Deadline Aware Scheduling for Input Queued Packet Switches

Transcription

1 Deadline Aware Scheduling for Input Queued Packet Switches Aditya Dua and Nicholas Bambos Department of Electrical Engineering, Stanford University 35 Serra Mall, Stanford, CA 9435 Phone: , Fax: Netlab Technical Report, SU-Netlab-5-6/ Abstract We study the problem of designing scheduling algorithms for deadline-aware input-queued (IQ) packet switches. While most research on scheduling algorithms for IQ switches has focused on maximizing throughput, packet deadlines are a key consideration in the context of real-time applications like multimedia streaming and video telephony. We consider traffic streams associated with service profiles, which reflect the inter-packet deadline (IPD) constraints between packets constituting the stream, and are determined by application layer requirements. To make the NP-hard problem of scheduling with strict deadline constraints tractable, we use soft deadlines as a modeling tool, and study the scheduling problem with soft deadlines in a dynamic programming (DP) framework. Our objective is to design a scheduler which minimizes aggregate soft deadline violation, or equivalently, preserves the service profiles associated with each traffic stream as faithfully as possible. We establish the optimality of a myopic or greedy scheduling policy for the canonical crossbar switch, in a finite-horizon setting. For bigger switches, we demonstrate using a Lyapunov function technique that a myopic policy, even though not optimal, maximizes the admissible region of the switch when the inter-packet deadlines are geometrically distributed. We develop low-complexity approximations to the near-optimal myopic policy, one based on the notion of neighborhood search, and two others based on convex relaxations of an integer programming problem. We demonstrate the efficacy of the proposed policies via simulations, using goodput as a performance metric. A key feature of the proposed policies is that they do not require any knowledge of traffic statistics (for example, periodicity of traffic streams), rendering them robust and amenable to implementation. Index Terms Input-queued switches, Scheduling, Soft Deadlines, Dynamic Programming, Myopic policy, Convex optimization, Lyapunov technique. 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 play a key role in QoS provisioning for such applications. Initial work on scheduling focused on output-queued (OQ) switches [], [], owing to their conceptual simplicity. For a K K OQ switch the switching fabric runs at K times the line rate, so that packets are readily available at the output ports for transmission upon entering the switch. While stringent QoS guarantees can be provided with OQ switches, scalability is a bottle-neck in their design. In contrast, the switching fabric runs at line rate for an input-queued (IQ) switch, and buffering occurs only at the input ports. Input queued (IQ) switches have received considerable interest both in the research community and industry, owing to their scalable architecture. On the flip side, sophisticated scheduling algorithms are needed to control the switching fabric to extract maximum benefit from IQ switches. A brief version of this technical report appears in IEEE GLOBECOM 6, San Francisco, CA, Nov. 6.

2 Most research on scheduling for IQ switches has focused on throughput and stability related issues. It is well known that the throughput of IQ switches is restricted to 58.6% due to head-of-line (HoL) blocking [3], an issue which can be eliminated by employing virtual output queues (VOQs) at the input ports. Several scheduling algorithms based on maximum weight matching (MWM) and its approximations and variants ( [4] [6] etc.) have been proposed in the literature, which achieve % throughput. The average delay performance of many throughput optimal algorithms has been explored both analytically and via simulations ( [5], [7] etc.). This vast body of results, while important in its own right, does not address the question of QoS provisioning for deadline constrained traffic. Liu et. al. [8] studied scheduling of multi-class periodic traffic flows through IQ switches. In their formulation, the deadline of a packet when it arrives at the switch is equal to the period of the flow to which it belongs. 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. [9] (nested period scheduling) and Rai et. al. [] (uniform weight round robin) also proposed heuristic scheduling policies for periodic flows. In other related work, Keslassy et. al. [] studied low-jitter scheduling based on a Birkhoff-von Neumann (BV) decomposition of the rate matrix, and Li et. al. [] presented a non-work conserving frame based scheduler with guaranteed delay and jitter bounds for leaky-bucket constrained traffic. Chang et. al. proposed schemes for providing delay guarantees in IQ switches based on BV decomposition in [3], and based on EDF for load balanced switches (see [4]) in [5]. 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 or end-user. For instance, in the context of video streaming, missed packet deadlines lead to playout freezes due to buffer underflow, and hence user annoyance. 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, preserve the service profile for each stream as faithfully as possible. Minimizing deviations from the service profile for every stream at every switch in the network ensures high overall QoS for deadline sensitive applications. IPDs provide a useful characterization of real-time traffic, especially in the context of applications like multimedia streaming. As an example, an IPD of ms between two successive video packets implies that the first packet requires ms for playout, and the second packet is not required at the receiver until the first packet has been consumed. IPDs are a measure of the stress a traffic stream induces in the network. Tighter or smaller the IPDs, more the resources required by the stream, and vice-versa. IPDs can be modulated by adaptive buffering and playout strategies adopted at the receiver, or variable rate encoding at the transmitter. For instance, if the network is congested, the receiver could slow down the video playout rate from 3 frames/sec to 5 frames/sec without perceptible quality degradation, and in process stretch out the IPDs by a factor of.. Alternatively, the transmitter could choose to encode the video with a lower resolution, and reduce the rate requirements of the video stream. These considerations make IPDs an attractive modeling tool. 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 up-stream 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 inter-packet deadline constraints is an NP-complete problem. 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 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 a switch. Soft deadlines can be thought of as a modeling device used to develop approximations to the scheduling problem with strict 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

