on the special case of quadratic cost functions, that is, φ i (k) = k 2, k Z, i. It is easily seen that the myopic policy schedules Q i (d,t) in time-slot t given deviation vector d (or equivalently, state vector n) if i { (d, t) = argmax (di x t i) }. (9) i=,...,n Note that since d =, x t = t, the myopic policy chooses Q if d i x t i > for i = 1,...,N, that is, all queues have positive deviation (leading). Otherwise, the myopic policy chooses the queue with the most negative deviation (largest lag). In other words, the myopic policy is equivalent to the Largest Lag First () policy. Recall our assumption that each traffic stream can be associated with an average rate, which is equal to the inverse of the average IPD between successive packets in the stream. Let ρ i denote the average rate of the traffic stream associated with Q i, and letρ = (ρ 1,..., ρ N ) denote the rate vector. Define Ψ SS {ρ : ρ,1 < 1, ρ i [, 1]}. It can be easily be shown that for any ρ / Ψ SS, that is ρ i > 1, the lag of at least of the queues will increase without bound, regardless of the scheduling policy employed. We call Ψ SS the admissible region for the single server parallel queue

7 system. For the special case of geometrically distributed IPDs (with mean 1/ρ i for the traffic stream associated with Q i ), the policy can guarantee bounded lags to all queues for any admissible rate vector ρ Ψ SS. We refer to this case as i.i.d. loading, because geometrically distributed IPDs for a traffic stream are equivalent to each entry of the associated ESS being an independent and identically distributed Bernoulli random variable (with parameter ρ i for the traffic stream associated with Q i ). Further, we call the loading uniform if ρ i = ρ i. Theorem 3: For all i.i.d. admissible loads, if the single server system with parallel queues is operated using the policy, then liminf t dt i > with probability 1 and lim inf t E[dt i ] > i. E. and switch scheduling The scheduling policy can support all admissible loads for the single server system with parallel queues. What are the implications of this result in the context of scheduling in IQ switches? Recall from Section IV that the deviation of a meta-queue is simply the minimum of the deviation of its constituent VOQs. Also recall that with single subset operation, each VOQ is associated with a unique meta-queue. Suppose that the switch is being operated using the configuration subset S v, generated by configuration vector v. There is a one-to-one correspondence between the N meta-queues in the single server model, namely Q 1,..., Q N, and the switch configurations in the set S v, namely, {v, C(v),..., C N 1 (v)}. That is, scheduling Q i in the single server model is equivalent to setting the switch in the configuration corresponding to vector C i 1 (v). The queue Q i is scheduled if it has maximum lag amongst all queues. Since the lag of Q i is the maximum of the lags of its constituent VOQs, the policy chooses the VOQ with the maximum lag. Since each VOQ is associated with exactly one configuration in the subset, choosing a VOQ uniquely determines the switch configuration. Analogous to the admissible region for the single server system, we define the admissible region for the IQ switch by Ψ IQ { r (i 1)N+j < 1, r (j 1)N+i < 1, i = j=1 j=1 1,...,N, r i [, 1], i = 1,...,N 2 }, where r i denotes the average rate associated with the i th VOQ, and r = (r 1,..., r N 2) is the corresponding rate vector. By virtue of our meta-queue construction, the average rate of the traffic stream associated with a meta-queue is the maximum of the average rates of the traffic streams associated with its constituent VOQs. Now, consider the special case of i.i.d. uniform admissible loading, where r = r 1 for some r < 1, that is, the N traffic streams associated with all VOQs follow a geometric distribution with parameter r/n. In this case, regardless of the choice of configuration subset, the average rate of the traffic stream associated with each meta-queue is r/n. Since the rate vector ρ = r N 1 Ψ SS, it follows from Theorem 3 that the lags of all VOQs remain bounded if the switch is operated using a single configuration subset, in conjunction with the policy. It is not evident whether single subset operation combined with can guarantee bounded lags to all VOQs under i.i.d. non-uniform admissible loading. The choice of the operational subset is a critical factor in this case. For instance, consider the load vector r = (1 ɛ,,,,, 1 ɛ,, 1 ɛ, ) for a 3 3 switch, for some ɛ (, 2/3). In this case, it is easy to check that using the first configuration subset in Fig. 3 (generated by v 1 ) cannot guarantee bounded lags to all VOQs (the meta-queue average rates sum to 3 3ɛ > 1), while using the second configuration subset can do so, by virtue of Theorem 3 (the meta-queue average rates sum to 1 ɛ < 1). Next, consider the load vector r = (c,,,, c/2, c/2,, c/2, c/2), where c = (1 ɛ)/2 for some ɛ (, 1/2). The meta-queue average