Service Control Policies for Packet Switches with Target Output Traffic Profiles

Transcription

1 Service Control Policies for Packet Switches with Target Output Traffic Profiles Aditya Dua and icholas Bambos Department of Electrical Engineering Stanford University 35 Serra Mall, Stanford CA Abstract We address the problem of packet scheduling for traffic streams with target profiles through input queued switches, arising in real-time applications like multimedia streaming. The target profiles specify the ideal inter-departure times of packets in each stream from the switch (inter-packet deadlines). The goal of the scheduler is to dynamically select service configurations for the switch so that output traffic streams adhere to their associated target profiles as accurately as possible. We develop service controls/schedules to minimize deviation of output streams from their desirable targets for real-time multimedia services with tight delay and jitter requirements. We formulate the problem within a dynamic programming framework and establish key structural properties of the optimal schedules. We propose the notion of configuration subset based controls, which enables us to design simple scheduling algorithms amenable to implementation in high performance packet switches. The proposed algorithms deliver excellent performance at low computational complexity and are robust with respect to statistics of input traffic traces. While our focus is on the input queued switching architecture, the idea of service trace control to match target profiles is more widely applicable to any scenario involving delay and jitter sensitive communication over a packet switched network. Index Terms Packet switching, Real-time Scheduling, Quality of Service, Dynamic Programming, Lyapunov Techniques. I. ITRODUCTIO Real-time services such as multimedia streaming, video on demand (VOD), video telephony etc. continue to gain popularity amongst Internet users. These applications have strict quality-of-service (QoS) requirements with regard to packet deadlines and jitter. Scheduling algorithms employed in packet switches/routers play a key role in QoS provisioning for real-time Internet applications. While early research on packet switching focused on the output-queued (OQ) switch architecture [], [], input-queued (IQ) switches have received considerable attention in recent times, owing to their scalable architecture. However, nontrivial scheduling/arbitration algorithms are needed to resolve contention between input traffic streams to ensure efficient operation of an IQ switch. Most research on IQ switch scheduling has revolved around performance metrics like throughput and average delay, which are conceived on macro time-scales (at the flow level). umerous scheduling algorithms based on maximum weight matching (MWM) and its variants ( [3]- [5] etc.) have been proposed in the literature, which provably guarantee % throughput. The average delay performance of these schedulers has been studied both analytically and experimentally ( [5], [6] etc.). This vast body of literature, while important in its own right, does not address the problem of QoS provisioning for deadline constrained real-time traffic, which entails performance engineering and control of the switch on micro time-scales (at the packet level). In this paper, we develop IQ switch scheduling algorithms for traffic streams associated with target stream profiles. The target profile associated with a traffic stream specifies the differential or inter-packet deadlines (IPD) between consecutive packets in the stream. Equivalently, the profile determines the ideal inter-departure times of packets in the stream from the output of the switch. In the absence of congestion, packets from a stream will depart the switch in accordance with the target profile. However, contention between competing input traffic streams for the shared switch fabric causes congestion in the switch. Consequently, the actual departure process of a stream deviates from the ideal departure process, as dictated by its target profile. In other words, the target profile of the stream gets distorted by the switch. Thus, the objective of the scheduler is to minimize distortion of target profiles of all streams traversing the switch, i.e, the scheduler must select service traces for streams such that their departure processes track their corresponding target profiles as accurately as possible. We call this the service trace control problem for an IQ switch. The driving motivation is to render packet switched networks like the Internet transparent to deadline sensitive multimedia traffic. The target profiles are determined by the times at which packets need to be delivered to end users to ensure uninterrupted multimedia playout (the playout profile). Thus, a high quality multimedia experience is guaranteed to end users if traffic streams negotiate all routers/switches along their destined path with minimal profile distortion. Of particular interest to us is the case where the switch does not drop delayed packets, but is penalized for violating deadlines. This case is representative of half-duplex applications like lossless multimedia streaming, where the end user would much rather wait for a delayed packet than miss viewing the media content encoded in the delayed packet (which would happen if the switch drops delayed packets). The switch is also penalized for being ahead of deadlines. This is done to prevent buffer overflows at downstream switches and the end user, and to avoid starvation of delay tolerant traffic (without target profiles) being served by the switch. In our framework, packets can be thought of as being associated with soft deadlines. Any deviation from the target

2 profile manifests itself as a soft deadline violation, which is associated with a penalty/cost. The softness of a deadline is quantified by the cost associated with its violation. The service trace control problem thus translates to minimization of aggregate soft deadline violation cost over all traffic streams. A. Related work The problem of scheduling periodic messages through IQ switches has been addressed in the literature. Under the periodic model, packets for each traffic stream are generated periodically, and the maximum time allowed for transmission of a packet is equal to the period of the stream. A schedule is deemed feasible if all messages meet their deadline requirements. ote that the periodic model is a special case of our general model, with constant inter-departure times (equal to the period of the stream). Inukai [7] showed that a feasible schedule can be constructed when the periods of all streams are equal and both input and output link utilization are less than. Liu et. al. [8] conjectured that Inukai s conclusion holds for traffic streams with arbitrary periods and also proposed heuristic scheduling algorithms based on the earliest deadline first (EDF) and minimum laxity first (MLF) policies. The performance of their heuristics degrades rapidly with switch size. In support of the conjecture, Giles et. al. [9] proposed the nested periodic scheduling (PS) rule, which finds a feasible schedule when each period divides all longer periods and link utilization is less than. PS also finds a feasible schedule for arbitrary message periods, provided the link utilization is no more than /4. The computational complexity of PS is O( 4 ) for an switch. Rai et. al. [] developed heuristic weighted round robin (WRR) scheduling policies for multiclass periodic traffic, with an online implementation complexity of O( 3 ) per time-slot. On a different strand of research, Li et. al. [] developed a 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 the Birkhoff-von eumann (BV) decomposition of the input rate matrix in [] and based on EDF for load balanced switches (see [3]) in [4]. Their schemes have an offline computational complexity of O( 4.5 ) and an online memory requirement of O( 3 log ). Keslassy et. al. [5] proposed a frame based scheduler based on the BV decomposition to guarantee low jitter, under the assumption that jitter sensitive traffic forms a small fraction of the overall switch load. A common feature of all works cited above is that they deal with scheduling of smooth/regular traffic (with fixed rates known to the scheduler). However, traffic arriving to a switch can be irregular due to the bursty nature of traffic sources (e.g. variable bit rate video), due to flow aggregation, or due to jitter induced by upstream switches. Further, rates of different streams are not always known to the scheduler. Also, these schemes have prohibitively high computational complexity, making them difficult to implement in high speed switches. We do not make any assumptions on the rate, periodicity etc. of traffic streams arriving to the switch. Consequently, the service trace controls we develop are agnostic to specifics of the input streams, making them adaptive and robust. We also address the issue of implementation complexity by developing a family of service trace control policies using the notion of subset based control. The key idea is to partition the huge set of possible switch configurations (of size!) into smaller subsets of size each, and operate the switch using only one subset in every time-slot. The resultant policies have a manageable complexity of O( ) per time-slot. For completeness we also mention two other peripherally related bodies of work, akin in spirit to our modeling approach. Our soft deadline model is reminiscent of the time/utility function (TUF) approach introduced by Jensen at. al. [6] to study scheduling in real-time operating systems. Our notion of target profiles for different traffic streams is reminiscent of the rich set of network calculus tools developed by Cruz ( [7] and several subsequent works with others) to study the problem of providing deterministic QoS guarantees in time-slotted virtual circuit networks, based on the notion of service curves. Conventional IQ switch scheduler design is geared toward delay tolerant traffic, where the objective is to push as much traffic as possible through the switch to maximize throughput. Our work signals a departure from this conventional approach. In our switching paradigm, the objective is to pull out traffic from the switch at just the right rate, determined by the target profiles of traffic streams. We believe that it is essential to adopt the pull approach to provide circuit switched like connections with QoS guarantees over the packet switched Internet, which is inherently a QoS unaware architecture. B. Contributions The key contributions of our work are two-fold. Firstly, we develop a novel deadline aware switching framework based on the idea of shaping the output process of the switch to match desired profiles. While we exclusively study this model in the context of an IQ switch, the core ideas are more widely applicable to any queuing system where competing users/jobs are associated with deadline constraints. Secondly, we develop low complexity control policies for IQ switches using the idea of subset based control. While we do so exclusively in the context of our service trace control problem, subset based control provides a low complexity approximation to the maximum weight matching problem, a classical problem in graph theory [8]. The work presented here is a substantial extension of the research thread initiated in [9] []. C. Organization of the paper The remainder of this paper is organized as follows: In Section II, we formulate the service trace problem for minimizing distortion of target stream profiles as a finite-horizon dynamic program []. We establish the optimality of a greedy policy for a switch and explore its feasibility as a heuristic policy for bigger switches. We introduce the notion of subset based service trace control in Section III. In Section IV, we develop the notion of meta-queues, which yields an alternative view of subset based control and also provides a general framework for designing different families of service trace control policies. We define the admissible region of the

