Optimal Bu er Sharing

We address the problem of designing optimal bu er management policies in shared memory switches when packets already accepted in the switch can be dropped pushed out Our goal is to maximize the overall throughput or equivalently to minimize the overall loss probability in the system For a system with two output ports we prove that the optimal policy is of push out with threshold type POT The same result holds if the optimality criterion is the weighted sum of the port loss probabilities For this system we also give an approximate method for the calculation of the optimal threshold which we conjecture to be asymptotically correct For the N ported system the optimal policy is not known in general but we show that for a symmetric system equal tra c on all ports it consists of always accepting arrivals when the bu er is not full and dropping one from the longest queue to accommodate the new arrival when the bu er is full Numerical results are provided which reveal an interesting and somewhat unexpected phenomenon While the overall improvement in loss probability of the optimal POT policy over the optimal coordinate convex policy is not very signi cant the loss probability of an individual output port remains approximately constant as the load on the other port varies and the optimal POT policy is applied a property not shared by the optimal coordinate convex policy This work was done while at the IBM T J Watson Research Center And Department of Electrical Engineering Technion Haifa Israel Part of the work was done while visiting the IBM T J Watson Research Center Introduction Shared memory fast packet switches are widely used in high speed wide area networks Such switches consist of a single large memory where packets arriving from all inputs are stored while they wait before being transmitted on their respective output s While this design presents a number of technical challenges in particular memory access and speed the sharing of a single memory by all input and output ports o ers numerous advantages One of them is improved bu er e ciency which translates into smaller memory sizes to satisfy a given loss probability requirement However despite this greater e ciency losses remain unavoidable and it is therefore still of interest to understand how they can be minimized Furthermore sharing of the memory also introduces new potential problems as individual inputs can now a ect the performance seen by others In this paper we focus on identifying how to best share the memory between the system ports so that overall system throughput is maximized There has been a number of prior works which have addressed this problem In particular Kamoun and Kleinrock analyzed several sharing schemes namely Complete Sharing CS in which an arriving packet is accepted if any storage space is available Complete Partitioning CP in which the entire storage is permanently partitioned among the output ports Sharing with Maximum Queue Lengths SMXQ in which a limit on the number of bu ers allocated to each output port is imposed Sharing with a Minimum Allocation SMA in which a minimum number of bu ers is always reserved for each output port and the remaining bu ers are shared between all output ports and Sharing with a Maximum Queue and Minimum Allocation SMQMA which is a combination of the SMXQ and SMA schemes Their study assumed independent Poisson arrivals and exponential service times and they obtained closed form expressions for the probability distribution of the bu er occupancy based on the fact that it has a well known product form solution From their numerical examples they showed that sharing can improve performance especially when little storage is available but that some restrictions should be imposed to avoid throughput degradation in asymmetric systems Additional numerical results for a CS policy but with bursty arrivals further supported this conclusion by showing that some outputs can become temporarily congested and monopolize the use of the shared memory The existence and the structure of an optimal sharing policy in the sense of minimum packet loss or maximum throughput was then rst investigated by Foschini and Gopinath They considered optimality within the class of policies that never drop a packet once they admit it in the bu er and hence have coordinate convex state space if x then x x xi xm for all i N These policies referred to as coordinate convex policies include the policies of For a switch with two output ports they proved that the optimal coordinate convex policy is to limit the queue length of output port i i to some xed level mi such that m m B where B is the bu er size For more than two ports they conjectured that the optimal policy is simple see de nition in Their proofs were based on the fact that the probability distribution of the bu er occupancy has a product form solution Wei et al suggested a sharing policy which allows for the dropping of accepted packets and therefore does not belong to the class of coordinate convex policies According to this policy named Drop on Demand or DoD an arriving packet is always accepted if there is an empty bu er If a packet destined for output port i arrives and nds the bu er full and output port l has more packets in the shared memory than any other ports the following action is taken if i l the arriving packet is dropped if i l the arriving packet joins the bu er and one port l packet is dropped In general policies which can accept an arriving packet by dropping another packet from the system are known as push out policies see e g Push out policies include coordinate convex policies never push out a packet as well as the DoD policy In numerical examples were provided showing that the DoD policy yields better throughput and lower packet losses than either the CS and CP policies However as we shall show this policy is optimal only for symmetric systems In this paper we consider a model similar to the one of The bu er size is denoted by B and the arrival and service processes of type i destined to output i packets are Poisson and exponential with rates i and i respectively Upon arrival of a packet the system can decide to either accept the packet or reject it or accept it and drop another packet from the system In other words we include pushout policies and our goal is to determine the policy which maximizes the overall throughput or equivalently minimize the overall loss probability For a two ported switch we prove that the optimal policy is of push out with threshold type POT i e whenever the bu er is non full the arrival should be accepted and whenever it is full an arrival from type i i is accepted and a type i the other type packet is pushed out if the number of type i packets is below some threshold k i where k k B The same result is true if the optimality criterion is the weighted sum of the port loss probabilities For and we also show that k
