496 IEEE COMMUNICATIONS LETTERS, VOL. 7, NO. 10, OCTOBER 2003 Maximum Size Matching is Unstable for Any Packet Switch Isaac Keslassy, Member, IEEE, Rui Zhang-Shen, Member, IEEE, and Nick McKeown, Senior Member, IEEE Abstract—Input-queued packet switches use a matching algo- rithm to configure a nonblocking switch fabric (e.g., a crossbar). Ideally, the matching algorithm will guarantee 100% throughput for a broad class of traffic, so long as the switch is not oversub- scribed. An intuitive choice is the maximum size matching (MSM) algorithm, which maximizes the instantaneous throughput. It was shown, by McKeown et al. in 1999, that with MSM the throughput can be less than 100% when , even with benign Bernoulli i.i.d. arrivals. In this letter, we extend this result to , and hence show it to be true for switches of any size. Index Terms—Instability, maximum size matching (MSM), switching algorithms. I. INTRODUCTION H IGH-SPEED Internet routers commonly use virtual output queueing (VOQ), a crossbar switch, and (inter- nally) fixed-size cells. Time is slotted with one cell transmission per time-slot. At each time-slot a matching algorithm finds a match between inputs and outputs ( , since there is no need for a matching algorithm when ), and cells are transferred according to this match. This letter is about switches that are unstable even though no input or output is over-subscribed. It is known that for a broad class of traffic, a switch is stable (for ) if the max- imum weight matching (MWM) algorithm is used [1], [3]. On the other hand, it is known that with the maximum size matching (MSM) algorithm, a switch can be unstable for [1] (if ties are broken randomly). 1 This is surprising because MSM max- imizes the instantaneous throughput by transferring the max- imum number of cells during each time-slot. The instability result in [1] is based on a counterexample that holds for . In this letter we extend the proof to , and hence prove that MSM is unstable for any switch. We also derive the exact throughput formula for the case. Our results are mainly of theoretical interest (it is unusual to build 2 2 switches); the letter completes existing results, extending them to switches of any size. Manuscript received March 9, 2003. The associate editor coordinating the review of this paper and approving it for publication was Prof. C. Douligeris. The authors are with the Computer Systems Laboratory, Stanford University, Stanford, CA 94305-9030 USA (e-mail: keslassy@stanford.edu; rzhang@stan- ford.edu; nickm@stanford.edu). Digital Object Identifier 10.1109/LCOMM.2003.817330 1 We assume here that MSM breaks ties randomly. In [2] it is shown that oth- erwise, MSM could be stable for . II. PROBLEM STATEMENT We will consider a packet switch with two inputs and two outputs, i.e., . Notation: Time-slot represents the interval . Let denote the VOQ at input destined to output . contains packets at the end of time-slot , with for all by convention. packets arrive at at the beginning of time-slot and packets depart from it at the end of the time-slot, with , . The service indicator is 1 if is serviced at time , and 0 otherwise. There is a departure from if it both receives a service and is nonempty. As a consequence, for , satisfies the following equation: (1) where the notation is equivalent to . Arrivals: For our counterexample, it is sufficient to assume that the arriving traffic follows a Bernoulli i.i.d. distribution with mean rate arriving to . We will consider the following type of traffic: (2) where and are positive constants. It is assumed that no input or output is oversubscribed, i.e., . Services: VOQs are serviced according to a MSM algo- rithm with ties broken randomly. Stability: A queue is said to be unstable if after a finite time, its occupancy never returns to zero with probability one. Note that with Bernoulli traffic, this is implied by the queue having a positive drift, which happens if the service rate is less than the incoming traffic rate. A switch is said to be unstable if any of its queues is unstable. III. INSTABILITY OF MSM WHEN Our approach is to find values of and such that the service rate of is less than its arrival rate. Lemma 1: At the end of a time-slot , at least one of the two queues and is empty (3) Proof: By induction. The case when is clear. As- sume that this property holds until the end of some time-slot . Consider the following two cases. 1089-7798/03$17.00 © 2003 IEEE