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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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References

Borovkov, A. A. (1972) Stochastic Processes in Queueing Theory. (in Russian) Nauka, Moskva.Google Scholar
Breiman, L. (1968) Probability. Addison-Wesley, Reading, Mass. Google Scholar
Brown, J. R. (1976) Ergodic Theory and Topological Dynamics. Academic Press, New York.Google Scholar
Geman, D. and Horowitz, J. (1973) Remarks on Palm measures. Ann. Inst. H. Poincaré 9, 215–232.Google Scholar
Grandell, J. (1976) Doubly Stochastic Poisson Processes. Lecture Notes in Mathematics 529, Springer-Verlag, Berlin.Google Scholar
Kallenberg, O. (1976) Random Measures. Academic Press, London.Google Scholar
Kingman, J. F. C. (1964) On doubly stochastic Poisson processes. Proc. Camb. Phil. Soc. 60, 923–930.Google Scholar
Kopocińska, I. (1980) The reliability of an element with alternating failure rate. Submitted to Zast. Mat. Google Scholar
König, D. and Schmidt, V. (1980) Stochastic inequalities between customer-stationary and time-stationary characteristics of queueing systems with point processes. J. Appl. Prob. 17, 768–777.Google Scholar
Loynes, R. M. (1962) The stability of a queue with non-independent inter-arrival and service times. Proc. Camb. Phil. Soc. 58, 497–520.Google Scholar
Mecke, J. (1967) Stationäre zufällige Masse auf lokalkompakten Abelschen Gruppen. Z. Wahrscheinlichkeitsth. 9, 36–58.CrossRefGoogle Scholar
Miyazawa, M. (1976) On the role of exponential distributions in queueing models. Research Report on Information Sciences No B-35, Tokyo Institute of Technology.Google Scholar
Miyazawa, M. (1977) Time and customer processes in queues with stationary inputs. J. Appl. Prob. 14, 349–357.Google Scholar
Pfanzagl, J. (1974) Convexity and conditional expectation. Ann. Prob. 2, 490–494.CrossRefGoogle Scholar
Rolski, T. (1981) Stationary Random Processes Associated with Point Processes. Lecture Notes in Statistics, Springer-Verlag, New York.Google Scholar
Ross, S. M. (1978) Average delay in queues with non-stationary Poisson arrivals. J. Appl. Prob. 15, 602–609.Google Scholar
Stidham, S. (1974) A last word on L = λ W. Operat. Res. 22, 417–422.Google Scholar
Stoyan, D. (1977) Further stochastic order relations among GI/GI/1 queues with a common traffic intensity. Math. Operationsforsch. Statist., Ser. Optimization 8, 541–548.Google Scholar
Szczotka, W. (1979) Factorization of the distribution function of waiting time. Zast. Mat. 3, 395–403.Google Scholar