Ronald W. Wolff

AN UPPER BOUND FOR MULTI-CHANNEL QUEUES

Moments of the delay distribution and other measures of performance for a multi-channel queue are shown to be bounded above by corresponding

A review of regenerative processes

The authors present an expository review of the theory of regenerative processes starting with the more traditional notions and then moving on to some of the more recent and modern developments. Fi...

An upper bound on the performance of queues with returning customers

Multiple channel queues with Poisson arrivals, exponential service distributions, and finite capacity are studied. A customer who finds the system at capacity either leaves the system for ever or may return to try again after an exponentially distributed time. Steady state probabilities are approximated by assuming that the returning customers see time averages. The approximation is shown to result in an upper bound on system performance.

PREDICTING DISPATCHING DELAYS ON A LOW SPEED, SINGLE TRACK RAILROAD

This paper presents queueing models for predicting dispatching delays on single track, low speed rail lines with widely spaced passing locations. Because of scheduling unpredictabilities, we assume Poisson arrivals of trains. Because of the slow transit speeds, we assume that trains traveling in the same direction can do so on close headways. We also assume siding capacity at passing locations is not limiting. Under these assumptions, we calculate expected delays on individual segments of single track and for segments of single track with an alternate route.

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

An Extension of Erlang's Loss Formula

Abstract : A two server loss system is considered with N classes of Poisson arrivals, where the service distribution function and server preferences are arrival class dependent. The stationary state probabilities are derived and found to be independent of the form of the service distributions.

Further results on ASTA for general stationary processes and related problems

We consider the equivalence of state probabilities of a general stationary process at an arbitrary time and at embedded epochs of a given point process, which is called ASTA (Arrivals See Time Averages). By using an event-conditional intensity, we give necessary and sufficient conditions for ASTA for a large class of state sets, which determines a state distribution. For a stationary pure-jump process with a point process, ASTA is obtained for all state sets. As an application. Anti-PASTA is obtained for a pure-jump Markov process and a certain class of generalized semi-Markov processes, where Anti-PASTA means that ASTA implies that the arrival process is Poisson

SYMMETRIC QUEUES WITH BATCH DEPARTURES AND THEIR NETWORKS

Batch departures arise in various applications of queues. In particular, such models have been studied recently in connection with production systems. For the most part, however, these models assume Poisson arrivals and exponential service times; little is known about them under more general settings. We consider how their stationary queue length distributions are affected by the distributions of interarrival times, service times and departing batch sizes of customers. Since this is not an easy problem even for single departure models, we first concentrate on single-node queues with a symmetric service discipline, which is known to have nice properties. We start with pre-emptive LIFO, a special case of the symmetric service discipline, and then consider symmetric queues with Poisson arrivals. Stability conditions and stationary queue length distributions are obtained for them, and several stochastic order relations are considered. For the symmetric queues and Poisson arrivals, we also discuss their network. Stability conditions and the stationary joint queue length

Idle period approximations and bounds for the GI/G/1 queue

The average delay for the GI/G/1 queue is often approximated as a function of the first two moments of interarrival and service times. For highly irregular arrivals, however, it varies widely among queues with the same first two moments, even in moderately heavy traffic. Empirically, it decreases as the interarrival time third moment increases. For GI/M/1 queues, a heavy-traffic expression for the average delay with this property has been previously obtained. The method, however, sheds little light on why the third moment arises. We analyze the equilibrium idle-period distribution in heavy traffic using real-variable methods. For GI/M/1 queues, we derive the above heavy-traffic result and also obtain conditions under which it is either an upper or lower bound. Our approach provides an intuitive explanation for the result and also strongly suggests that similar results should hold for general service. This is supported by empirical evidence. For any given service distribution, it has been conjectured that the expected delay under pure-batch arrivals, where interarrival times are scaled Bernoulli random variables, is an upper bound on the average delay over all interarrival distributions with the same first two moments. We investigate this conjecture and show, among other things, that pure-batch arrivals have the smallest third moment. We obtain conditions under which this conjecture is true and present a counterexample where it fails. Arrivals that arise as overflows from other queues can be highly irregular. We show that interoverflow distributions in a certain class have decreasing failure rate.

A NOTE ON THE EXISTENCE OF REGENERATION TIMES

We rigorously prove that for a stochastic process, X = {X(t) : t - 0), the existence of a first regeneration time, R1, implies the existence of an infinite sequence of such times, {R1, R2, - - ), and hence that the definition of regenerative process need only demand the existence of a first regeneration time. Here we include very general processes up to and including processes where cycles are stationary but not necessarily independent and identically distributed.