3 3 switch in Section V and show using Lyapunov techniques that subset based service trace control, in conjunction with an appropriate subset selection rule, guarantees finite delay for all traffic streams, under any admissible load. We experimentally evaluate the performance of our proposed policies in Section VI, and conclude our paper in Section VII. D. otations and conventions Some notations and conventions employed throughout the paper are enumerated here for convenience. All vectors and sequences are denoted in boldface. denotes the set of natural numbers and Z denotes the set of integers. denotes the all zeros vector and denotes the all ones vector. e i denotes the i th unit vector in R, i.e., a vector with a in the i th location and s elsewhere. Further, e =. The inner product between two vectors x and y is denoted x,y. Finally, the big-oh notation f() = O(g()) is used to indicate that c > such that f() cg() for large enough. II. SERVICE TRACE COTROL TO MIIMIZE DISTORTIO OF TARGET STREAM PROFILES A. Definitions and preliminaries Consider an IQ switch with virtual output queues (VOQs) at all input ports to prevent head-of-line (HOL) blocking. There are VOQs in an switch with input and output ports, as shown in Fig.. Both input and output ports are indexed,...,. The i th VOQ stores packets destined from input port (i )/ + to output port (i ) mod + and is denoted Q i. The switch operates in slotted time and is governed by the crossbar constraints, i.e., every input (output) port can be connected to at most one output (input) port in a time-slot. An switch can be set into! possible configurations. Each configuration is associated with a unique configuration vector of length. Let v i = (v i, v i,...v i, ) V denote the i th configuration vector, where V is the set of all possible configuration vectors. Then, v i,j = if Q j is served when the switch is set in configuration v i and v i,j = else. For example, the two possible configuration vectors for a switch are v = ( ) and v = ( ). Every VOQ is associated with a traffic stream, characterized by a Target Stream Profile (TSP). A TSP is a sequence of s and s which specifies the inter-packet deadline (IPD) constraints between packets in the stream. Let s = (s, s,...) denote the TSP for a typical traffic stream. Suppose that the k th in s occurs at location τ and the (k +) st occurs at location τ + δ k. The interpretation is that the IPD between the k th and (k + ) st packets of the stream is δ k time-slots. Equivalently, the ideal inter-departure time (from the switch) between the k th and (k+) st packets of the stream is δ k timeslots. From the TSP we derive the Cumulative Target Stream t Profile (ctsp), denoted S = (S, S,...), where S t τ= is the number of packets of the stream which should ideally have departed the switch by the end of the t th time-slot. For We will use configuration and configuration vector interchangeably. s τ VOQs Fig.. Switch fabric Input ports STC Input Queued Switch Output ports example, for periodic traffic with fixed IPDs, i.e., δ k = δ k, we have S t = t/δ. To describe the service provided by the switch, we associate with every stream a Received Service Trace (RST), also a sequence of s and s. Let r = (r, r,...) denote the RST associated with a typical stream. Then, r τ = if the switch serves a packet from the stream in the τ th time-slot, and r τ = else. Similar to the ctsp, we derive the Cumulative Received Service Trace (crst), denoted R (R, R,...), t where R t = r τ is the number of packets of the stream τ= which departed the switch by the end of t th time-slot. Ideally, for every stream we desire R t = S t t, which implies that every stream traverses the switch without experiencing any distortion of its target profile. However, this goal is not always realizable due to congestion caused by contention between competing streams for the shared switch fabric. If for a particular stream R t > S t, the stream has received more service than it requires to satisfy its IPD constraints and is said to be leading at time t. If R t < S t, the stream has received less than its desired amount of service and is said to be lagging at time t (see left side of Fig. ). To quantify distortion of target profiles due to congestion, we track for every traffic stream its deviation d t R t S t, which quantifies the excess or deficiency in service catered to the stream as a function of time. A negative deviation (lag) indicates missed packet deadlines and is undesirable from a QoS perspective. On the other hand, a positive deviation (lead) is undesirable because it can cause buffer overflows at downstream switches and the end user, and can also lead to starvation of delay tolerant flows traversing the switch. B. Dynamic Programming (DP) formulation Consider a finite horizon of T > time-slots indexed by t {,...,T }. Let s i = (s i,...,st i ) and S i = (Si,..., ST i ) denote the the first T entries of the TSP and ctsp of Q i, respectively. Define x t (s t,...,s t ) and X t (S t,..., St )}. Thus, x t (X t ) is a vector of the t th

4 4 Lag Lead crst ctsp Cost Lag Cost Lead Cost Time Deviation Fig.. The left side illustrates the notions of ctsp, crst and deviation. The right side depicts a typical deviation cost function. entries of the TSP (ctsp) of each of the streams. ext, define d t (d t,...,dt ) as the state of the switch in the t th time-slot, where d t i is the deviation of Q i in the t th time-slot. Since deviations from target profiles are undesirable, they are associated with a cost. In particular, to the i th stream we assign the cost function φ i (k), which reflects the cost associated with a deviation k Z. We assume that φ i () = (no deviation is desirable), φ i (k) is non-negative and increasing for both k > and k < (since both leads and lags are undesirable), and φ i (k) is convex. A typical cost function is depicted on the right side of Fig.. An example of a cost function which we will often use in this paper is the quadratic cost function φ i (k) = k. Finally, let Φ(d t ) φ i (d t i) denote the sum of deviation cost of all VOQs. Remarks: It is important to note that unlike the interpacket deadline constraints, the cost functions are not an inherent part of the problem (i.e., they are not explicitly specified by the transmitter/receiver), but are instead extraneously assigned by the switch for the purpose of service trace control. Thus, the service trace controller has the freedom to tune these cost functions in order to optimize switch performance. Remarks: In our modeling framework, packets can be thought of as being associated with soft deadlines. For the more conventional case of strict deadlines, the value of a packet is constant prior to its deadline and zero thereafter. As a result, a packet is dropped if it has not departed the queue before its due date. In our context, where the motivating application is multimedia streaming, on-time and lossless delivery of packets is sought. The value of a packet reaches its peak at its target delivery time (as dictated by the target stream profile). The packet is treated as less valuable (but not dropped) if received either before or after its target time. In this sense, the deadline constraints, and their softness is quantified by the steepness of the cost functions. In every time-slot, the STC controls the evolution of service traces for various traffic streams by setting the switch in one of! possible configurations (chosen from the set V) or idling the switch. We define an admissible policy Π T = {v t V {}, t =,...,T } as a sequence of configurations selected by the STC in time-slots t =,...,T. Given the initial state d, we are interested in computing the optimal (minimum cost) finite horizon policy Π T which satisfies { T } Π T = argmin Φ(d t Π T ), () Π T t= i= where d t Π T denotes the state of the switch at the beginning of the t th time-slot under policy Π T. We will adopt the methodology of dynamic programming to compute Π T. Suppose Π T chooses configuration v t = v j in the t th timeslot. The deviation of the Q i increases by at the end of the t th time-slot if it is served by configuration v j, i.e., v j,i =. Also, the deviation of the Q i decreases by if its TSP has a nonzero entry in the t th location, i.e., x t i =. More compactly, the new deviation vector at the beginning of the (t+) st time-slot is given by d t+ Π T = d t Π T + v t x t. Let V t (d) be the cost incurred by Π T over time-slots t,...,t, starting in state d at the beginning of the t th timeslot. In DP terminology, V t ( ) is referred to the as the cost-togo function, and is recursively computed from the following DP equations for t =,...,T V t (d) = min v V {} V t+ (d + v x t ) }{{} Cost-to-go in the next time-slot + Φ(d + v x t ) }{{}, Instantaneous cost () and the boundary conditions V T+ (d) = d. We will henceforth refer to d x t as the deviation vector. C. Myopic/Greedy service trace control Observe from () that the optimal decision in state d in the t th time-slot is determined by the cost-to-go in the (t + ) st time-slot, as well as the instantaneous cost. ow consider the myopic policy, which is greedy with respect to the instantaneous cost. In particular, the myopic policy chooses configuration v in the t th time-slot in state d such that v { = argmin Φ(d + v x t ) }. (3) v V {} There is no reason for the myopic policy to be optimal in general. However, numerical results strongly suggest that the myopic policy is very close to optimal. In fact, for =, the myopic policy is provably optimal! Theorem : The optimal finite horizon policy Π T for a switch ( = ) is myopic. Proof: See Appendix VIII-A. Quadratic cost functions: For concreteness, let us assign quadratic cost functions to all traffic streams, i.e., φ i (k) = k i. For any v V {} we have Φ(d + v x t ) = d x t,d x t + v,v + d x t,v. }{{}}{{} Policy independent Policy dependent : ς(v) (4)

