Upper bounds on work in system for multichannel queues

Previously derived sample path upper bounds for multi-channel work in system and work in queue are shown to be false. A new proof is given for the corresponding stochastic bounds on these quantities. STOCHASTIC UPPER BOUND; SAMPLE PATH; INEQUALITIES; CYCLIC ASSIGNMENT Recently, Daley (personal communication) has raised serious questions about the validity of (9) and (10) in Wolff (1977). (See also the Acknowledgment.) These results purport to be all-realizations (sample path) upper bounds on work in system and on work in queue for multichannel queues. These bounds are used in Wolff(1977) to derive stochastic upper bounds (orderings) on the same quantities. A number of recent papers (e.g. Wolff( 1984), Wolfson (1984)), have used the stochastic bounds to derive other results. The methods introduced in Wolff (1977) have also been used to derive similar results for other models, see e.g. Smith and Whitt (1981). Should the stochastic bounds turn out to be false or, at best, unproven, considerable subsequent work would be undermined. The purpose of this note is to clear up this matter. In Section 2, we explain why (9) and (10) in Wolff (1977) are in fact false. In Section 3, we show that, nevertheless, the stochastic bounds are true! It turns out that (9) and (10) may be replaced by an all-realizations result that holds for any fixed epoch (time). 1. The model and a device for comparing realizations As in Wolff (1977), the original system is a conventional c-channel (server) system, c > 2, fed by a single queue, where customers depart from the queue in their order of arrival (FIFO). The arrival process is arbitrary, where t,n 0, n > 1, is the nth ordered arrival epoch. The service times are i.i.d. and are independent of the arrival process. Initially, the system is empty. Received 27 August 1985; revision received 8 April 1986. * Postal address: Department of Industrial Engineering and Operations Research, University of California, Berkeley, CA 94720, USA. 547 This content downloaded from 157.55.39.208 on Fri, 14 Oct 2016 04:12:47 UTC All use subject to http://about.jstor.org/terms