rates sum up to 2(1 ɛ) > 1 for both configuration subsets in this case. Thus, single subset operation cannot guarantee bounded VOQ lags. There are (N 1)! configuration subsets, each comprised of N configurations. Every configuration within a subset corresponds to a meta-queue in the single server model. The average rate of the traffic stream associated with a metaqueue is the maximum of the average rates of the traffic streams associated with the constituent VOQs. The policy with single subset operation can guarantee bounded lags to all meta-queues (and hence, to all VOQs) if the average rates for the N meta-queues associated with the chosen configuration subset sum to less than 1. However, verifying this condition involves searching over the entire space of configuration subsets. Moreover, it requires knowledge of the average rates of all traffic streams traversing the switch. V. AND SINGLE SUBSET SCHEDULING We saw in Section IV that with single subset operation the policy is optimal can guarantee bounded lags to all VOQs under uniform admissible loading conditions. Under non-uniform admissible loading, the choice of the operational configuration subset is dictated by the average rates of the traffic streams associated with various VOQs. It may not even be possible to find a subset, operating on which can guarantee bounded lags to all VOQs. Nevertheless, single subset based scheduling is a promising technique for designing low complexity scheduling algorithms with good performance. We now propose another class of scheduling policies, which combine the notion of single subset based scheduling and maximum weight matching. Suppose the switch is being operated using the configuration subset S v, generated by configuration vector v. The subset comprises of configurations vectors C k (v), k =,...,N 1. Given the deviation vector d in time-slot t, consider a scheduling policy which chooses configuration vector C k (v) such that k = arg max k=,...,n 1 { (d t x t ), C k (v) }. (1) We refer to the policy in (1) as MaxProj (Maximum Projection) because it selects the configuration vector with

8 the largest projection on the (negative of the) deviation vector. Note than MaxProj can be interpreted as computed only over configurations in subset S v. Also, observe that MaxProj based switch scheduling is equivalent to scheduling in a single server system with parallel metaqueues, where the deviation of a meta-queue is defined as the sum of the deviations of its constituent meta-queues (rather than the minimum, as in ). To make MaxProj comparable to, the switch is idled if all configuration vectors in S v have a positive projection on the deviation vector. It can be shown using a Lyapunov technique that just like, MaxProj guarantees bounded lags to all VOQs under i.i.d. uniform admissible loading conditions, with single subset operation. Theorem 4: For all i.i.d. uniform admissible loads, if the switch is operated using the MaxProj policy, then lim inf t dt i > i with probability 1 and liminf t E[dt i ] > i. As in the case of, it is easy to construct non-uniform admissible loading scenarios where single subset operation fails to guarantee bounded lags to all VOQs. However, a randomized subset selection rule, in conjunction with MaxProj, can ensure bounded lags for all VOQs under any i.i.d. admissible load. Denote the k th configuration subset by S k, and its corresponding generator vector by v k. The configuration vectors in S k are C i (v k ), i =,...,N 1. Given a load vector r Ψ IQ, consider its BV decomposition to get r = (N 1)! k=1 ζ ik C i (v k ), (N 1)! k=1 ζ ik = ζ < 1. (11) Define a probability distribution on the subsets of configuration vectors by ψ k 1 ζ ik, k = 1,...,(N 1)!. Now, ζ consider the following two-step scheduling policy: (a) Choose a configuration subset S k at random, according to probability distribution {ψ k }. (b) Either choose one of possible N configurations from subset S k based on MaxProj, or idle the switch if all configuration vectors have a positive projection on the deviation vector. We call the above policy Randomized Maximum Projection (RandMaxProj). Note that single subset based MaxProj is a special case of RandMaxProj, where the entire probability mass is concentrated on one subset. Theorem 5: For all i.i.d. admissible loads, if the switch is operated using the RandMaxProj policy, then lim inf t dt i > i with probability 1 and liminf t E[dt i ] > i. VI. TWO FAMILIES OF HEURISTIC POLICIES We have studied two types of switch scheduling policies based on single configuration subset operation, namely and MaxProj. Both these policies can provably guarantee bounded lags to all VOQs under uniform loading conditions. This is not necessarily true under non-uniform loading, where a dynamic subset selection rule is needed in conjunction with a scheduling policy to keep all lags finite. A provably good subset selection rule typically requires knowledge of the input rate vector, as we saw in the