5 5 The myopic policy in this case reduces to { v ṽ ; d x = t,ṽ + ; else, where ṽ { arg min d x t,v }. v V (5) due to unavailability of packets in advance. However, MSL(l) is pertinent in a scenario where the switch resides at the egress of a multimedia server, where all traffic streams are pre-cached at the input of the switch. In this case, the switch can furnish a lead of up to l to provide a cushion against possible congestion in the downstream network. D. Partial configurations and the MSL policy So far we have assumed that the STC either selects a complete configuration (every input port is connected to an output port) or idles the switch. However, operating the switch using complete configurations only is not sufficient to exercise individual control on service traces of different streams. Example: Consider a switch where the streams for Q and Q 4 are periodic with periods and 4 respectively, and Q and Q 3 are idle. The policy in (5) either selects v = ( ) or in every time-slot. It is easily verified that the lag of Q and lead of Q 4 grow without bound. The sum of their deviations, however, never exceeds in magnitude. To exercise individual control over service traces, we allow the STC to use partial configurations. Suppose configuration v V serves VOQs indexed by the set I = {i,...,i }. Any partial configuration v extracted from v is characterized by an -length vector ξ = (ξ,...,ξ ), where ξ j = if v serves Q ij and ξ j = if v idles Q ij. Thus, partial configurations can be extracted from any complete configuration. For example, the partial configuration set {( ), ( ),,v} can be extracted from v = ( ), for a switch. Quadratic cost functions: Let us revisit the myopic service trace control policy for the case of quadratic cost functions, allowing for partial configurations this time. Recall from (4) that the policy dependent part in Φ is ς(v) = v,v + d x t,v. If v serves VOQs indexed by set I, ς(v) can be rewritten as vj + v j (d j x t j). Since v is a complete j I j I configuration, v j = j I. ow, split I into two disjoint subsets, I + and I, where I + = {j I : d j x t j } and I = {j I : d j x t j < }. Clearly, ς(v) can be strictly decreased by setting v j = j I +. Doing so is equivalent to extracting a partial configuration from v by idling all VOQs with non-negative deviation. We therefore get the following two-step STC policy, which we refer to as the Maximum Sum of Lags (MSL) policy. ) Select ṽ { = argmin d x t,v }. v V ) Extract a partial configuration from ṽ by idling all VOQs with non-negative deviation. The computational complexity of MSL is O( 3 ) per timeslot, since Step involves an MWM operation. Remarks: Step of MSL can be generalized to construct a broader class of policies, namely MSL(l), indexed by l {}. Under MSL(l), a VOQ served by the chosen complete configuration is idled only when its deviation is l or more. By this token, MSL MSL(). ote that it is not feasible to implement MSL(l) for arbitrary l, since the switch may not be able to provide a lead of l even in the absence of congestion, Key properties of some service trace control policies proposed in the paper are collected in Table I. III. SUBSET BASED SERVICE TRACE COTROL To address the issue of high computational complexity associated with optimal service trace control, we propose the novel approach of subset based control in this section. The key idea is to partition the configuration set V of size! into smaller subsets of size each and operate the switch using configurations from only one of these subsets in any time-slot. A. Subset construction It follows from the crossbar constraints that all configuration vectors are necessarily of the form v = [e π() e π()...e π() ], where π is a permutation of {,,..., }. ow define the circular shift operator C by C(v) [e π() e π() e π()...e π( ) ]. (6) Recursively define C k (v) C(C k (v)), k, which corresponds to applying the circular shift operator k times to v. By convention, C (v) = v. Also, note that C k (v) = C k mod (v). Thus, starting with any configuration vector v V, we can generate a set of distinct configuration vectors by applying C to v times. We say that v generates the configuration subset S v = {v, C(v),..., C (v)} V and refer to v as the generator vector. Following the outlined procedure, we can partition V into ( )! disjoint configuration subsets of size each. Configuration subsets for a 3 3 switch are depicted in Fig. 3. For any v V, v, C k (v) = k. Physically, this implies that no VOQ is served by more than one configuration in a subset. Geometrically, this mean that all configuration vectors within a subset are orthogonal to each other. We therefore say that the generated subsets are orthogonal. Also, note that every VOQ is served by some configuration within a subset. Consequently, we say that every subset is complete. Combining the orthogonality and completeness properties we see that every VOQ is associated with exactly one configuration vector in every subset, implying j= C j (v) = v V. (7) B. Single subset operation the MSL-SS policy Let S v = {C i (v)} i= be the configuration subset generated by v. ow consider operating the switch such that the STC is allowed to choose configurations from S v alone, rather than from V. In particular, consider a restriction of the MSL policy to the configuration subset S v. ow, in Step, the STC selects configuration C i (v) S v such that i = arg min i=,..., { d x t, C i (v) }. (8)