3 3 packets are not dropped. Such applications abound on the internet today. As we mentioned earlier, most work on scheduling for IQ switches focuses 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 We set up a mathematical model for the IQ Switch scheduling problem with soft deadlines in Section II. Next, we analyze this model for the canonical IQ switch within a dynamic programming (DP) framework [6] in Section III. Our key result is that the optimal finite-horizon policy is a myopic policy. Leveraging our knowledge of the optimal policy for a switch, we develop three low-complexity approximations to the myopic scheduling policy for bigger crossbar switches (which is near-optimal, with O(K!) complexity for a K K switch) in Section IV, one based on the idea of neighborhood search and other two based on convex relaxations of an integer programming problem. We provide a geometric interpretation (based on maximum projection) for the myopic policy in Section V. We also demonstrate using a Lyapunov technique that the myopic policy maximizes the admissible region of the switch, when the inter-packet deadlines of all traffic streams follow a geometric probability distribution. We show the efficacy of the proposed policies via simulations in Section VI, and provide concluding remarks in Section VII. II. MODEL CONSTRUCTION Let us consider an input-queued switch with virtual output queues (VOQs) at the input ports to prevent head-of-line blocking. A K K IQ switch has K VOQs at the input (K at each input port). Each VOQ is associated with a traffic stream, comprised of packets arriving to that queue. Traffic streams with deadline constrained packets can be described using a vector x = (x, x,...), with x i {, }. We will call this the expected service vector (ESV) for a stream. The ESV essentially encapsulates the notion of service profile introduced in Section I. Suppose that the k th in the traffic vector occurs at location m and the (k + ) st occurs at location m + d k, that is, x m =, x m+ =,...,x m+dk =, x m+dk =. The interpretation is that if the stream is being serviced in accordance with its expected service vector, the inter-departure time between the k th and (k + ) st packets is exactly d k time-slots. Equivalently, d k is the inter-packet deadline (IPD) between the k th and (k + ) st packets of the stream. We define the cumulative expected service vector (CESV) as X = (X, X,...), where n X n = x i, n =,,... () X n represents the total number of packets of the stream which should ideally have departed the switch by the end of time-slot n. We assume X n n +, that is, successive packets in a stream have an IPD of at least one time-slot. For large n, we can think of X n /(n + ) as the average rate associated with the stream. For example, if the IPDs are constant (say d), X n = (n + )/d and the average rate is /d. We will assume for the remainder of this paper that lim n Xn /(n + ) exists for every traffic stream, so that the notion of average rate is well defined. Due to congestion caused by resource contention for multiplexed output links at the switch, a stream will not always receive service in accordance with its ESV. To quantify this effect, we associate with each traffic stream a received service vector (RSV) y = (y, y,...) with y i {, }. Then, y i = if the switch forwards a packet of this stream in the i th time-slot, and else. Similar to CESV, we define the cumulative received service vector (CRSV) as Y = (Y, Y,...), where n Y n = y i, n =,,... () Y n represents the number of packets that have been forwarded from the traffic stream till the end of time-slot n. Ideally, we would like Y n = X n n. If Y n > X n, the stream has received more service than it requires. In this case, we say that the Throughout this paper, vectors are denoted in boldface.

4 4 stream is leading. This is not desirable from a fairness perspective, especially if other traffic streams are starved for service. If Y n < X n, the stream has received less than its required share of service. In this case, we say that the stream is lagging. This is not desirable from a QoS perspective, since a lag corresponds to missed packet deadlines, which for instance translates into playout freezes (and hence user annoyance) in the context of video streaming. Each traffic stream at the input of the switch is subject to this trade-off between fairness and QoS. The trade-off can be quantified using the notion of deviation, D n Y n X n. The objective of the scheduler is to keep the deviation D n as close to as possible at all times, for every traffic stream. The idea is pictorially depicted in Fig. (on the left side). Since we allow for deviations from expected service, we can think of packets 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. III. THE CANONICAL SWITCH OUTPUT PORTS INPUT PORTS (a) (b) Fig.. Two possible configurations of a crossbar switch: (a) Configuration C and (b) Configuration C. Let us study the scheduling problem for the canonical IQ switch (See Fig. ). The input and output ports are labeled and. There are four VOQs at the input, one corresponding to each input-output pair. VOQ Q ij holds packets destined from input port i to output port j. In each time-slot, the switch can be set in configuration C or C. In configuration C, input ports, are connected to output ports, respectively, while in configuration C they are connected to output ports, respectively. The switch speed-up is assumed to be unity, that is, at most one packet can be transferred to each output from one of the inputs in a time-slot. A. Cost Structure We define the state of system as the four-tuple s = (s, s, s, s ), where s ij is the deviation associated with Q ij. We denote x n = (x n, x n, x n, x n ), where x n ij {, } is the nth element of the ESV of the traffic stream associated with Q ij. Similarly, X n = (X n, Xn, Xn, Xn ) where Xn ij is the nth element of the CESV of the traffic stream associated with Q ij. We denote a = (,,, ), b = (,,, ) and = a b = (,,, ). With Q ij we associate the cost function φ ij (k) = f + ij (k +) + f ij (k ), which measures the cost associated with deviation k Z, where k + = max(k, ) and k = min(k, ). We assume that f + ij is a convex non-negative non-decreasing function and f ij is a convex non-negative non-increasing function. Thus, f+ ij measures the cost of positive deviation (lead) and f ij measures the cost of negative deviation (lag). A typical cost function is depicted in Fig., on the right side. Recall that a convex function g : Z R satisfies We define the sum cost function g(k + ) g(k) g(k) g(k ), k Z. φ(s) = i= j= φ ij (s ij ), (3) as the sum of deviation cost of all four VOQs. Finally, we define the discrete derivative of φ ij (k) as ψ ij (k) = φ ij (k) φ ij (k ). (4)

