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Queues with non-stationary input stream: Ross's conjecture

Published online by Cambridge University Press:  01 July 2016

Tomasz Rolski
Tomasz Rolski*
Affiliation:
Wrocław University
*
∗Postal address: Mathematics Institute, Wrocław University, pl. Grunwaldski 2/4, 50–384 Wrocław, Poland.
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Abstract

Characteristics of queues with non-stationary input streams are difficult to evaluate, therefore their bounds are of importance. First we define what we understand by the stationary delay and find out the stability conditions of single-server queues with non-stationary inputs. For this purpose we introduce the notion of an ergodically stable sequence of random variables. The theory worked out is applied to single-server queues with stationary doubly stochastic Poisson arrivals. Then the interarrival times do not form a stationary sequence (‘time stationary’ does not imply ‘customer stationary’). We show that the average customer delay in the queue is greater than in a standard M/G/1 queue with the same average input rate and service times. This result is used in examples which show that the assumption of stationarity of the input point process is non-essential.

Keywords

FIFO SINGLE-SERVER QUEUESTATIONARY DELAYDELAYERGODIC STABILITYDOUBLY STOCHASTIC POISSON PROCESSBOUND FOR DELAYNON-STATIONARY ARRIVALS
Type
Research Article
Information
Advances in Applied Probability , Volume 13 , Issue 3 , September 1981 , pp. 603 - 618
Copyright
Copyright © Applied Probability Trust 1981 

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