6 6 v C(v ) C (v ) v C(v ) C (v ) Fig. 3. Two configuration subsets (of size 3 each) for a 3 3 switch. The three leftmost configurations are generated by v = [e e e 3 ] and the three rightmost configurations are generated by v = [e e 3 e ]. In Step, the STC extracts a partial configuration from C i (v) by idling all VOQs with non-negative deviation. We name this two-step policy the Maximum Sum of Lags-Single Subset (MSL-SS) policy. The per time-slot computational complexity for MSL-SS is O( ), in contrast to O( 3 ) for MSL. Two crucial questions arise at this point: ) What is the performance loss (if any) incurred by operating the switch using only one configuration subset? ) Can we compensate for the loss (if needed), without sacrificing the advantage of low complexity? We will address these questions in the remainder of the paper. IV. META-QUEUE BASED SERVICE TRACE COTROL In this section, we study subset based service trace control in a broader framework, based on the notion of meta-queues. We will recover the MSL-SS policy proposed in Section III-B as a special case of the meta-queue framework. A. Meta-queue construction Setting the switch in configuration v = [e π()...e π() ] is equivalent to serving VOQs indexed by the set I = {(i ) + π(i), i =,...,}. Thus, every complete configuration serves VOQs concurrently, which we group together to form a meta-queue. Let us focus on a single subset, say S v. Since S v is orthogonal and complete, each configuration vector in S v can be associated with a unique meta-queue, constructed by grouping distinct VOQs. ote that all VOQs are assigned to a group, each one exactly once. The HOL meta-packet of a meta-queue is constructed by grouping the HOL packets of its constituent VOQs. Subsequent meta-packets in every meta-queue are similarly constructed. With this construction, choosing a switch configuration is equivalent to serving the HOL meta-packet of the corresponding meta-queue. While grouping concurrently served VOQs to form a metaqueue seems quite natural, the relation between the deviation of a meta-queue and the deviations of its constituent VOQs is not immediately evident. In fact, we have the freedom to choose a mapping Γ : Z Z, which relates the deviation of a meta-queue to the deviation of its constituent VOQs. Given a mapping Γ, the problem of subset based control of an IQ switch (using a single subset of size ) turns into a problem of scheduling parallel meta-queues on a single server. The latter is an important and interesting scheduling problem in its own right. We now briefly digress from the service trace control problem for the IQ switch to study the single server scheduling problem. Subsequently, we will show that by appropriately choosing Γ, one can construct good, low complexity service trace control policies for the switch. B. The single server scheduling problem The formulation is similar in spirit to the formulation for an IQ switch (Section II-B), and so is the notation. Consider a system comprised of + parallel meta-queues and a single server. The i th meta-queue is denoted M i, i =,...,. In every time-slot the scheduler serves the HOL meta-packet of one of the meta-queues, chosen according to some scheduling policy. While M,...,M are physical meta-queues, M is a dummy meta-queue, scheduling which is tantamount to idling the server. Each meta-queue is associated with a traffic stream characterized by a TSP. The TSP associated with M has all zero entries. We denote by d t = ( d t,..., d t ) the deviation vector for the system in the t th time-slot, where d t i is the deviation for M i. Define x t ( s t,..., s t X ) and t ( S, t..., S t ), where st i and S i t are respectively the tth elements of the TSP and ctsp of M i. To M i we assign the cost function ψ i (k), which measures the cost of deviation k Z. We assume that ψ i (k) is non-negative, convex, and increasing for both k > and k <, and ψ (k) = k. Finally, let Ψ(d t ) ψ i (d t i) denote the sum of deviation i= costs of all meta-queues. We confine our attention to a finite horizon of T time-slots. At the beginning of every time-slot, the scheduler selects one of the + meta-queues for service. The configuration vector corresponding to scheduling M i is e i. An admissible policy Π T for the single server scheduling problem is a sequence of scheduling decisions {i t } T t=, corresponding to scheduling meta-queue M it in the t th time-slot. Let dt ΠT denote the deviation vector at the beginning of the t th time-slot under scheduling policy Π T. Our goal is to compute the optimal finite horizon policy Π T which satisfies { T } Π T = arg min Ψ(d t Π T ). (9) Π T t= We specify the state of the system at the end of the t th timeslot by n t = (n t,...,n t ), where nt i is the number of times M i has been served within the first t time-slots. Since the

7 7 server is allowed to idle, n t, t t. The system state and deviation vector are uniquely related by d t+ = d +n t X t. If the state at the beginning of the t th time-slot is n and the scheduler chooses M i in the t th time-slot, the new state at the beginning of the (t + ) st time-slot is n + e i. Letting Ṽ t (n) denote the cost-to-go at the beginning of the t th time-slot in state n, we have the following DP equations for t =,...,T Ṽ t (n) = min t+ (n + e i ) + Ψ( n + e i=,..., Ṽ }{{} i X }{{} t ), ew state ew deviation vector () and the boundary conditions Ṽ T+ (n) = n. For notational convenience, define Ω t (n) Ṽ t (n)+ψ(n X t ). Also, define the pairwise decision functions γ t ij (n) Ωt+ (n + e i ) Ω t+ (n + e j ), i j. () It follows that Π T prefers M i over M j in the t th time-slot in state n if γ t ij (n), and prefers M j else. The pairwise decision functions satisfy the following: Lemma (Monotonicity of γ): γij t (n) is a non-decreasing function of n i and a non-increasing function of n j for i, j {,...,}, i j and t =,...,T. Proof: See Appendix VIII-B. Lemma implies that any two-dimensional subspace of the -dimensional state-space is partitioned into + connected decision regions by Π T. The states in the ith decision region are those in which Π T schedules M i in the t th time-slot. Unfortunately, this structural insight does not immediately yield a low complexity approximation of the optimal policy. The myopic policy: The complexity of computing Π T by solving the DP equations increases exponentially in both T and. However, the per time-slot complexity of the myopic policy associated with the problem is only O(). The myopic policy schedules M i in the t th time-slot in state n such that i = argmin{ψ(n + e j X t )}. () j=,..., umerical results suggest that once again the myopic policy is near-optimal. In fact, the myopic policy is provably optimal for = (proof similar to Theorem ). The degree of suboptimality of the myopic policy can be quantified in this case. Doing so, however, is beyond the scope of this paper. Quadratic cost functions: Consider the case of quadratic cost functions for concreteness. It is easy to show that the myopic policy in this case selects M i such that { i ĩ ; ( dĩ x = t ) + < ĩ ; else, (3) where ĩ argmin{ d j x t j }. We call the policy in (3) j=,..., Largest Lag First (LLF) because it chooses the meta-queue with the most negative deviation (or largest lag). LLF idles the server if the deviations of all meta-queues are non-negative. C. Meta-queue based service trace control Having studied the salient features of the single server scheduling problem, we revert our attention to the service trace control problem for the IQ switch. Suppose that the STC chooses a switch configuration from subset S v alone, and VOQ deviations are mapped to meta-queue deviations through Γ. Let I i = {i,...,i } denote the set of VOQs which are served by the i th configuration in S v, namely C i (v). Given deviation vector d for the switch, we denote the deviation of the i th meta-queue by Γ(d; I i ). The LLF policy then schedules the j th meta-queue such that j = arg min i=,..., {Γ(d; I i )}. Once a meta-queue is chosen, a partial configuration is extracted by idling VOQs with non-negative deviation. We now examine two special choices of Γ. ) Γ(d; I) = j I d j : With this choice of Γ, we recover the MSL-SS policy. Thus, the service trace control problem for an IQ switch with single subset operation (Section III-B) is equivalent to the single server scheduling problem if all cost functions (both for the VOQs and the meta-queues) are quadratic and the deviation of a meta-queue is defined as the sum of the deviations of its constituent VOQs. ) Γ(d; I) = min {d j}: With this choice of Γ, the STC j I in effect selects the VOQ with the largest lag. Since every VOQ is associated with a unique configuration in S v, selecting a VOQ immediately specifies the configuration. We call this policy the Largest Lag First-Single Subset (LLF-SS) policy. The per time-slot complexity of LLF-SS is O( ), since it involves computing the maximum of an unsorted list of numbers. It can be reduced to O() through parallelization or efficient data structures (for maintaining dynamic lists). Remarks: There is a very natural interpretation for the above choice of Γ. Suppose that the switch is operated using only complete configurations and has been set in configuration C i (v) τ times by the end of the t th time-slot. Denote by Si t j the t th entry of the TSP of Q ij. It follows that the deviation of the Q ij in the t th time-slot is d t i j = τ Si t j, since C i (v) serves VOQs indexed by I i = {i,..., i }. ow, define S i t max j=,..., {St i j }. If configuration C i (v) is chosen at least S i t times by the end of the t th time-slot (τ S t ), all VOQs indexed by set I i have a non-negative deviation (no lag). We therefore let S i = ( S i, S i,...) be the ctsp of the meta-queue generated by grouping VOQs indexed by set I i. It follows that the deviation of the i th meta-queue in the t th time-slot is d t i = τ max j=,..., {St i j }, i.e., d t i = min j=,..., {dt i j }. Remarks: The meta-queue construction provides a general framework for designing service trace control policies under single subset operation. While we have illustrated the idea with two specific examples here, different families of policies with varying performance tradeoffs can be constructed by appropriately selecting the mapping Γ, as well as the metaqueue selection policy (we chose LLF here). For instance, we can set Γ(d; I) = j I c j d j, where {c j } are weight parameters chosen to provide differentiated QoS to VOQs.