5 5 Lead Cost Lag Expected Service Lead Cost Received Service Lag Cost Time Deviation Fig.. The left side of the figure depicts the notion of expected and received service, and deviation from expected service. The right side of the figure depicts a typical deviation cost function associated with a VOQ. Since φ ij is convex, ψ ij is a non-decreasing function of k. B. Dynamic Programming (DP) Formulation We will consider a finite time-horizon of N time-slots. To facilitate exposition, we will assume that all VOQs are infinitely backlogged. Consequently, all VOQs are potential scheduling candidates over the entire time-horizon N. We define an admissible policy Π N {C, C } N as a sequence of switch configurations in time-slots,...,n. Letting s n Π N denote the state at the beginning of time-slot n under policy Π N, the objective is to find a policy Π N such that { Π N = argmin Π N Φ N (Π N ) = N φ ( s n ) } Π N, (5) given the initial state s. The set of admissible policies has cardinality N. Clearly, an exhaustive search for the optimal policy which involves evaluating the cost incurred by all admissible policies is not feasible even for moderately large N. Instead, we adopt the methodology of dynamic programming [6] to compute the optimal policy Π N. Let us first examine the evolution of the state under an admissible policy Π N. In state s n at the beginning of time-slot n, Π N can either choose configuration C or configuration C. If Π N chooses C, one packet each is transferred from input ports and to output ports and respectively. The deviations of Q and Q either stay the same or increase by, depending on whether x n and xn are or, respectively. The deviations of Q and Q either stay the same or decrease by, depending on whether x n and x n are or, respectively. Thus, the state of the system changes from s n = (s n, s n, s n, s n ) to (s n+ + x n, sn + xn, sn xn, sn xn ), or in vector notation sn+ = s n + a x n. By the same token, if Π N chooses configuration C in time-slot n, the new state is s n+ = s n + b x n. Let V n (s) denote the cost-to-go at the beginning of time-slot n, starting in state s. V n (s) is the cost incurred by the optimal policy Π N from time-slot n to time-slot N (end of horizon), starting in state s in time-slot n. By the principle of optimality, the cost-to-go function can be computed from the following set of non-linear dynamic programming equations for n =,...,N: and the boundary conditions V N (s) = s, where n= V n (s) = min{ω n+ (s + a x n ), Ω n+ (s + b x n )}, }{{}}{{} Configuration C Configuration C Ω n (s) V n (s) + φ(s). (7) It is easy to check, V (s ) = Φ N (Π N ). Thus, solving the DP equations given by (6) to compute V (s ) is equivalent to finding the minimum cost policy Π N. Now define the decision function γ n (s) Ω n+ (s + a x n ) Ω n+ (s + b x n ). (8) (6) Configuration C is optimal in state s in time-slot n if γ n (s), and configuration C is optimal else. We denote, { C n C ; γ n (s) (s) = C ; γ n (s) >. (9)

6 6 C. Properties of the Optimal Policy Having formulated the soft-deadline scheduling problem for a switch within a DP framework, let us now explore key structural properties of the optimal scheduling policy Π N. The two possible states in time-slot n + starting in state s in time-slot n are s + a x n and s + b x n, which differ by a b = (for any policy). The following two lemmas become important in the light of this observation. Lemma : φ(s + ) φ(s) φ(s) φ(s ), s. Proof: Recalling the definition of ψ ij, φ(s + ) φ(s) = ψ (s + ) + ψ (s + ) ψ (s ) ψ (s ) φ(s) φ(s ) = ψ (s ) + ψ (s ) ψ (s + ) ψ (s + ). Since ψ ij (s ij + ) ψ ij (s ij ), the result follows. Lemma : V n (s + ) V n (s) V n (s) V n (s ), n N, s. Proof: The proof is by induction. Since V N (s) = s, the lemma is trivially true for n = N. We will assume that the lemma is true for m = n + < N, and show that it holds for m = n. In particular, we assume Setting t = s + a x n in () and invoking Lemma we get V n+ (t + ) V n+ (t) V n+ (t) V n+ (t ), t. () Ω n+ (s + + a x n ) Ω n+ (s + a x n ) Ω n+ (s + a x n ) Ω n+ (s + b x n ). () Similarly, setting t = s + b x n in () we get Ω n+ (s + a x n ) Ω n+ (s + b x n ) Ω n+ (s + b x n ) Ω n+ (s + b + x n ). () It follows from the definition of γ n, () and () that In view of (3), four distinct cases can arise: γ n (s + ) γ n (s) γ n (s ). (3) (a) γ n (s + ) γ n (s) γ n (s ): In this case, C n (s ) = C n (s) = C n (s + ) = C. We have, V n (s + ) = Ω n+ (s + + a x n ) V n (s) = Ω n+ (s + a x n ) V n (s ) = Ω n+ (s + b x n ). The claim of the lemma follows from (). (b) γ n (s + ) γ n (s) γ n (s ) : In this case, C n (s ) = C n (s) = C n (s + ) = C. We have, V n (s + ) = Ω n+ (s + a x n ) V n (s) = Ω n+ (s + b x n ) V n (s ) = Ω n+ (s + b x n ). The claim of the lemma follows from (). (c) γ n (s + ) γ n (s) γ n (s ): In this case, C n (s ) = C n (s) = C, C n (s + ) = C. We have, The claim of the lemma follows because γ n (s). V n (s + ) = Ω n+ (s + a x n ) V n (s) = Ω n+ (s + a x n ) V n (s ) = Ω n+ (s + b x n ).