B In general the determination of the threshold k is computationally intensive but for the two ported system we develop a simple and yet reasonably accurate heuristic to obtain its value The results for the two ported system establish the non optimality of DoD for asymmetric systems For the symmetric N ported system with identical arrival rates and identical transmission rates we show that the optimal policy is to accept an arrival whenever the bu er is non full or the queue corresponding to the type of the arriving packet is not the largest in the second case a packet from the longest queue is dropped This establishes the optimality of DoD for the N ported symmetric system The proofs of the results for both the two ported and N ported systems are based on the theory of Markov decision processes The behavior of the optimal policies are then investigated for the two ported case by means of numerical examples which reveal an interesting and somewhat unexpected phenomenon While the overall improvement in loss probability of the optimal POT policy over the optimal coordinate convex policy is found to be relatively minor a signi cant di erence is observed when focusing on the loss probability of an individual output port The use of the optimal POT policy results in an approximately constant loss probability on a given port as the load on the other varies In contrast signi cant variations can be observed with the optimal coordinate convex policy The insensitivity of individual losses is clearly a desirable feature but nevertheless surprising given the global nature overall throughput of our optimization For the two ported system we also investigate the heuristic method for determining the threshold of the optimal POT policy which based on the numerical results obtained is conjectured to be asymptotically correct as the bu er size B increases Numerical comparisons further show that the approximation is very good for most practical scenarios Finally we note that the structure of the Markov process arising from a POT policy permits the development of an e cient method for the computation of the state probabilities The method consists in reducing the B system of equations to the solution of a system of B equations The paper is organized as follows In section we introduce the system model and provide the formulation of the optimization problem In section we investigate the structure of the optimal policy We rst focus on the two ported system for which we prove that the optimal policy is of POT type We also derive a number of interesting properties of the optimal policy and describe the approximation we propose to compute the optimal threshold Next we consider the more general N ported system and also identify the optimal policy but only for the symmetric case arrival and service rates are the same at all ports Section is devoted to numerical comparisons between the performance of the optimal POT and coordinate convex policies We concentrate again on the two ported system for which we also study the accuracy of the proposed threshold approximation Section brie y summarizes the ndings of the paper and suggests some open problems Finally appendix A provides proofs of lemmas used in section and appendix B outlines an e cient method for computing state probabilities in a two ported system operating under the POT policy The Model and Problem Formulation The system consists of a bu er shared by packets destined to any of N output ports Packets are said to be of type i i N if they are destined to port i Type i i N packets arrive to the bu er according to a Poisson process with rate i and are transmitted by output port i with a transmission time which is exponentially distributed with rate i We assume that i i so that only a nite number of transitions can occur in any nite interval of time and that packet inter arrival and transmission times from all sources are mutually independent The total bu er size is taken to be B packets and a packet occupies its bu er until it has been completely transmitted Our goal is to determine how the B bu ers are best shared among packets of di erent types so that the overall system throughput is maximized This amounts to identifying rules that specify when and how packets of di erent types are allowed to occupy a space in the shared bu er In this paper acceptable rules include accepting or rejecting an arriving packet as well as discarding pushing out an already stored packet to accommodate an arriving one Because the state of the system can be represented by a Markov chain the rules or policy governing the sharing of the bu er can be expressed as a continuous time Markov decision process Decision epochs correspond to arrivals and departures from the system where at each epoch a decision is made as to whether a current or future packet should be accepted rejected or accepted by pushing out another packet from the system Next we proceed with a precise formulation of this process Let x n x n x n xN n be the state of the system at decision epoch n where xi n i N denotes the number of type i packets in the system at decision epoch n Let X f Bg be the state space of the system De ne the following operators to denote the rejection acceptance and push out of packets respectively upon arrival epoch Pr x x Pai x x ei x Dom Pai i N P p j i x x ei ej x Dom Ppj i i j N where ei i N denotes the vector with all components zero except the ith which is equal Dom Pai f PN j xj Bg and Dom pi fxj g Let U fu u u uN ui fr ai p j i j i Ngg be the set of possible decisions and U x fu U x i Dom Pui g be the set of all admissible actions when the system state is x The speci cation of the continuous time Markov decision process is then completed once we have de ned the length of time between successive decision epochs and the transition probability function When the system is in state x the length of time until the next decision epoch is an exponential random variable with transition rate PN i i i fxi g and the transition probability to the next state is given by Pr x n Pui x n j x n x u n u i PN i i i fxi g i N Pr x n Di x n j x n x u n u i fxi g PN i i i fxi g i N where Di x n indicates a departure of a type i packet when the system is in state x at time n As mentioned earlier the objective of the optimization is to minimize the overall loss probability or equivalently to maximize the overall throughput of the system That is denoting by n T the number of decision epochs up to time T we are interested in the policy that for any initial state x x maximizes