8 8 V. ADMISSIBLE REGIO AD SUBSET SELECTIO Consider a traffic stream with TSP s and ctsp S. The average distance between consecutive s in s can be interpreted as the average inter-packet deadline associated with the traffic stream. By definition, S t is the number of s in the TSP in the first t time-slots. We assume that the limit λ = lim exists for every traffic stream, and refer to t /λ as the average IPD for the traffic stream. For example, consider periodic traffic with period δ, i.e., a fixed IPD of δ time-slots. Clearly, S t = t/δ and λ = /δ. Larger the λ for a stream, the smaller are its IPDs (on an average), and more the service it requires from the switch to meet its target profile. Thus, λ can be thought of as the load imposed by a stream on the switch. Letting λ i denote the load imposed by the stream at the i th VOQ, we define λ (λ,..., λ ) as the load vector for the switch. We consider a special case where the IPDs for the traffic stream associated with the i th VOQ are geometrically distributed with parameter λ i (, ). Equivalently, every entry in the TSP of Q i is an independent identically distributed (i.i.d.) Bernoulli random variable with mean λ 3 i. We refer to this scenario as i.i.d. loading. Further, we refer to the scenario λ i = λ i, i.e., λ = (λ/ ) as uniform i.i.d loading. ext, we define the admissible region for the IQ switch as the set of all load vectors for which some service trace control policy guarantees finite lags to all VOQs, at all times. Alternatively, if the switch is subject to a load vector not contained in the admissible region, the lag of at least one VOQ grows without bound, regardless of the service trace control policy employed. The above discussion, coupled with the crossbar constraints, leads to the following admissible region for an IQ switch: Λ {λ : t S t λ (i )+j <, j= λ j( )+i <, j= i =,...,, λ i (, ), i =,..., }. (4) Ideally, we would like our control policies to support (ensure finite lags for all VOQs) all load vectors contained in Λ. A policy which satisfies this property will be said to have % admissibility. The MSL(l) policy possesses this property. Theorem : liminf t E[dt i] > i under the MSL(l) policy, for any admissible i.i.d. load. Proof: See Appendix VIII-C. Subset based control was proposed in Section III to combat the high complexity of MSL(l). We are now ready to answer the two questions raised at the end of Section III regarding the efficacy of subset based control. We focus on two particular subset based policies, namely, MSL-SS (Section III-B) and LLF-SS (Section IV-C). Our answer to the first question is that by restricting operation to a single subset of configurations, not 3 The theory developed here can be extended to the case where TSP entries are generated from a Markov modulated Bernoulli (MMB) process. This is done by considering a multi-step drift of the Lyapunov function, rather than the one-step drift we consider in our proofs. See, for example, [3]. all admissible load vectors in Λ can be supported. However, all uniform loads can be supported. In particular, Theorem 3: liminf t E[dt i] > i under the LLF-SS policy, for any admissible uniform i.i.d load, independent of the choice of operational subset. Proof: See Appendix VIII-D. Theorem 4: liminf t E[dt i ] > i under the MSL(l)-SS policy, for any admissible uniform i.i.d. load, independent of the choice of operational subset. Proof: See Appendix VIII-E. There are several non-uniform load vectors (even near the boundary of Λ) under which LLF-SS guarantees bounded lags to all VOQs, if the operational subset is suitably chosen. Example: Consider λ = ( ɛ,,,,, ɛ,, ɛ, ) for a 3 3 switch, for some ɛ (, /3). In this case, the first subset in Fig. 3 cannot guarantee bounded lags to all VOQs under LLF-SS, while the second subset can. It is certainly possible to construct non-uniform i.i.d. loading scenarios under which LLF-SS cannot guarantee bounded lags to all VOQs, irrespective of the choice of operational subset. Example: Consider λ = (c,,,, c/, c/,, c/, c/) where c = ɛ for some ɛ (, /). The LLF-SS policy cannot guarantee bounded lags to all VOQs, irrespective of the choice of operational subset. Similar examples can be constructed for the MSL-SS policy. A. Randomized subset selection Service trace control based on a single subset is not good enough to support all admissible loads. However, subset operation in conjunction with an appropriate subset selection policy can achieve the desired objective. We propose one such randomized subset selection policy in this section. To this end, denote the k th configuration vector by v k and the corresponding generated subset by S k = {C i (v k )} i=. Consider the BV decomposition of λ given by λ = ( )! i= ζ ik C i (v k ), ( )! i= ζ ik = ζ <. (5) Define a probability distribution on the configuration subsets by θ k ζ ik. ow, consider the following two-step service trace control policy, namely Maximum Sum of Lags(l)- ζ i= Random Subset (MSL-RS): ) Select configuration subset S k with probability θ k. ) Select a configuration from S k based on MSL(l)-SS. The computational complexity of MSL(l)-RS is O( ) per time-slot, since MSL(l)-SS has complexity O( ) and the BV decomposition of λ contains at most + nonzero terms []. The LLF(l)-RS rule policy is constructed analogously, by combining randomized subset selection with LLF(l)-SS. Theorem 5: liminf t E[dt i] > i under MSL(l)-RS, for any admissible i.i.d. load. Proof: See Appendix VIII-F.

9 9 Policy Brief description Complexity % admissibility eed to know λ MSL Pick configuration (from V) with maximum sum of VOQ lags O( 3 ) MSL-SS Pick configuration (from single subset) with maximum sum of VOQ lags O( ) LLF-SS Pick configuration (from single subset) with most lagged VOQ O( ) MSL-RS Randomized subset selection + MSL-SS O( ) LLF-RS Randomized subset selection + LLF-SS O( ) MSL-pSEL(P) Periodic subset selection (with period P ) + MSL-SS O( ) LLF-pSEL(P) Periodic subset selection (with period P ) + LLF-SS O( ) TABLE I KEY PROPERTIES OF SOME SERVICE TRACE COTROL POLICIES PROPOSED I THE PAPER. B. Periodic subset selection MSL(l)-RS can support all admissible loads and has low computational complexity. However, the policy requires knowledge of the load vector λ. We, on the other hand, are interested in designing robust control policies which do not rely on statistical knowledge about the input traffic streams. Thus, to eliminate dependence on λ, we propose the following periodic subset selection policy: Suppose the switch is being operated using subset S v. Every P > time-slots, select a complete configuration vector v based on Step of the MSL policy. If v S v, continue operating the switch with configuration subset S v, else start operating the switch in the configuration subset generated by v. We refer to this subset selection rule as psel(p) (Periodic Selection with period P ). We now combine psel(p) with MSL(l)-SS to propose the MSL-pSEL(P) service trace control policy. Every P timeslots, MSL-pSEL(P) selects a configuration subset based on the periodic psel(p) rule. In the intermediate time-slots, MSL-pSEL(P) operates the switch using the MSL(l)-SS policy. MSL-pSEL(P) has all the desired traits: a manageable complexity of O( ) (for P = O() or bigger), no dependence on load vector λ, and as the next result tells us, ability to support all load vectors in the admissible region Λ. Theorem 6: liminf t E[dt i] > i under MSL-pSEL(P), for any admissible i.i.d. load. Proof: See Appendix VIII-G. The LLF-pSEL(P) service trace control policy is defined analogously, by combining psel(p) with LLF(l)-SS. VI. PERFORMACE EVALUATIO In this section, we evaluate the performance of our proposed service trace control policies via simulations. All results presented here are for a 6 6 IQ switch. We contrast the performance of the following policies: MSL, MSL-SS, LLF- SS, MSL-pSEL(6) and LLF-pSEL(6). All policies require a two step implementation. A complete switch configuration is selected in the first step and all VOQs with non-negative deviations are idled to extract a partial configuration in the second step. Thus, no VOQ can lead under any of the policies under consideration 4. We considered the following two performance metrics: 4 We also simulated the performance of policies which allow VOQs to acquire a lead of up to l >, but did not observe any relative difference in the performance of different policies for fixed l. ) Average Deviation: Empirical mean of VOQ deviations, averaged over all 6 = 56 VOQs. ) Variance: Empirical variance of VOQ deviations, averaged over all 56 VOQs. Clearly, we want both the mean and variance of the deviations to be close to zero, for all traffic streams. A mean close to zero with large variance is not good enough, since it indicates severe instantaneous positive or negative distortion of the target profile (which averages out to zero over a long run). Quite obviously, a small variance with non-zero mean is also undesirable. Every point on the performance curves depicted here was generated by averaging over 5, packets. We report simulation results for four distinct loading scenarios. ) Uniform i.i.d. loading: In this case, all streams have geometrically distributed IPDs with parameter λ/6. The load per input port is 6 λ/6 = λ. The performance of MSL, MSL-SS, and LLF-SS is depicted in Fig. 4. Subset selection is not needed in this scenario, because single subset operation based on MSL or LLF delivers good performance. ) Diagonal i.i.d loading: In this case, all traffic streams associated with diagonal VOQs have geometrically distributed IPDs with parameter λ and off-diagonal VOQs have geometrically distributed IPDs with parameter λ < λ. We call a VOQ diagonal if it stores packets destined from the i th input port to the i th output port for i {,..., 6}, and offdiagonal else. We fixed λ and varied λ to vary the load per input port, viz. λ +5λ. Performance results are depicted in Fig. 5. For MSL-SS and LLF-SS, we selected the configuration subset generated by the configuration which concurrently serves all diagonal VOQs. Since this configuration needs to be selected frequently, especially as λ increases, single subset operation based policies perform quite well. 3) Cross i.i.d. loading: Once again, all traffic streams have geometrically distributed IPDs. However, VOQs Q i,i+ (i odd) and Q i,i (i even) are more heavily loaded (parameter λ ) than other VOQs (parameter λ < λ ). We allude to this scenario as cross loading because the configuration which serves the more heavily loaded VOQs forms a cross pattern with eight crosses. We fixed λ and varied λ to vary the load per input port, viz. λ + 5λ. For MSL-SS and LLF- SS, we used the same configuration subset as for diagonal i.i.d. loading. Since the cross pattern is not contained in this subset, the performance of single subset based policies degrades severely as λ increases. However, periodic subset selection (P = 6), in conjunction with single subset operation ensures good performance. The results are depicted in Fig. 6.