7 7 (d) γ n (s + ) γ n (s) γ n (s ): In this case, C n (s ) = C, C n (s) = C n (s + ) = C. We have, V n (s + ) = Ω n+ (s + a x n ) V n (s) = Ω n+ (s + b x n ) V n (s ) = Ω n+ (s + b x n ). The claim of the lemma follows because γ n (s) >. Since the four cases are collectively exhaustive, the proof is complete. The following corollaries are immediate consequences of Lemma and. Corollary : Ω n (s + ) Ω n (s) Ω n (s) Ω n (s ), n N, s. Proof: Adding the results of Lemma and Lemma and invoking the definition of Ω n, we get the desired result. Corollary : γ n (s + ) γ n (s), n N, s. Proof: From Corollary, Ω n (t + ) Ω n (t) Ω n (t) Ω n (t ), n N, t. Setting t = s + a x n and invoking the definition of γ n, we get the desired result. D. A Different Perspective, and More Properties of the Optimal Policy Given the initial state s, we say that a state s is reachable in time-slot n if there exists a sequence of switch configurations in time-slots,...,n which drive the switch to state s in time-slot n. The reachable states in time-slot n constitute the set S n = {s + ka + (n k)b X n, k =,...,n}. We set s n k = s + ka + (n k)b X n. The state s n k is reached if the optimal policy Π N chooses configuration C in k of the time-slots from,...,n, and configuration C in the remaining n k time-slots. From our earlier discussion, it follows that the states reachable in time-slot n+ starting from state s n k in time-slot n are sn k +a xn+ = s n+ k+ (if configuration C is chosen) and s n k + b xn+ = s n+ k (if configuration C is chosen). Equivalently, we can identity the state in time-slot n by the index k, which increases by in the next time-slot if configuration C is chosen and remains the same if configuration C is chosen. Root Index k Time Index k increments by if configuration C is chosen Index k remains unaltered if configuration C is chosen 3 Fig. 3. A DAG based representation of the reachable states in successive time-slots. The solid lines represent choice of configuration C, while the dotted lines represent choice of configuration C. The inset depicts the evolution of state index k, based on the choice of configuration in a state. The index increments by if C is chosen, and remains unaltered if C is chosen. Observe that the state s n k is a sum of two components: a static component s + ka + (n k)b, which is independent of the input traffic streams and a dynamic component, X n which is determined by the CESVs (and hence inter-packet deadlines) of the input traffic streams. The temporal evolution of the static component depends on the choice of scheduling policy, while the evolution of the dynamic component does not. The evolution of the static component can be elegantly represented using

8 8 a directed acyclic graph (DAG) (see Fig. 3). The root of the DAG is s. Nodes of the DAG at a depth n correspond to the static component of the reachable states in time-slot n. There are n + nodes at depth n, and they are ordered from right to left in increasing order of index k. Each node has exactly two children. The right child corresponds to choosing configuration C in the state corresponding to that node, while the left child corresponds to choosing configuration C. As discussed earlier, choosing C in time-slot n in state s n k is tantamount to incrementing the index k by in time-slot n +, while choosing C leaves the index k unaltered. The leaf nodes correspond to the terminal states at the end of the time-horizon. The goal of the optimal policy is to traverse the least-cost path from the root node to one of the leaf nodes. The DAG structure provides an efficient means of solving the DP equations to compute the optimal scheduling policy Π N. Also, it is a useful aid for establishing several structural properties of the optimal policy, as we shall see in the remainder of this section. Lemma 3 (Threshold Property): For n =,...,N, there exists a k n (n + ) such that Cn (s n k ) = C k k n and C n (s n k ) = C k > k n. Proof: By definition, s n k+ sn k =. Also, from Corollary we have γn (s + ) γ n (s). It follows that γ n (s n k+ ) γ n (s n k ), or equivalently, γn (s n k ) is non-decreasing in k. Recall that the optimal action in state s at time n is completely determined by the sign of γ n (s). For fixed n, γ n (s n k ) can change sign at most once as k increases from to n. In other words, kn such that γn (s n k ) k k n (implying Cn (s n k ) = C ) and γ n (s n k ) > k > k n (implying Cn (s n k ) = C ). In words, the lemma states that the set of reachable states in time-slot n, when arranged in increasing order of index k, can be partitioned by a threshold k n into two subsets, such that C is optimal on one subset, and C on the other. Thus, if we traverse the DAG at depth n from right to left (that is, from k = to k = n), there exists a k n n such that configuration C is optimal in all nodes (states) to the right of k n, and configuration C is optimal in all nodes (states) to the left of k n. Let us set the initial state to s = (,,, ) and gain further intuition into the threshold property. We have s n = nb Xn = ( X n, Xn, n Xn, n Xn ) and sn n = na Xn = (n X n, n Xn, Xn, Xn ). In state sn, Q and Q have a negative deviation while Q and Q have a positive deviation. If configuration C is chosen, the deviation of the former two queues can only increase, while the deviation of the latter two queues can only decrease. Thus, the deviation of all four queues will move closer to (desirable). Choosing configuration C, on the other hand, will drive the deviation of all four queues away from (undesirable). On the other extreme, choosing configuration C in state s n n will drive the deviation of all four queues closer to, while choosing configuration C will drive the deviation of all four queues away from. We see that C is the optimal configuration for k =, while C is the optimal configuration for k = n; we can expect the existence of a threshold k n at which the choice of optimal configuration switches over from C to C. The next result states that for fixed n, V n achieves its minimum at k n. Lemma 4 (Minima of V n ): argmin V n (s n k ) = k n. k n Proof: It follows from Lemma that V n (s n k+ ) V n (s n k ) V n (s n k ) V n (s n k ). (4) By definition of n k, C(sn kn +) = C and C(s n k ) = C. Thus, V n (s n n kn +) = V n (s n k ) = n Ωn+ (s n+ k Setting k = k n+). n and then k = kn + in (4) we get V n (s n kn +) V n (s n kn +) = V n (s n k ) V n (s n n kn ). Inductively, we conclude that V n (s n k ) is non-increasing for k kn and non-decreasing for k > k n, which is precisely the claim of the lemma. A direct consequence of Lemma 4 is the following corollary: Corollary 3 (Minima of Ω n ): arg min Ω n (s n k ) = k n. k n Equipped with various results regarding φ, V n, Ω n and γ n, we are now ready to present the most important result regarding the optimal policy Π N, which is the optimality of a myopic policy. What does it mean for the optimal policy to be myopic? It implies that the optimal decision in state s in time-slot n can be made by merely choosing the configuration with the lowest instantaneous cost, rather than solving the DP equations over the horizon n,..., N to compute the optimal configuration C n (s). Equivalently, the result of one joint N-period optimization is the same as the result of N successive one-period optimization problems. A myopic policy is defined more formally as a part of the proof of Theorem.

