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A note on stochastic decomposition in a GI/G/1 queue with vacations or set-up times

Published online by Cambridge University Press:  14 July 2016

Bharat T. Doshi
Bharat T. Doshi*
Affiliation:
AT&T Bell Laboratories
*
∗Postal address: AT&T Bell Laboratories, Holmdel, H04K428, NJ 07733, USA.
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Abstract

In this note we prove some stochastic decomposition results for variations of the GI/G/1 queue. Our main model is a GI/G/1 queue in which the server, when it becomes idle, goes on a vacation for a random length of time. On return from vacation, if it finds customers waiting, then it starts serving the first customer in the queue. Otherwise it takes another vacation and so on. Under fairly general conditions the waiting time of an arbitrary customer, in steady state, is distributed as the sum of two independent random variables: one corresponding to the waiting time without vacations and the other to the stationary forward recurrence time of the vacation. This extends the decomposition result of Gelenbe and Iasnogorodski [5]. We use sample path arguments, which are also used to prove stochastic decomposition in a GI/G/1 queue with set-up time.

Type
Research Papers
Information
Journal of Applied Probability , Volume 22 , Issue 2 , June 1985 , pp. 419 - 428
Copyright
Copyright © Applied Probability Trust 1985 

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References

[1] Cooper, R. B. (1970) Queues served in cyclic order: waiting times. Bell System Tech. J. 49, 399–413.Google Scholar
[2] Feller, W. (1966) An Introduction to Probability Theory and its Applications, Vol. 2. Wiley, New York.Google Scholar
[3] Franken, P., König, D., Arndt, U. and Schmidt, V. (1981) Queues and Point Processes. Akademie-Verlag, Berlin.Google Scholar
[4] Fuhrmann, S. W. (1983) A note on the M/G/1 queue with server vacations. Operat. Res. Google Scholar
[5] Gelenbe, E. and Iasnogorodski, R. (1980) A queue with server of walking type (autonomous service). Ann. Inst. H. Poincaré 16, 63–73.Google Scholar
[6] Heyman, D. P. (1977) The T-policy for the M/G/1 queue. Management Sci. 23, 775–778.Google Scholar
[7] Lemoine, A. (1975) Limit theorems for generalized single server queues: the exceptional system. SIAM J. Appl. Math. 28, 596–606.CrossRefGoogle Scholar
[8] Levy, H. and Kleinrock, L. (1984) A queue with starter: delay analysis. Operat. Res. Google Scholar
[9] Levy, Y. and Yechiali, U. (1975) Utilization of idle time in an M/G/1 queueing system. Management Sci. 22, 202–211.Google Scholar
[10] Minh, D. (1980) Analysis of the exceptional queueing system by use of regenerative processes and analytical methods. Math. Operat. Res. 5, 147–159.Google Scholar
[11] Ott, T. J. (1984) On the M/G/1 queue with additional inputs. J. Appl. Prob. 21, 129–142.Google Scholar
[12] Pakes, A. G. (1972) A GI/M/1 queue with a modified service mechanism. Ann. Inst. Statist. Math. 24, 589–597.Google Scholar
[13] Pakes, A. G. (1973) On the busy period of the modified GI/G/1 queue. J. Appl. Prob. 10, 192–197.Google Scholar
[14] Revuz, D. (1975) Markov Chains. North-Holland, Amsterdam.Google Scholar
[15] Scholl, M. and Kleinrock, L. (1983) On the M/G/1 queue with rest period and certain service-independent queueing disciplines. Operat. Res. 31, 705–719.CrossRefGoogle Scholar
[16] Servi, L. (1983) D/G/1 queue with vacations. Operat. Res. Google Scholar
[17] Sonderman, D. (1980) Comparing semi-Markov processes. Math. Operat. Res. 5, 110–120.CrossRefGoogle Scholar
[18] Van Der Duyn Schouten, F. A. (1978) An M/G/1 queueing model with vacation times. Z. Operat. Res. 22, 95–105.Google Scholar
[19] 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.Google Scholar
[20] Whitt, W. (1981) Comparing counting processes and queues. Adv. Appl. Prob. 13, 207–220.Google Scholar
[21] Wolff, R. (1970) Work conserving priorities. J. Appl. Prob. 7, 327–337.Google Scholar
[22] Wolff, R. (1982) Poisson arrivals see time averages. Operat. Res. 30, 223–231.Google Scholar
[23] Yeo, G. F. (1962) Single server queues with modified service mechanisms. J. Austral. Math. Soc. 3, 499–507.Google Scholar