Hostname: page-component-77c89778f8-swr86 Total loading time: 0 Render date: 2024-07-17T01:33:40.009Z Has data issue: false hasContentIssue false
  • English
  • Français

Single-server queueing systems with uniformly limited queueing time

Published online by Cambridge University Press:  09 April 2009

D. J. Daley
D. J. Daley
Affiliation:
Department of Statistics, University of Melbourne, and Statistical Laboratory, University of Cambridge
Rights & Permissions [Opens in a new window]

Summary

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The paper considers the queueing system GI/G/1 with a type of customer impatience, namely, that the total queueing-time is uniformly limited. Using Lindiley's approach [10], an integral equation for the limiting waiting- time distribution is derived, and this is solved explicitly for M/G/1 using an expansion of the Pollaczek-Khintchine formula. It is also solved, in principle for Ej/G/l, and explicitly for Ej/Ek/l. A duality noted between GIA(x)/GB(x)/l and GIB(x)/GA(x)/l relates solutions for GI/Ek/l to Ek/G/l. Finally the equation for the busy period in GI/G/l is derived and related to the no-customer-loss distribution and dual distributions.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 4 , Issue 4 , November 1964 , pp. 489 - 505
Copyright
Copyright © Australian Mathematical Society 1964

References

[1]Barrer, D. Y., Queueing with impatient customers and indifferent clerks, Operat. Res. 5 (1957), 644–649.CrossRefGoogle Scholar
[2]Barrer, D. Y., Queueing with impatient customers and ordered service, Operat. Res. 5 (1957), 650–656.CrossRefGoogle Scholar
[3]Doob, J. L., Stochastic Processes. Wiley, New York (1953).Google Scholar
[4]Finch, P. D., A limit theorem for Markov chains with continuous state space, J. Austral. Math. Soc. 3 (1963), 351–358.CrossRefGoogle Scholar
[5]Ghosal, A., Emptiness in the finite dam, Ann. Math. Statist. 31 (1960), 803–808.CrossRefGoogle Scholar
[6]Karlin, S., Miller, R. G. and Prabhu, N. U., Note on a moving single server problem, Ann. Math. Statist. 30 (1959), 243–246.CrossRefGoogle Scholar
[7]Kendall, D. G., Some problems in the theory of queues, J. R. Statist. Soc. B 13 (1951), 151–185.Google Scholar
[8]Kovalenko, I. N., A study of the queueing system M/M/s with ordered service and limited time of staying in the system (In Russian), Ukrain. Mat. Z. 12 (1960), 471–476.Google Scholar
[9]Kovalenko, I. N., Some queueing problems with restrictions (In Russian), Teor. Veroyat. Primen. 6 (1961), 222–228.Google Scholar
[10]Lindley, D. V., The theory of queues with a single server, Proc. Camb. Phil. Soc. 48 (1952), 277–289.CrossRefGoogle Scholar
[11]Loynes, R. M., The stability of a queue with non-independent inter-arrival and service times. Proc. Camb. Phil. Soc. 58 (1962), 497–520.CrossRefGoogle Scholar
[12]McMillan, B. and Riordan, J., A Moving single server problem, Ann. Math. Statist. 28 (1957), 471–478.CrossRefGoogle Scholar
[13]Moran, P. A. P., A probability theory of dams and storage systems, Aust. J. Appl. Sci. 5 (1954), 116–124.Google Scholar
[14]Prabhu, N. U., On the integral equation for thefinite dam, Quart. J. Math. (2). 9 (1958), 183–188.CrossRefGoogle Scholar
[15]Prabhu, N. U., Application of storage theory to queues with Poisson arrivals, Ann. Math. Statist. 31 (1960), 475–482.CrossRefGoogle Scholar
[16]Saaty, T. L., Elements of Queueing Theory with Applications. McGraw-Hill, New York (1961).Google Scholar
[17]Takacs, L., Investigation of waiting-time problems by reduction to Markov processes, Acta Math. Acad. Sci. Hung. 6 (1955), 101–129.CrossRefGoogle Scholar
[18]Takacs, L., Introduction to the Theory of Queues. Oxford University Press, New York (1962).Google Scholar