9 9 Theorem (Optimality of Myopic Policy): The finite-horizon optimal policy Π N is myopic. Proof: In terms of our notation, optimality of a myopic policy is equivalent to argmin{ω n+ (s + a x n ), Ω n+ (s + b x n )} = argmin{φ(s + a x n ), φ(s + b x n )}. (5) An equivalent form of (5), which we will prove, is the following: sgn { Ω n+ (s + a x n ) Ω n+ (s + b x n ) } = sgn {φ(s + a x n ) φ(s + b x n )}, (6) where sgn(y) = + if y and sgn(y) = if y <. Consider state s in time-slot n and assume that C n (s) = C. We will only present arguments for this case. The proof for the case C n (s) = C is symmetric. The states reachable (from s) in time-slot n + are s = s +a x n and s = s +b x n. Four different cases can arise, depending on whether C n+ (s ) and C n+ (s ) are C or C. ) C n+ (s ) = C, C n+ (s ) = C : Since V N (s) = s, the result is trivially true for n = N. Let us consider n < N. It follows by definition that V n+ (s ) = V n+ (s ) = V n+ (s+a+b x n x n+ ). Thus, Ω n+ (s ) Ω n+ (s ) = φ(s ) φ(s ), implying (6). ) C n+ (s ) = C, C n+ (s ) = C : Again, the result is trivially true for n = N. Let n < N. Several possibilities can arise. In state s, the optimal action is C. Thus, starting in state s in time-slot n+, the optimal state in time-slot n+ is s + b x n+. The next state is determined by the optimal action in state s + b x n+ in time-slot n +, and so on. In general, we can construct a chain of states which the system visits under the optimal policy Π N starting at s. The chain terminates for one of following two reasons: (a) The finite time-horizon N is reached. (b) A state is reached where the optimal configuration is C. For all states constituting the chain except possibly the last, the optimal action is C. The optimal action in state s in time-slot n + is also C. Thus, we can construct a similar chain of states originating from s, which terminates for one of the two reasons (a) or (b). What do these two chains of states look like? Since action C is optimal in all states belonging to these chains (except possibly the last state), the chain originating from s comprises of states of the type (in time-slots n + j, j =,,...) n+j t j = s + jb x m = s + jb X n+j + X n, and the chain originating from s comprises of states of the type (in time-slots n + j, j =,,...) m=n n+j t j = s + jb x m = s + jb X n+j + X n. m=n Thus, in time-slot n + j < N, the chains originating from s and s are respectively in states t j and tj. Note, tj tj =. The threshold property (Lemma 3) implies that the chain originating from s cannot terminate before the chain originating from s due to reason (b). Lemma implies If both chains terminate due to reason (a), we have φ(t j ) φ(tj ) φ(s ) φ(s ) = δ φ. (7) Ω n+ (s ) = N n φ(t j ), Ωn+ (s ) = Ω n+ (s ) Ω n+ (s ) = N n φ(t j ). N n [φ(t j ) φ(tj )] (N n + )δ φ, where the first inequality follows because C n (s) = C and second inequality follows from (7). Clearly, (6) holds.

