Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-23T13:21:42.862Z Has data issue: false hasContentIssue false
  • English
  • Français

The GI/G/1 queue with uniformly limited virtual waiting times; the finite dam

Published online by Cambridge University Press:  01 July 2016

Do Le Minh
Do Le Minh*
Affiliation:
Clemson University
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper studies the GI/G/1 queueing system in which no customer can stay longer than a fixed interval D. This is also a model for the dam with finite capacity, instantaneous water supply and constant release rule. Using analytical method together with the property that the queueing process ‘starts anew’ probabilistically whenever an arriving customer initiates a busy period, we obtain various transient and stationary results for the system.

Keywords

GI/G/1 QUEUELIMITED VIRTUAL WAITING TIMESFINITE DAMREGENERATIVE PROCESSESACTUAL WAITING TIMESLOST SERVICE TIMESVIRTUAL WAITING TIMESBUSY CYCLETRANSIENT RESULTSSTATIONARY RESULTS
Type
Research Article
Information
Advances in Applied Probability , Volume 12 , Issue 2 , June 1980 , pp. 501 - 516
Copyright
Copyright © Applied Probability Trust 1980 

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

1. Afanas'eva, L. G. (1965) Existence of a limit distribution in queueing systems with bounded sojourn time. Theory Prob. Appl. 10, 515–522.CrossRefGoogle Scholar
2. Afanas'eva, L. G. and Martynov, A. V. (1967) On ergodic properties of queues with constraints. Theory Prob. Appl. 12, 104–109.CrossRefGoogle Scholar
3. Brown, M. and Ross, S. M. (1972) Asymptotic properties of cumulative processes. SIAM J. Appl. Math. 22, 93–105.CrossRefGoogle Scholar
4. Cohen, J. W. (1969) The Single Server Queue. North-Holland, Amsterdam.Google Scholar
5. Cohen, J. W. (1976) On Regenerative Processes in Queueing Theory. Lecture Notes in Economics and Mathematical Systems 121, Springer-Verlag, Berlin.CrossRefGoogle Scholar
6. Crane, M. A. and Iglehart, D. L. (1974) Simulating stable stochastic systems, I: General multi-server queues. J. Assoc. Comput. Mach. 21, 103–113.CrossRefGoogle Scholar
7. Daley, D. J. (1964) Single-server queueing systems with uniformly limited queueing time. J. Austral. Math. Soc. 4, 489–505.CrossRefGoogle Scholar
8. Daley, D. J. (1965) General customer impatience in the queue GI/G/1. J. Appl. Prob. 2, 186–205.CrossRefGoogle Scholar
9. Ghosal, A. (1960) Emptiness in the finite dam. Ann. Math. Statist. 31, 803–808.CrossRefGoogle Scholar
10. Kingman, J. F. C. (1962) On the algebra of queues. J. Appl. Prob. 3, 285–326.CrossRefGoogle Scholar
11. Kovalenko, I. N. (1961) Some queueing problems with restrictions. Theory. Prob. Appl. 6, 204–208.CrossRefGoogle Scholar
12. Lemoine, A. J. (1974) Limit theorems for generalized single server queues. Adv. Appl. Prob. 6, 159–174.CrossRefGoogle Scholar
13. Minh, D. L. (1980) Analysis of the exceptional queueing system by the use of regenerative processes and analytical methods. Math. Operat. Res. 5.CrossRefGoogle Scholar
14. Pollaczek, F. (1952) Fonctions caractéristiques de certaines répartitions définies au moyen de la notion d'ordre. Application à la theorie des attentes. C.R. Acad. Sci. Paris. 234, 2334–2336.Google Scholar
15. Prabhu, N. U. (1965) Queues and Inventories. Wiley, New York.Google Scholar
16. Ross, S. M. (1970) Applied Probability Models with Optimization Applications. Holden Day, San Francisco.Google Scholar
17. Smith, W. L. (1953) On the distribution of queueing times. Proc. Camb. Phil. Soc. A 49, 449–461.CrossRefGoogle Scholar
18. Smith, W. L. (1955) Regenerative stochastic processes. Proc. R. Soc. London A 232, 6–31.Google Scholar
19. Stidham, S. Jr. (1972) Regenerative processes in the theory of queues, with applications to the alternating-priority queue. Adv. Appl. Prob. 4, 542–577.CrossRefGoogle Scholar
20. 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. Sankhyā A 25, 91–100.Google Scholar
21. Takács, L. (1970) A fundamental identity in the theory of queues. Ann. Inst. Statist. Math. 22, 339–348.CrossRefGoogle Scholar
22. Takács, L. (1972) On a linear transformation in the theory of probability. Acta. Sci. Math. (Szeged) 33, 15–24.Google Scholar
23. Takács, L. (1974) A single-server queue with limited virtual waiting time. J. Appl. Prob. 11, 612–617.CrossRefGoogle Scholar
24. Takács, L. (1976) On fluctuation problems in the theory of queues. Adv. Appl. Prob. 8, 548–583.CrossRefGoogle Scholar