Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-05T21:45:49.269Z Has data issue: false hasContentIssue false
  • English
  • Français

A formal approach to queueing processes in the steady state and their applications

Published online by Cambridge University Press:  14 July 2016

Masakiyo Miyazawa
Masakiyo Miyazawa*
Affiliation:
Science University of Tokyo
*
∗Postal address: Department of Information Sciences, Faculty of Science and Technology, Science University of Tokyo, Noda City, Chiba 278, Japan.
Get access Rights & Permissions [Opens in a new window]

Abstract

Using the theory of point processes, we give a formal treatment to queueing processes in the steady state. Based on this result, we obtain invariance relations between several quantities in G/G/c queues. As applications, the finiteness of their moments is discussed for G/G′/c queues. The basic results and notations used in this paper are contained in the author's previous paper (Miyazawa (1977)).

Keywords

FCFS MANY-SERVER QUEUESSTATIONARY INPUTINVARIANCE RELATIONSEXISTENCE OF MOMENTSSTEADY-STATE DISTRIBUTIONSPOINT PROCESSPALM MEASURE
Type
Research Papers
Information
Journal of Applied Probability , Volume 16 , Issue 2 , June 1979 , pp. 332 - 346
Copyright
Copyright © Applied Probability Trust 1979 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barlow, R. E. and Proschan, F. (1965) Mathematical Theory of Reliability. Wiley, New York.Google Scholar
Borovkov, A. A. (1972) Stochastic Processes in Queueing Theory, Nauka, Moscow. English translation (1976), Springer-Verlag, Berlin.Google Scholar
Brumelle, S. L. (1971) On the relation between customer and time averages in queues. J. Appl. Prob. 8, 508–520.Google Scholar
Brumelle, S. L. (1972) A generalization of L = ?W to moments of queue length and waiting times. Opns Res. 20, 1127–1136.Google Scholar
Franken, P. (1976a) Einige Anwendungen der Theorie zufälliger Punktprozesse in der Bedienungstheorie I. Math. Nachr. 70, 303–319.CrossRefGoogle Scholar
Franken, P. (1976b) A generalization of regenerative processes with applications to stable single server queues. Stoch. Proc. Appl. To appear.Google Scholar
Franken, P. (1976c) A new approach to investigation of stationary queueing systems. Maths Opns Res. To appear.Google Scholar
Haji, R. and Newell, G. F. (1971) A relation between stationary queue and waiting time distribution. J. Appl. Prob. 8, 617–620.Google Scholar
Kalähne, U. (1976) Existence, uniqueness and some invariance properties of stationary distributions for general single server queues. Math. Operationsforsch. Statist. 7, 557–575.CrossRefGoogle Scholar
Kiefer, J. and Wolfowitz, J. (1955) On the theory of queues with many servers. Trans. Amer Math. Soc. 80, 470–501.Google Scholar
Kiefer, J. and Wolfowitz, J. (1956) On the characteristics of the general queueing process, with applications to random walk. Ann. Math. Statist. 27, 147–161.Google Scholar
König, D., Rolski, T., Schmidt, V. and Stoyan, D. (1978) Stochastic processes with imbedded marked point processes (PMP) and their applications in queueing. Math. Operationsforsch. Statist. 9, 125–141.Google Scholar
Krakowski, M. (1973) Conservation methods in queueing theory. Rev. Française Automat. Informat. Recherche Opérat. Ser. Verte 1, 63–84.Google Scholar
Lemoine, A. J. (1974) On two stationary distributions for the stable GI/G/1 queue. J. Appl. Prob. 11, 894–852.Google Scholar
Loynes, R. M. (1962) The stability of a queue with non-independent interarrival and service time. Proc. Camb. Phil Soc. 58, 494–520.Google Scholar
Marshall, K. T. and Wolff, R. W. (1971) Customer average and time average queue lengths and waiting times. J. Appl. Prob. 8, 535–542.Google Scholar
Mecke, J. (1967) Stationäre zufällige Masse auf lokalkompakten Abelschen Gruppen, Z. Wahrscheinlichkeitsth. 9, 36–58.CrossRefGoogle Scholar
Miyazawa, M. (1976a) Stochastic order relations among GI/G/1 queues with a common traffic intensity. J. Opns Res. Japan, 19, 193–208.Google Scholar
Miyazawa, M. (1976b) Conservation laws in queueing theory and their applications to the finiteness of moments. Research Report on Information Sciences B-27, 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
Mori, M. (1974) A note on a (j, X)-process with continuous state space and its applications to the queueing theory. Research Report on Information Sciences B-2, Tokyo Institute of Technology.Google Scholar
Saks, S. (1937) Theory on the Integral, 2nd edn., trans. Young, L. C. Dover, New York.Google Scholar
Takács, L. (1963) The limiting distribution of the virtual waiting time and the queue size for a single-server queue with recurrent input and general service times. Sankya A 25, 91–100.Google Scholar