10 Now, suppose the chain originating at s terminates in time-slot M < N due to reason (a), that is, C M (t M n ) = C. We have two further sub-cases - (i) C M (t M n ) = C and (ii) C M (t M n ) = C. For sub-case (i), It follows from (7) that Ω n+ (s ) = Ω n+ (s ) = M n M n φ(t j ) + ΩM+ (t M n+ ), φ(t j ) + ΩM+ (t M n+ + ). }{{} t M n+ Ω n+ (s ) Ω n+ (s ) (M n + )δ φ, implying δ φ > and hence (6). For sub-case (ii), we have C M (t M n ) = C M (t M n ) = C, so Corollary 3 implies that Ω M (t M n ) Ω M (t M n ), or equivalently γ M (t M n ). That is, the optimal action in state t M n is C, which leads to a contradiction since the construction of the chain originating from s necessitates C as the optimal action in state t M n. Consequently, sub-case (ii) cannot arise. 3) C n+ (s ) = C, C n+ (s ) = C : This case violates the threshold property (Lemma 3) and hence cannot arise. 4) C n+ (s ) = C, C n+ (s ) = C : This case leads to a contradiction similar to that in sub-case (ii) of case () and hence cannot arise. By considering a set of collectively exhaustive cases when C n (s) = C, we have shown that (6) holds. Symmetric arguments can be presented when C n (s) = C. The conclusion is that to make the optimal decision in state s at time n, comparing Ω n+ (s + a x n ) and Ω n+ (s + b x n ) is equivalent to comparing φ(s + a x n ) and φ(s + b x n ). Consequently, the optimal policy Π N is a myopic policy. Is the optimality of a myopic policy a useful property? Yes. Computing the optimal policy over an N-period horizon from the DP equations in (6) requires O(N ) operations (if the DAG structure is exploited), while computing the myopic policy (which is also the optimal policy) requires only O(N) operations. E. Beyond : Modular Switches Before we leverage our knowledge of the optimal policy for a switch to design heuristics for bigger crossbar switches, we pause to mention that the results are important in their own right in the context of modular switches. Arbitrary sized switches can be realized by interconnecting several smaller switches (say ) in a cascade fashion [7]. Two examples are - Clos networks and Distributed Buffer Switches (for example, Banyan networks). The optimal scheduling policy Π N can be used at each element in this bigger switch. QoS guarantees at each stage of the switch (ensured by Π N ) translate into end-to-end QoS guarantees for the entire switch. IV. HEURISTIC SCHEDULING POLICIES We have shown that the optimal finite-horizon scheduling policy for a IQ switch is a myopic policy. Generalizing our DP formulation to a K K switch for K > will result in the state-space Z K (one for each VOQ) and K! (number of possible matchings of input to output ports) configurations in each state. Our numerical experiments show that a myopic policy (which chooses the configuration with the smallest instantaneous cost), although not optimal, is near-optimal for K >. Computing the myopic policy requires O(NK!) operations for a K K switch over a time-horizon of N time-slots, which is impractical even for mildly large K (for K = 6, K! 3 ). In some special cases (quadratic cost functions), computing the myopic policy reduces to the well-studied problem of finding the maximum weight matching (MWM) on a bipartite graph, using the (negative of) deviations of the VOQs as edge weights. However, MWM, with a computational complexity of O(K 3 ), is also quite unfriendly from an implementation perspective. Motivated by this discussion, we will now focus our attention on developing low-complexity approximations to the myopic policy (which is near-optimal). We will evaluate the performance of the proposed approximations experimentally in Section VI, and show that they mimic the myopic policy quite well, with much lower complexity.

11 A. Neighborhood Search (a) (b) (c) (d) Fig. 4. (a) A possible configuration of a 3 3 switch, and (b)-(d) Neighbors of the configuration depicted in (a). As discussed earlier, there are K! permissible configurations in each state. The myopic policy would have to evaluate the cost of all possible configurations, before choosing the configuration associated with the lowest cost. This is computationally prohibitive even for moderately large K. The neighborhood search heuristic scheduling algorithm is based on the assumption that the optimal configuration of the switch cannot change drastically from one time-slot to another. The notion was introduced in [6]. We can think of a switch configuration as a permutation π : {,..., K } {,...,K }, such that input port i is connected to output port π(i). A neighbor of permutation π is constructed by connecting input port i to output port π(j) and input port j to output port π(i), for some i j. An ( example ) for a 3 3 switch is depicted in Fig. 4. We denote the set of K neighbors of configuration π by N π. Clearly, N π =. Given the current system state and switch configuration π, the neighborhood search algorithm involves searching for the lowest-cost configuration over the set of configurations {π} {N π }. The per-time-slot complexity is a tractable O(K ), a vast improvement over the O(K!) complexity for computing the myopic policy precisely. More sophisticated notions of neighbors can be used, based for instance on geometric interpretations of the action space. See, for example, [8]. B. Convex Relaxation A Computing the optimal myopic policy is equivalent to solving an integer-programming (IP) problem in each time-slot (with K! feasible solutions), which can be formulated in time-slot n as follows: minimize φ ij (s n ij xn ij + α ij) subject to α ij =, j =,...,K α ij =, i =,...,K α ij {, }, i =,...,K, j =,...,K. The indicator variable α ij is if input port i is connected to output port j, and else. The constraints dictate that each input (output) port is connected to exactly one output (input) port. While the above IP is NP-hard, we can relax the constraint α ij {, } to α ij [, ] to obtain a constrained convex optimization problem with O(K ) constraints and K variables. Note that φ ij are convex (by choice) and all other constraints are linear (and hence convex). A convex formulation is attractive since numerous computationally efficient methods are available for obtaining the optimal solution [9]. The relaxation will have non-integer solutions in general. How do we convert this fractional solution to a valid switch configuration? For each input port i, we can think of the list ( α i,,..., α i,n ) as a preference list. We map input port i to output port j if j = argmax α ik. Contentions can be resolved randomly or by port numbers. k