10 Average deviation.5.5 MSL.5 MSL SS LLF SS Average deviation.5.5 MSL MSL psel(6) LLF psel(6) MSL MSL SS LLF SS.5 MSL MSL psel(6) LLF psel(6) Variance 3 Variance Fig. 4. Performance of MSL, MSL-SS and LLF-SS under uniform i.i.d. loading for an 6 6 switch Fig. 6. Performance of MSL, MSL-pSEL(6) and LLF-pSEL(6) under cross i.i.d. loading for an 6 6 switch Average deviation.5.5 MSL MSL SS LLF SS Variance MSL MSL SS LLF SS Fig. 5. Performance of MSL, MSL-SS and LLF-SS under diagonal i.i.d. loading for an 6 6 switch Average deviation Variance MSL.6 MSL SS LLF SS MSL MSL SS LLF SS Fig. 7. Performance of MSL, MSL-SS and LLF-SS under uniform periodic loading for an 6 6 switch 4) Uniform periodic loading: In this case, all traffic streams have identical IPDs equal to δ. The streams are however allowed to be offset relative to each other, i.e., the TSPs of all streams are translated versions of each other. We generated the offsets randomly and varied δ to vary the load per input port, viz. 6/δ. The performance results are depicted in Fig. 7. The efficacy of single subset based operation is evident. We also studied the proposed policies under non-uniform periodic loading and Markovian loading (TSP entries generated from an MMB process). The performance was observed to be qualitatively invariant to the statistics of the input traffic streams, underlining the robustness of the proposed policies. Remarks: The simulation results strongly indicate that it is possible to render packet switches nearly transparent to deadline sensitive traffic streams by ensuring minimal distortion of their target profiles. Moreover, this can be accomplished with low complexity online algorithms. The results are particularly impressive when the switch is moderately loaded (< 6%), which is a very relevant regime, since a switch is unlikely to be saturated (loaded to capacity) with real-time traffic. For example, at 5% load, for all four loading scenarios reported here, the average deviation under all proposed policies is between and -.3, with a variance of less than.. Moreover, the proposed policies do not require any prior knowledge of the statistics of the input traffic. For instance, they do not require the streams to be periodic or be constrained by a leaky bucket, as long as the offered load is within the admissible region of the switch. Also, the O( ) complexity of the proposed policies renders them more amenable to implementation in high performance packet switches vis-àvis other schedulers proposed in the literature for multiclass periodic traffic, which have complexity O( 3 ) or O( 4 ). VII. COCLUSIO This paper examined service trace control for minimizing aggregate distortion of target profiles of deadline sensitive traffic streams traversing an IQ packet switch. The study was motivated by the desire to guarantee QoS for real-time multimedia traffic over wired packet networks like the Internet. The notion of subset based control was leveraged to design robust, low complexity, near optimal algorithms amenable to

11 implementation in high performance packet switches. This paper signals a departure from conventional scheduler design principles. A key underlying message of the work is that it is essential to adopt the proposed pull based approach to scheduling, in contrast to the traditional push based approach, in order to support deadline sensitive multimedia applications over the existing Internet infrastructure, which is inherently QoS unaware by design. A. Proof of Theorem VIII. APPEDIX Proof: For ease of exposition, we only treat the case where the switch is never idled. A switch can be set into two possible configurations, v = ( ) and v = ( ). Given initial state d, we say that state d is reachable in the t th time-slot if there is a sequence of configurations which drive the switch to state d in t timeslots. The reachable states in the t th time-slot constitute the set R t = {d t k d + kv + (t k)v X t } t k=. The states reachable in the (t + ) st time-slot given state d t k in the tth time-slot are d t k + v x t+ = d t+ k+ and dt k + v x t+ = d t+ k. Equivalently, we can identify the state in the t th timeslot by the index k, which increases by in the next time-slot if v is chosen and remains unaltered if v is chosen. Observe that d t k is a sum of two components (a) d +kv +(t k)v, the evolution of which is determined by the service trace control policy and (b) X t, the evolution of which is determined by the IPDs of the input streams. The former can be represented using a directed acyclic graph (DAG). The root of the DAG is at d. odes of the DAG at depth t correspond to the policy dependent component of the reachable states in the t th time-slot. There are t + nodes at depth t, ordered in increasing order of index k from right to left (Fig. 8). The optimal policy traverses the least cost path from the root to one of the leaves. We use C(d t k ) = i to denote that Π T chooses configuration v i in the t th time-slot in state corresponding to index k. Also, for compactness, we define Ω t (d) V t (d) + Φ(d). We need the following auxiliary results 5 to prove the optimality of the myopic policy: AR. For any state d, Φ(d+v v ) Φ(d) Φ(d) Φ(d v + v ). AR. For t =,..., T, kt {,...,t + } such that C(d t k ) = k k t and C(d t k ) = k > k t. AR3. arg min Ω t (d t k ) = k t. k=,...,t Equipped with the above results, we will show that arg min{ω t+ (d + v x t ), Ω t+ (d + v x t )} = arg min{φ(d + v x t ), Φ(d + v x t )}, (6) which implies that the myopic policy is optimal. Say the switch is in state d in the t th time-slot and C(d) =. The states reachable from d in the (t + ) st time-slot are d = d + v x t and d = d + v x t. Four cases arise, depending on whether C(d ) and C(d ) are or. 5 See [] for proofs of these results. ) C(d ) =, C(d ) = : It follows by definition that V t+ (d ) = Ω t+ (d + v x t+ ) and V t+ (d ) = Ω t+ (d +v x t+ ). However, d +v x t+ = d +v x t+ = d+v +v x t x t+. Thus, Ω t+ (d ) Ω t+ (d ) = Φ(d ) Φ(d ), implying (6). ) C(d ) =, C(d ) = : Several possibilities can arise. Since C(d ) =, the state in the (t + ) nd time-slot is d + v x t+. The next state is determined by the optimal choice of configuration in state d + v x t+ in the (t + ) nd time-slot, and so on. In general, we can construct a chain of states which the switch visits under Π T, starting in state d in the (t + ) st time-slot. The chain terminates for one of the following two reasons: ) The end of the time horizon T is reached. ) A state is reached where the optimal choice is v. For all states constituting the chain except possibly the last, the optimal configuration is v. The optimal configuration in state d in the (t + ) st time-slot is also v. Thus, we can construct a similar chain of states originating at d, which terminates for one of the two reasons cited above. The chain originating in state d comprises of the states of the type t j = d + jv + X t X t+j (j =,,...) and the chain originating in state d comprises of the states of the type t j = d + jv + X t X t+j (j =,,...). AR implies that the chain originating in state d cannot terminate before the chain originating in state d due to the reason. AR implies Φ(t j ) Φ(tj ) Φ(d ) Φ(d ) δ Φ. If both chains terminate due to reason (), we can show Ω t+ (d ) Ω t+ (d ) (T t+)δ Φ, thereby implying (6). ow, suppose the chain originating in state d terminates in the τ th time-slot (τ < T ) due to reason (). We have two further sub-cases: (i) C(t τ t ) = and (ii) C(t τ t ) =. For sub-case (i), we can show Ω t+ (d ) Ω t+ (d ) (τ t+)δ Φ, thereby implying (6). Sub-case (ii) cannot arise because we reach a contradiction by virtue of AR3. 3) C(d ) =, C(d ) = : This case violates AR and therefore cannot arise. 4) C(d ) =, C(d ) = : This case leads to a contradiction similar to the one obtained in sub-case (ii) of case () and therefore cannot arise. By considering a set of collectively exhaustive cases we have shown that (6) holds when C(d) =. Analogous arguments can be constructed for the case C(d) =. It follows that the optimal finite horizon policy for = is myopic. B. Proof of Lemma Proof: The proof is by induction. We will establish monotonicity of γ ij as a function of n i. The proof for monotonicity of γ ij as a function of n j follows similarly. ) Base Case (t = T ): By definition of γij T and our choice of boundary conditions, γij T(n + e ) γij T (n) = ψ i(n i + X i T) ψ i(n i + X i T)] + ψ i(n i X i T ), where the inequality follows from the convexity of ψ i. ) Inductive Step: ow, assume that γ t+ ij (n + e i ) (n) for some t < T and i j. We will establish that γ t+ ij

