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Idle and busy periods in stable M/M/k queues

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

D. J. Daley  and
L. D. Servi
D. J. Daley*
Affiliation:
Australian National University
L. D. Servi*
Affiliation:
GTE Laboratories Incorporated
*
∗Postal address: Stochastic Analysis Group, Centre for Mathematics and its Applications, School of Mathematical Sciences, Australian National University, ACT 0200, Australia. Email address: [email protected]
∗∗Postal address: GTE Laboratories Incorporated, 40 Sylvan Rd., Waltham MA 02254, USA. Email address: [email protected]
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Abstract

This paper finds the first and second moments of the number of arrivals in a stable M/M/k queue during an idle period, i.e. in a period when at least one server is idle. These and similar results are used along with renewal theory asymptotics to find the first and second moments of the proportion of arrivals occurring while there are at least ξ customers waiting for service. The asymptotics of the second moment for large ξ are established.

Keywords

Many-server queuebusy periodidle period

MSC classification

Primary: 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 4 , December 1998 , pp. 950 - 962
Copyright
Copyright © Applied Probability Trust 1998 

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Footnotes

Work done with support from GTE while visiting GTE Laboratories Incorporated.

References

Cohen, J. W. (1982). The Single Server Queue revised edn, North-Holland Amsterdam.Google Scholar
Conolly, B. W. (1971). The busy period for the infinite capacity service system M/G/∞. Studii di Probabilità, Statistica e Ricerca Operativa in Onore di G. Pompilj, Instituto di Calcolo delle Probabilità, Università di Roma, Oderisi Gubbio, Rome, pp. 128–130.Google Scholar
Conolly, B. W. (1974). The generalised state-dependent queue: the busy period Erlangian. J. Appl. Prob. 11, 618–623.CrossRefGoogle Scholar
Conolly, B. W., and Chan, J. (1977). Generalised birth and death queuing processes: recent results. Adv. Appl. Prob. 9, 125–140.CrossRefGoogle Scholar
Daley, D. J. (1968). The serial correlation coefficients of waiting times in a stationary single server queue. J. Austral. Math. Soc. 8, 683–699.CrossRefGoogle Scholar
Daley, D. J. (1969). Monte Carlo estimation of the mean queue size in a stationary GI/M/1 queue. Operat. Res. 16, 1002–1005.CrossRefGoogle Scholar
Daley, D. J., and Servi, L. D. (1993). A two-point Markov chain boundary-value problem. Adv. Appl. Prob. 25, 607–630.CrossRefGoogle Scholar
De Smit, J. H. A. (1973a). Some general results for many server queues. Adv. Appl. Prob. 5, 153–169.CrossRefGoogle Scholar
De Smit, J. H. A. (1973b). On the many server queue with exponential service times. Adv. Appl. Prob. 5, 170–182.CrossRefGoogle Scholar
Feller, W. (1968). An Introduction to Probability Theory and its Applications Vol. 1, 3rd edn. Wiley, New York.Google Scholar
Ghahramani, S. (1986). Finiteness of moments of partial busy periods for M/G/c queues. J. Appl. Prob. 23, 261–264.CrossRefGoogle Scholar
Ghahramani, S. (1990). On remaining full busy periods of GI/G/c queues and their relation to stationary point processes. J. Appl. Prob. 27, 232–236.CrossRefGoogle Scholar
Keilson, J. (1965). A review of transient behavior in regular diffusion and birth–death processes II. J. Appl. Prob. 2, 405–428.CrossRefGoogle Scholar
Kiefer, J. C., and Wolfowitz, J. (1956). On the characteristics of the general queuing process with applications to random walk. Ann. Math. Statist. 27, 147–161.CrossRefGoogle Scholar
McCormick, W. P., and Park, Y. S. (1992). Approximating the distribution of the maximum queue length for M/M/s queues. Queuing and Related Models, ed. Bhat, U. N. and Basawa, I. V., Clarendon Press, Oxford, pp. 240–261.Google Scholar
Natvig, B. (1975). On the waiting-time and busy period distributions for a general birth-and-death queuing model. J. Appl. Prob. 12, 524–532.CrossRefGoogle Scholar
Omahen, K., and Marathe, V. (1978). Analysis and application of the delay cycle for the M/M/c queuing system. J. Assoc. Comp. Mach. 25, 283–303.CrossRefGoogle Scholar
Ramalhoto, M. F. (1984). Bounds for the variance of the busy period of the M/G/∞ queue. Adv. Appl. Prob. 16, 929–932.CrossRefGoogle Scholar
Rosenlund, S. I. (1978). The queue G/M/m/N: busy period and cycle, waiting time under reverse and random order of service. Naval Res. Logist. Quart. 25, 107–119.CrossRefGoogle Scholar
Stadje, W. (1985). The busy period of the queuing system M/G/∞. J. Appl. Prob. 22, 697–704. Addendum: (1987). 24, 290.CrossRefGoogle Scholar
Wiens, D. P. (1989). On the busy period distribution of the M/G/2 queuing system. J. Appl. Prob. 26, 858–865.CrossRefGoogle Scholar