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Time-dependent process of M/G/1 vacation models with exhaustive service

Part of: Computer system organization Operations research and management science Special processes

Published online by Cambridge University Press:  14 July 2016

Hideaki Takagi
Hideaki Takagi*
Affiliation:
IBM Research, Tokyo Research Laboratory
*
∗Postal address: IBM Japan, Ltd., No. 36 Kowa Building, 5–19 Sanban-cho, Chiyoda-ku, Tokyo 102, Japan.
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Abstract

Generalized M/G/1 vacation systems with exhaustive service include multiple and single vacation models and a setup time model possibly combined with an N-policy. In these models with given initial conditions, the time-dependent joint distribution of the server's state, the queue size, and the remaining vacation or service time is known (Takagi (1990)). In this paper, capitalizing on the above results, we obtain the Laplace transforms (with respect to time) for the distributions of the virtual waiting time, the unfinished work (backlog), and the depletion time. The steady-state limits of those transforms are also derived. An erroneous expression for the steady-state distribution of the depletion time in a multiple vacation model given by Keilson and Ramaswamy (1988) is corrected.

Keywords

SERVER VACATIONSN-POLICYTIME-DEPENDENT QUEUESUNFINISHED WORKDEPLETION TIME

MSC classification

Primary: 60K25: Queueing theory
Secondary: 90B22: Queues and service 68M20: Performance evaluation; queueing; scheduling
Type
Research Papers
Information
Journal of Applied Probability , Volume 29 , Issue 2 , June 1992 , pp. 418 - 429
Copyright
Copyright © Applied Probability Trust 1992 

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References

Boxma, O. J. (1989) Workloads and waiting times in single-server systems with multiple customer classes. QUESTA 5, 185–214.Google Scholar
Doshi, B. T. (1986) Queueing systems with vacations - A survey. QUESTA 1, 29–66.Google Scholar
Doshi, B. T. (1990) Single-server queues with vacations. In Stochastic Analysis of Computer and Communication Systems, ed. Takagi, H., pp. 217–265, Elsevier North-Holland, Amsterdam.Google Scholar
Keilson, J. and Ramaswamy, R. (1988) The backlog and depletion-time process for M/G/1 vacation models with exhaustive service discipline. J. Appl. Prob. 25, 404–412.CrossRefGoogle Scholar
Keilson, J. and Servi, L. D. (1987) Dynamics of the M/G/1 vacation model. Operat. Res. 35, 575–582.CrossRefGoogle Scholar
Keilson, J. and Sumita, U. (1983) The depletion time for M/G/1 systems and a related limit theorem. Adv. Appl. Prob. 15, 420–443.CrossRefGoogle Scholar
Kleinrock, L. and Scholl, M. O. (1980) Packet switching in radio channels: new conflict-free access schemes. IEEE Trans. Comm. 28, 1015–1029.CrossRefGoogle Scholar
Levy, Y. and Yechiali, U. (1975) Utilization of idle time in an M/G/1 queueing system. Management Sci. 22, 202–211.CrossRefGoogle Scholar
Shanthikumar, J. G. (1981) Optimal control of an M/G/1 priority queue via N-control. Amer. J. Math. Management Sci. 1, 191–212.Google Scholar
Takács, L. (1962) Introduction to the Theory of Queues. Oxford University Press, New York.Google Scholar
Takagi, H. (1990) Time-dependent analysis of M/G/1 vacation models with exhaustive service. QUESTA 6, 369–390.Google Scholar
Welch, P. D. (1963) Some Contributions to the Theory of Priority Queues. , Department of Mathematical Statistics, Columbia University; IBM Research Report RC-922, IBM Research Center, Yorktown Heights, New York.Google Scholar
Welch, P. D. (1964) On a generalized M/G/1 queueing process in which the first customer of each busy period receives exceptional service. Operat. Res. 12, 736–752.CrossRefGoogle Scholar