12 Root Index k Time Index k increments by if configuration v is chosen Index k remains unaltered if configuration v is chosen 3 Fig. 8. A DAG based representation of the reachable states in successive time-slots, for a switch. The solid lines represent choice of configuration v, while the dotted lines represent choice of configuration v. The inset depicts the evolution of state index k, based on the choice of configuration in a state. this assumption implies γij t (n + e i) γij t (n). By definition, γ t ij(n + e i ) = Ω t+ (n + e i ) Ω t+ (n + e i + e j ) γ t ij (n) = Ωt+ (n + e i ) Ω t+ (n + e j ). (7) Several possible cases arise, depending on the optimal decision in states n+e i, n+e i +e j, n+e i and n+e j in the (t+) st time-slot. Our inductive assumptions (monotonicity of γ t+ ij ) imply that the (n t+ i, n t+ j ) plane gets partitioned into + distinct connected decision regions by the optimal policy Π T. This greatly restricts the number of possible cases we need to consider. We will illustrate a representative case where all four states are in the interior of the decision region corresponding to M k. All cases in which one or more of states of interest occur at the boundary of two decision regions can be treated as a combination of the representative cases. From (7), Ω t+ (n + e i ) = Ω t+ (n + e i +e k ) + Ψ(n + e i X t+ ), Ω t+ (n +e i +e j ) = Ω t+ (n +e i +e j +e k ) + Ψ(n + e i + e j X t+ ), Ω t+ (n + e i ) = Ω t+ (n + e i + e k ) + Ψ(n + e i X t ), and Ω t+ (n + e j ) = Ω t+ (n + e j + e k )+Ψ(n+e j X t ). It follows that γij t (n+e i) = γ t+ ij (n+ e i +e k ) + Ψ(n + e i X t+ ) Ψ(n +e i +e j X t+ ) and γij t (n) = γt+ ij (n +e k ) + Ψ(n +e i X t ) Ψ(n +e j X t ). Also, we have from the base case, γij T (n + e i) = Ψ(n + e i X t+ ) Ψ(n + e i + e j X t+ ) and γij T (n) = Ψ(n + e i X t ) Ψ(n + e j X t ). Combining, we get γij t (n + e i) γij t (n) = γt+ ij (n + e i + e k ) γij t+ (n + e k ) }{{} by our inductive assumptions as desired. C. Proof of Theorem + γij T (n + e i) γij T }{{ (n), } by our base case Proof: Define the time and state dependent quadratic Lyapunov function L t (d t ) d t,d t in state d t in the t th time-slot. Given d t = d, let v = argmin{ d x t,v }. v V MSL(l) extracts a partial configuration v from v by idling all VOQs with deviation l or more. Given that MSL(l) selects partial configuration v in the t th time-slot in state d, the deviation vector in the (t + ) st time-slot is d t+ = d t x t + v. ow define the conditional one step expected drift of the Lyapunov function by δ t L(d) E [ L t+ (d t+ ) L t (d t ) d t = d ]. (8) It follows by definition that L t+ (d t+ ) = L t (d t ) + d t, v x t + v x t, v x t. (9) Conditioning on {d t = d} and taking expectation on both sides of (9) we get δ t L(d) = d, v λ, v d, λ + λ, + v, v. () We now make the following observations: ) v, v and λ, v. ) From the definition of MSL(l), d x t,v d x t,v v V. Since V =!,! d x t,v d x t, v. It is easily verified that v = ( )!, v V v V implying d x t,v d xt, d,. 3) Consider the BV decomposition of λ given by! λ = α k v k,! α k = α <. () Combining () with the definition of MSL(l) we get! d, λ d x t, λ = α k d x t,v k! α k d x t,v = α d x t,v.!! 4) λ, = α k v k, = α k v k, = α. () 5) Since all VOQs which are idled under MSL(l) have nonnegative updated deviation, d x t, v < d x t,v. 6) x t, v. Combining )-6) and substituting in () we obtain δl(d) t ( α) d, + (3 + α). (3) } {{}}{{} B ɛ

13 3 Taking expectations on both sides of (3), using the law of iterated expectations, summing up both sides for τ =,...,T, assuming d = and using L we get T E[d τ ], BT T ɛ. (4) τ= Dividing both sides of (4) by T and taking limsup T lim sup T T T E[ d τ i ] B ɛ. (5) τ= i= Since the deviation of any VOQ under MSL(l) is upper bounded by l, we get from (5) j lim sup T T T τ= E[ d t j] B ɛ + ( )l <, (6) implying lim inf t E[dt j ] > j, as desired. D. Proof of Theorem 3 Proof: It can be shown using Lyapunov methods that LLF guarantees finite lags to all meta-queues in the single server model of Section IV-B if the inverse of the average IPDs of all meta-queues sum to less than. We will use this result, but skip its proof in the interest of space. Bounded lags for all meta-queues in the single server model imply bounded lags for all VOQs in the switch, since the lag of a metaqueue is the maximum of the lag of its constituent VOQs (see Section IV-C). Under uniform i.i.d. loading, i.e., λ = (λ/ ) for some λ <, the average IPD for every VOQ and hence for every meta-queue in every configuration subset is /λ. Consequently, the sum of the inverse of the average metaqueue IPDs sum to λ <, implying the desired result. E. Proof of Theorem 4 Proof: Consider L t (d t ) and δl t (d), as defined in the proof of Theorem. Assume that the switch operates in configuration subset S v = {C k (v)} k=. Let k = argmin d x t, C k (v). (7) k=,..., MSL(l)-SS selects a partial configuration v which is extracted from the complete configuration C k (v). It follows that δ t L(d) = λ d, λ + d, v + ( + λ), where λ (, ) is such that λ = λ. Using observations similar to those in the proof of Theorem in conjunction with the special property of configuration subsets given by (7) we can establish δl(d) t ( λ) d, + (3 + λ). (8) } {{}}{{} B ɛ The desired result now follows from arguments similar to those presented in the proof of Theorem. F. Proof of Theorem 5 Proof: Define δl t (d, k) as the expected drift in the Lyapunov function L t ( ) conditioned on the deviation vector d and the choice of configuration subset S k in the t th timeslot. If partial configuration v (k) derived from configuration C i (k) (v k ) S k is chosen in the t th time-slot, δl t (d, k) is given by (), with v replaced by v (k). Unconditioning with respect to the choice of configuration subset S k yields δ t L(d) = ( )! θ k δ t L(d, k) ( )! = d, λ θ k C i (k) (v k ), λ }{{} ( )! + θ k d, C i (k) (v k ) + λ, + } {{ } W d, λ + W + ( + α). (9) We now make the following observations: ) For every k, it follows from the definition of MSL(l)-RS ) d x t, C i (k) (v k ) d x t, C i (v k ) i. (3) We invoke (7) to get d x t, C i (k) (v k ) d,. ( )! θ k C i (k) (v k ),x t. 3) From ) and ) W d, +. 4) From (5) and the definition of MSL(l)-RS ( )! d, λ ζ θ k d x t, C i (k) (v k ) ζw. (3) Combining our observations and substituting in (9) we get δl(d) t ( ζ) d, + (3 + ζ), (3) } {{}}{{} B ɛ The desired result now follows from arguments similar to those presented in the proof of Theorem. G. Proof of Theorem 6 Proof: Suppose MSL(l) is used in the t th time-slot, followed by MSL-SS(l) in the (t + ) st,...,(t + P ) th time-slots, and MSL(l) again in the (t + P) th time-slot. We are interested in computing the P + step conditional expected drift in the Lyapunov function L t (d t ) of Theorem, given by δ t,p L E[Lt+P+ (d t+p+ ) L t (d t ) d t = d] P = E[L t+p+ (d t+p+ ) L t+p (d t+p ) d t (33) = d]. }{{} p= µ p The key idea is to bound each term µ p individually and then obtain a bound on their sum of the form (3). For ease of