12 C. Convex Relaxation B for Quadratic Cost Functions We propose another convex relaxation of the integer-program formulated in the preceding sub-section. While convex relaxation A was applicable for arbitrary choice of convex cost functions, we will now restrict our attention to quadratic cost functions, that is, φ ij (k) = k i, j. In particular, we solve the following constrained convex optimization problem minimize (s n ij x n ij + β ij ) subject to β ij =, j =,...,K β ij =, i =,...,K The advantage of this approach is that the solution can be computed analytically in closed form using Lagrange multipliers. It can be shown that the solution to the above constrained problem is given by where A i = B j = βij = (s n ij x n ij) + }{{} K (A i + B j S + ), (8) s n ij (s n ij xn ij ) = s n ij, i =,..., K S = K (s n ij xn ij ) = (s n ij xn ij ) = K s n ij, j =,...,K Observe that βij depends on the state of all VOQs contending for output port j, and all VOQs being served by input port i. βij can be interpreted as the affinity of input port i for output port j. Thus, β ij > β i j implies that it is more desirable to connect input port i than input port i to output port j. Similarly, βij > β ij implies that it is more desirable to connect input port i to output port j than output port j. Once again, we can use approach similar to the one used for convex relaxation A for converting our fractional solution into a valid switch configuration. We will re-visit convex relaxation B in Section V-C and highlight an interesting property. i= s n ij. V. MAXIMUM WEIGHT MATCHING, MAXIMUM PROJECTION POLICY AND ADMISSIBLE REGION A. Maximum Weight Matching and Maximum Projection Once again, we will restrict our attention to quadratic costs, that is, φ ij (k) = k i, j. The indicator variable α ij is set to if input port i is connected to output port j and else. The goal of the myopic policy in time-slot n is to choose α ij {, }, such that the cost function (s n ij xn ij + α ij) = is minimized. The cost function can be expressed as ( s n ij ) + α ij } {{ } Policy independent + ( s n ij + α ij) s n ij α ij. } {{ } Policy dependent The first term in the above summation is independent of the choice of scheduling policy. The second term is always K if perfect matchings are used, that is, every input port is connected to an output port. Thus, the optimal myopic action in

13 3 time-slot n is determined entirely by the third term. It follows that the myopic policy is a maximum weight matching policy, with s n ij serving as the weight for VOQ Q ij. With each admissible switch configuration, we can associate a K -length configuration vector. The k th entry of this configuration vector is if input port k/k is connected to output port mod (k, K), when this configuration is chosen. There are K! such configuration vectors. For instance, when K =, the possible configuration vectors are (,,, ) and (,,, ), corresponding to configurations C and C, respectively. With this interpretation, the myopic configuration (vector) in time-slot n is obtained via a maximum projection (MaxProj) policy as α (n) = arg max s n, α, (9) α where s n is the vectorized form of { s n ij }. Let us now define the state and time-dependent quadratic Lyapunov function By definition, L(n + ) = Now, letting δ L (n) = L(n + ) L(n), we see that L(n) = (s n ij). () (s n ij x n ij + α ij ) = ( s n ij). δ L (n) = s n x n, α s n,x n + x n,x n + K, () where x,y denotes the inner-product between vectors x and y. Recall that our goal is to keep the deviation of each VOQ as close to as possible, which is tantamount to keeping the quadratic function L( ) as close to as possible. With this interpretation, δ L can be thought of as the drift associated with the system. In our context, we would like the drift to be as negative as possible. If some of the VOQs have positive deviations (leading), their deviation (and hence the drift) can be driven back to by idling them for a few time-slots. Of course, this implies that we are allowing for configurations resulting from imperfect matchings, corresponding to configuration vectors α satisfying α, < K, where is a K -length vector with all unit entries. The more interesting case is one in which all VOQs have negative deviation (lagging). In this case, the myopic configuration will always correspond to a perfect matching. Henceforth, we will assume that all VOQs have a negative deviation. Since all terms in () barring the first one are independent of α, we have an alternative interpretation of (9) - the MaxProj policy minimizes the drift δ L (n) in each time-slot. B. Admissible Region We have assumed that we can associate with each traffic stream an average rate, which is equal to the inverse of the mean IPD between successive packets in the stream. It is clear that as the average rates of the input traffic streams increase (mean IPDs decrease), the switch becomes increasingly loaded, and deviations of the VOQs are more likely to be negative (lagging streams). Beyond a certain threshold load, the switch is unable to support the input traffic anymore, in the sense that at least one of the traffic streams gets infinitely lagged (its deviation approaches unboundedly). We denote the input load vector by r = (r,..., r K ), where /r i is the mean IPD for the input stream to the i th VOQ. We say that r is in the admissible region of the switch if there is a scheduling policy which ensures that all VOQs are finitely lagged at all times under load r. The admissible region is described by K r (i )K+j <, i =,...,K, j= K r (j )K+i <, i =,...,K, r i [, ), i =,...,K. () j= We call the loading uniform if r i = p, i for some p < /K. This means that all traffic streams have a mean IPD of /p. For a special case of i.i.d. uniform loading, viz. a geometric distribution of mean /p on the IPDs, we will demonstrate in A matrix {x ij } i, is vectorized into a K vector by juxtaposing its rows (from to K ) and taking the transpose. M For M vectors x and y, the inner product x,y = x i y i.