case of RandMaxProj. However, we are interested in designing scheduling algorithms which are agnostic to input traffic statistics, as discussed in Section I. To this end, we propose the following heuristic subset selection rule: Suppose the switch is operating in configuration subset S v generated by configuration vector v. Every τ > time-slots, compute the maximum weight matching over the entire configuration set V. Suppose the maximum is attained by configuration vector v. If v S v, continue operating the switch in subset S v, else start operating the switch in the configuration subset generated by v. We refer to this subset selection rule as p(τ ) (periodic maximum weight matching with period τ). We now propose two families of subset based scheduling algorithms, which can be employed in conjunction with the subset selection rule p(τ): (a) p(τ )-: Scheduling decisions are based on the rule. The operational subset is chosen every τ time-slots based on p(τ). (b) p(τ )-MaxProj: Scheduling decisions are based on the MaxProj rule. The operational subset is chosen every τ time-slots based on p(τ). The period τ is a design parameter which captures the complexity v/s performance trade-off. The complexity of is O(N 2 ) per time-slot (N +1 max operations over lists of length N). The complexity of MaxProj is also O(N 2 ) per time-slot (N 2 add operations, and one max operation on a list of size N). In both cases, the complexity can be reduced to O(N) by introducing parallelism in computation. The complexity of is O(N 3 ). Thus, the complexity of both p(n β )- and p(n β )-MaxProj is O(N 2 +N 3 β ) without parallelism, and O(N +N 3 β ) with parallelism. The former reduces to O(N 2 ) for β = 1, 2, while the latter reduces to O(N 2 ) for β = 1 and O(N) for β = 2. Both proposed policies offer a huge benefit over in terms of computation complexity. VII. SIMULATION RESULTS In this section, we evaluate the efficacy of the proposed policies, namely p(τ)- and p(τ)-maxproj via simulations, and contrast it to the benchmark scheduling policy. All results presented in this section are for a 4 4 IQ switch. The inter-packet deadlines for all streams were generated using a geometric distribution. Each VOQ was initially assumed to contain 1 4 packets. No further packet arrivals occurred to any VOQ. We considered three performance metrics: (a) Goodput the average fraction of packets which meet their deadlines, (b) Lateness average delay per late packet, and (c) Idle time fraction of time for which the switch idles. We simulated three different scenarios:

9 p(n) p(n 2 ) Goodput (%) Fraction of idle time slots p(n) p(n 2 ) 6.1 Fig. 6. Uniform Loading Scenario Goodput Fig. 8. Uniform Loading Scenario Idle Time 35 3 p(n) p(n 2 ) 1 95 Average lateness (# time slots) Goodput (%) MaxProj 65 Fig. 7. Uniform Loading Scenario Lateness Fig. 9. Non-uniform Loading Scenario A Goodput 1) Uniform i.i.d. loading: The parameter of the geometric distributions governing the IPDs is the same for all traffic streams in this case (say, r < 1/N). Thus, the load per input port is Nr. The parameter r was varied from.125 to.245 to vary the load per input port from.5 to.98. Fig. 6, Fig. 7, and Fig. 8 respectively depict the goodput, lateness, and idle time for and p(τ)- (for τ = N, N 2, ). Note that τ = is equivalent to based scheduling over a single subset in all time-slots. Observe that p(τ)- marginally outperforms in terms of goodput when the switch is moderately loaded. Further, the idle times of p(n)- and p(n 2 )- are about 1% more than that of at moderate loads. The idle time of the switch can be used to schedule best-effort traffic. Thus, for a given system load, a scheduler with higher idle time is more desirable. The performance of the p(τ)-maxproj class of policies is very similar to both and p(τ)-, and is therefore not depicted. 2) Non-uniform i.i.d. loading A: We call a VOQ diagonal if it contains packets destined from input port i to output port i for i = 1,..., N, and off-diagonal else. In this scenario, the average rate of traffic streams associated with off-diagonal VOQs was set to.1 and the average rate of traffic streams associated with diagonal VOQs was varied from.2 to.68, to vary the load per input port from.5 to.98. Fig. 9 and Fig. 1 respectively depict the goodput and lateness for,, and MaxProj. The configuration vector which serves the diagonal VOQs is given by v = [e T 1 et 2 et 3 et 4 ]T. For both and MaxProj, the switch was operated in the configuration subset generated by v. Since the diagonal VOQs are more heavily loaded, both these policies perform well. In fact, the policy comfortably outperforms, except when the load is close to 1%. 3) Non-uniform i.i.d. loading B: The average rates of the traffic streams associated with the VOQs containing packets destined from input port 1 to output port 2, 2 to 1, 3 to 4, and 4 to 3 were varied from.2 to.68, while the average rate of all other traffic streams was set to.1. Thus, the total load per input port varied from.5 to.98. For and MaxProj, the switch was operated in the configuration subset generated by v, as in non-uniform scenario A. Note that the configuration vector which serves the more heavily loaded VOQs is not contained in the subset generated by v. Consequently, the performance of these policies is poor, especially under heavy loads, as depicted in Fig. 11 and Fig. 12. However, combining or with p(τ) for subset selection yields significant performance benefits. In particular, the p(n)- policy provides performance superior to at O(N 2 ) complexity, compared to O(N 3 ) for. In summary, and MaxProj perform well with single subset operation under uniform loading, and in some cases