14 4 exposition, we illustrate the case P =. The proof extends in straightforward fashion to P > (with more algebra). Suppose (partial) configuration v τ is selected in the τth time-slot 6. From Theorem, we have µ d, v t d, λ + ( + α). (34) From the definition of L and d t+ = d t + v t xt, it follows µ d, v t+ d, λ + 3( + α). (35) By definition, MSL(l)-SS is used in the (t + ) st timeslot on the subset generated by (the complete configuration corresponding to) v t. This can be used to show d, v t+ v t < α. It follows that µ d, v t d, λ α. (36) ext, since MSL(l) is used in the (t + ) nd slot, it follows µ ( α) d, α, (37) Finally, from Theorem, d, v t d, λ ( α) d, +. Combining the above inequalities, we get L 6 ( α) d, α. (38) } }{{}{{} B ɛ δ t, The desired result now follows from arguments similar to those presented in the proof of Theorem. A similar bound can be (P + )( α) established for any P >, with ɛ = and a constant B which is an increasing function of P. REFERECES [] 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. etworking, vol., pp , Jun [] 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. etworking, vol., pp. 37-5, Apr [3]. McKeown, A. Mekkittikul, V. Anantharam, and J. Walrand, Achieving % throughput in an input-queued switch, IEEE Trans. Commun., vol. 47, no. 8, pp. 6-67, Aug [4] P. Giaccone, B. Prabhakar, and D. Shah, Randomized scheduling algorithms for high-aggregate bandwidth switches, IEEE JSAC, vol., no. 4, pp , May 3. [5] D. Shah and M. Kopikare, Delay bounds for approximate maximum weight matching algorithms for input queued switches, Proc. IEEE IFOCOM, pp. 4-3, ew York, Y, Jun.. [6] K. Ross and. Bambos, Dynamic QoS control in packet switch scheduling, Proc. IEEE ICC, Seoul, S. Korea, May 5. [7] T. Inukai, An efficient SS/TDMA time slot assignment algorithm, IEEE Trans. Commun., vol. 7, no., pp , Oct [8] I.R. Philip and J.W.S Liu, SS/TDMA scheduling of real-time periodic messages, Proc. ICTS, pp. 44-5, Mar [9] J. Giles and B. Hajek, Scheduling multirate periodic traffic in a packet switch, Proc. CISS, Baltimore, MD, Mar [] I.A. Rai and M. Alanyali, Uniform weighted round robin scheduling algorithms for input queued switches, Proc. IEEE ICC, pp. 8-3, Helsinki, Finland, Jun.. [] S. Li and. Ansari, Input-queued switching with QoS guarantees, Proc. IEEE IFOCOM, pp. 5-59, ew York, Y, Mar So far, we have been suppressing the time dependence of v in all the proofs, since we were only considering one-step drifts. [] C.S. Chang, W.J. Chen, and H.Y. Huang, Birkhoff-von eumann input-buffered crossbar switches for guaranteed-rate services, IEEE Trans. Commun., vol. 49, no. 7, pp , Jul.. [3] C.S. Chang, D.S. Lee, and C.Y Yue, Providing guaranteed rate services in the load balanced Birkhoff-von eumann switches, Proc. IEEE IFOCOM, pp. 6-63, San Francisco, CA, Apr. 3. [4] C.S. Chang, D.S. Lee, and Y.S. Jou, Load balanced Birkhoff-von eumann switches: part I: one-stage buffering, Comp. Commun., vol. 5, no. 6, pp. 6-6, Apr.. [5] I. Keslassy, M. Kodialam, T.V. Lakshman, and D. Stiliadis, On guaranteed smooth scheduling for input-queued switches, Proc. IEEE IFOCOM, pp , San Francisco, CA, Apr. 3. [6] E.D. Jensen, C.D. Locke, and H. Tokuda, A time-driven scheduling model for real-time systems, Proc. IEEE RTSS, pp. -, Dec [7] R.L. Cruz, Quality of service guarantees in virtual circuit switched networks, IEEE JSAC, vol. 3, no. 6, pp , Aug [8] T.H. Cormen, C.E. Leiserson, R.L. Rivest, and C. Stein, Introduction to Algorithms, nd Ed., MIT Press and McGraw Hill, 99. [9] A. Dua and. Bambos, Low-jitter scheduling algorithms for deadlineaware packet switches, Proc. IEEE Globecom, San Francisco, CA, Dec. 6. [] A. Dua and. Bambos, Scheduling with soft deadlines for input queued packet switches, Proc. 44 th Allerton Conference on Commun., Comp., and Contr., Allerton, IL, Sep. 6. [] A. Dua and. Bambos, Deadline aware scheduling for input queued packet switches, etlab Technical Report, SU-etlab-5-6/, Stanford University. [] D. Bertsekas, Dynamic Programming and Optimal Control, vol. &, nd Ed., Athena Scientific,. [3] M.J. eely, E. Modiano, and C.E. Rohrs, Dynamic power allocation and routing for time-varying wireless networks, IEEE JSAC, vol. 3, no., pp. 89-3, Jan. 5. [4] P.R. Kumar and S.P. Meyn, Stability of queuing networks and scheduling policies, IEEE Trans. on Automat. Contr., vol. 4, no., pp. 5-6, Feb Aditya Dua received his B.Tech degree and M.Tech degree in electrical engineering from the Indian Institute of Technology, Bombay, India in. He is currently a Ph.D. candidate in electrical engineering at Stanford University, California, USA, and expects to receive his degree in June 7. His current research interests encompass both theoretical and practical aspects of modeling, analysis, architecture design, and performance engineering of communication networks and computing systems. His recent focus has been on online resource scheduling problems for supporting deadline sensitive multimedia traffic over wireless and wired packet networks. In the past, he has studied physical layer design issues like multiuser detection and equalization for wireless communication systems. ick Bambos received his Ph.D. in electrical engineering and computer science from U.C. Berkeley in 989, after graduating in electrical engineering from the ational Technical University of Athens, Greece in 984. He served on the electrical engineering faculty of UCLA from 99 to 995 and joined Stanford University in 996, where he is now a professor in the electrical engineering department and the management science and engineering department. His current research interests are in performance engineering of communication networks and computing systems, including queuing and scheduling issues in wireless an wireline networks, as well as ergodic random processes, queuing theory and adaptive control of stochastic processing networks.