14 4 Theorem that all VOQs remain finitely lagged if the switch is operated using the MaxProj policy (9). It is intuitively clear that the deviation of at least one queue will drift to if p > /K, irrespective of the scheduling policy employed. More formally, consider E (s n+ ij s n ij) = α ij + E[x n ij] = K( Kp). If p > /K, the right-most term in the above equation is negative, implying that the sum of deviations has a negative expected drift in each time-slot, irrespective of the scheduling policy used. From () we obtain, E[δ L (n) s n ] = s n, α pk p s n, + K + pk. (3) It immediately follows from (3) that a maximum projection policy (which maximizes the projection s n, α ) minimizes the expected conditional drift in each time-slot. In particular, the policy is given Observe that the policy in (4) is different from the MaxProj policy in (9). Theorem : For p < /K and L(n) defined by (), we have lim sup n under the MaxProj policy given by (9). α (n) = arg max s n, α, (4) α L(n) < a.s., lim sup E[L(n)] <, n Proof: Let us first consider the maximum projection scheduling policy given by (4). With the choice of configuration α under (4) in time-slot n, the expected conditional drift is given by E[δ L (n) s n ] = s n, α pk p s n, + K + pk, It is easy to show that s n, K s n, α. (5) Using (5) and the assumption that p < /K, we get E[δ L (n) s n ] K ( pk) sn, + (K ) = K ( pk) sn, + (K ). Since s n ij are negative (by assumption) and integer valued, it follows that s n, s n,s n = L(n). Consequently, E[δ L (n) s n ] K ( pk) L(n) + (K ). ( ) K(K ) Observe that E[δ L (n) s n ] < when L(n) > = c(p, K), that is the expected conditional drift is negative for pk L(n) large enough, under the policy given by (4). Now consider the MaxProj policy given by (9). Note that s n, α = s n, α x n, α. Since x n ij {, } and α, = K, we get s n, α s n, α s n, α + K. Thus, the MaxProj policy always produces a bigger projection than the projection produced by the policy in (9). The implication is that the above bounding techniques are directly applicable to MaxProj. We now invoke standard arguments from the paper by Kumar and Meyn [] to claim that the hypothesis of the theorem is true. The implication of Theorem is that the deviation of every VOQ remains finite under i.i.d. uniform loading if p < /K, and the MaxProj scheduling policy is used. In other words, the MaxProj policy maximizes the admissible region or stability region of a deadline aware IQ switch under i.i.d. uniform loading. Analytically, the result is very similar to the stability or % throughput proof of the MWM scheduling policy for IQ switches under i.i.d. Bernoulli traffic [4]. However, due to the difference in application context and design objectives, our result is very different in spirit. Also, the notion of stability is

15 5 very different in the context of deadline aware switches. While stability of a scheduler under non-real-time traffic addresses its ability to maintain finite backlogs for all flows (under any admissible load), stability in our context refers to the ability of the scheduler to serve deadline constrained flows with bounded deviation from their service profiles (under any admissible load). As a final remark, the result of Theorem, which assumes uniform loading, can be extended to incorporate all i.i.d. admissible non-uniform loads by considering a Birkhoff von Neumann (BV) decomposition of the input load vector r [3]. The results can also be extended to the case of weighted quadratic cost functions, that is, φ ij (k) = λ ij k, λ ij >. C. Revisiting Convex Relaxation B Recall our convex relaxation B formulated in Section IV-C as a low-complexity approximation to the myopic policy. We computed the optimal fractional solution {βij } in closed form. One possible way of converting this fractional solution into a legitimate switch configuration is to compute a maximum weight matching with βij s as the edge weights, or equivalently, a maximum projection on permissible configuration vectors. Indeed, the resultant solution is exactly the same as the one obtained from the MaxProj schedule in (9). To see this, consider a typical switch configuration {(i, π(i), i =,..., K }, where π is a permutation. By definition, π(i) π(j) if i j. Let α denote the associated K -length configuration vector. With the solution of convex relaxation B as edge weights, the projection of β (vectorized form of {βij }) on the vector associated with this configuration is given by β, α = By definition of A i, B j and the fact that Since S is independent of α, βi,π(i) = B π(i) = s n i,π(i) + K A i + K B j, it follows that β, α = s n, α + + S. arg max β, α = arg min s n, α. α α B π(i) + S. While this is a useful interpretation of the solution of convex relaxation B, it does not reduce the computational complexity of scheduling. However, if a greedy heuristic is used to compute the maximum weight matching, we empirically observed that using βij as weights is more effective than using sn ij as weights. The greedy heuristic works as follows: Edge weights are arranged in a K K matrix with the i th row corresponding to the i th input port and the j th column corresponding to the j th output port. For computing the maximum weight matching, the largest entry in this matrix is chosen and the corresponding row and column are eliminated. The procedure is repeated K times, and results in a perfect matching. Ties are broken randomly or by port number. As an illustration, consider the matrix of S = { s n ij } and the corresponding B = {βij } given by S = , B = /3 /3 /3 /3 5/3 /3 The optimal configurations are {(, ), (, ), (, )} and {(, ), (, ), (, )}. Using the matrix S with the greedy heuristic produces one of the optimal configurations with probability /4, if ties are broken randomly. However, using the matrix B with the greedy heuristic produces one of the optimal configurations with probability. Thus, in some sense, the βij s provide a better description of the system state, rather than the state variables s n ij themselves. Finally, note that the form of β ij s is reminiscent of the longest-port-first (LPF) scheduling algorithm proposed in [] for throughput-aware switches.. VI. SIMULATION RESULTS In this section, we investigate the performance of proposed policies experimentally. All our results are for a 4 4 IQ switch. The switch has 4 = 6 VOQs, and 4! = 4 possible configurations in every time-slot. We considered two different kinds of probability distributions on the inter-packet deadlines of the input streams (a) Geometrically distributed IPDs, and (b) IPDs generated using a two-state Markov chain (MC). For each type of IPD distribution, we considered two possible loading