10 Average lateness (# time slots) MaxProj Average lateness (# time slots) p(n) p(n 2 ) p(n) MaxProj p(n 2 ) MaxProj 1 1 Fig. 1. Non-uniform Loading Scenario A Lateness Fig. 12. Non-uniform Loading Scenario B Lateness 1 Goodput (%) Fig. 11. p(n) p(n 2 ) p(n) MaxProj p(n 2 ) MaxProj MaxProj 2 Non-uniform Loading Scenario B Goodput under non-uniform loading if the operational configuration subset is carefully chosen (scenario A). However, their performance can be arbitrarily bad if the operational subset is poorly chosen (scenario B). Combining or MaxProj policies with the p(τ) rule for subset selection yields substantial performance gains relative to the benchmark scheduler, and that too at lower complexity. Furthermore, as we saw for the uniform loading case, the switch tends to idle more frequently when operated using or MaxProj, compared to. The idle time can be used to schedule deadline insensitive best-effort traffic. Thus, for a given level of QoS for deadline constrained traffic, the proposed policies can provide higher throughput than for best-effort traffic traversing the switch. VIII. CONCLUSIONS We studied the problem of packet scheduling with soft deadlines for IQ switches. By partitioning the set of switch configurations into orthogonal and complete subsets, we reduced the switch scheduling problem to one of scheduling parallel queues with soft deadlines on a single server. We analyzed the latter within a DP framework and established key properties of the optimal and myopic policies. We proposed two classes of heuristic scheduling policies which deliver superior performance at only O(N 2 ) complexity per timeslot, a huge improvement over the O(N 3 ) complexity of. It would be interesting to explore the proposed configuration subset based scheduling approach in the context of throughput aware switches and two-stage load balanced switches. REFERENCES [1] A.K. Parekh and R.G. Gallager, A generalized processor sharing approach to flow control in integrated services networks: single node case, IEEE/ACM Trans. Networking, vol. 1, pp , Jun [2] A.K. Parekh and R.G. Gallager, A generalized processor sharing approach to flow control in integrated services networks: multiple nodes case, IEEE/ACM Trans. Networking, vol. 2, pp , Apr [3] N. McKeown, A. Mekkittikul, V. Anantharam and J. Walrand, Achieving 1% throughput in an input-queued switch, IEEE Trans. Communications, vol. 47, no. 8, pp , Aug [4] P. Giaccone, B. Prabhakar and D. Shah, Randomized scheduling algorithms for high-aggregate bandwidth switches, IEEE Journal on Sel. Areas in Commun., vol. 21, no. 4, pp , May 23. [5] D. Shah and M. Kopikare, Delay bounds for approximate maximum weight matching algorithms for input queued switches, IEEE INFOCOM 22, pp , New York, NY, Jun. 22. [6] K. Ross and N. Bambos, Dynamic QoS control in packet switch scheduling, IEEE ICC 25, Seoul, S. Korea, May 25. [7] I.R. Philip and J.W.S Liu, SS/TDMA scheduling of real-time periodic messages, Intl. Conf. on Telecomm. Systems, pp , Mar [8] J. Giles and B. Hajek, Scheduling multirate periodic traffic in a packet switch, CISS 1997, Baltimore, MD, Mar [9] I.A. Rai and M. Alanyali, Uniform weighted round robin scheduling algorithms for input queued switches, IEEE ICC 21, pp , Helsinki, Finland, Jun. 21. [1] S. Li and N. Ansari, Input-queued switching with QoS guarantees, IEEE INFOCOM 1999, pp , New York, NY, Mar [11] C.S. Chang, W.J. Chen and H.Y. Huang, Birkhoff-von Neumann input-buffered crossbar switches for guaranteed-rate services, IEEE Trans. Commun., vol. 49, no. 7, pp , Jul. 21. [12] C.S. Chang, D.S. Lee and C.Y Yue, Providing guaranteed rate services in the load balanced Birkhoff-von Neumann switches, IEEE INFOCOM 23, pp , San Francisco, CA, Apr. 23. [13] C.S. Chang, D.S. Lee and Y.S. Jou, Load balanced Birkhoff-von Neumann switches: part I: one-stage buffering, Comp. Commun., vol. 25, no. 6, pp , Apr. 22. [14] D. Bertsekas, Dynamic Programming and Optimal Control, vol. 1 & 2, 2 nd Ed., Athena Scientific, 2. [15] P.R. Kumar and S.P. Meyn, Stability of queueing networks and scheduling policies, IEEE Trans. Auto. Control, vol. 4, no. 2, pp , Feb [16] A. Dua and N. Bambos, Low-jitter scheduling algorithms for deadline-aware packet switches, IEEE GLOBECOM 26, to appear.