Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-12T21:14:39.115Z Has data issue: false hasContentIssue false

Bounds for the variance of the busy period of the M/G/∞ queue

Published online by Cambridge University Press:  01 July 2016

M. F. Ramalhoto*
Affiliation:
Instituto Superior Técnico, Lisbon
*
Postal address: Department of Mathematics, Instituto Superior Técnico, Av. Rovisco Pais, 1000 Lisboa, Portugal.
Rights & Permissions [Opens in a new window]

Abstract

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.

Some bounds for the variance of the busy period of an M/G/∞ queue are calculated as functions of parameters of the service-time distribution function. For any type of service-time distribution function, upper and lower bounds are evaluated in terms of the intensity of traffic and the coefficient of variation of the service time. Other lower and upper bounds are derived when the service time is a NBUE, DFR or IMRL random variable. The variance of the busy period is also related to the variance of the number of busy periods that are initiated in (0, t] by renewal arguments.

Type
Letters to the Editor
Copyright
Copyright © Applied Probability Trust 1984 

Footnotes

Research partially supported by the Centro de Estatística e Aplicaçõ es do INIC under the Applied Stochastic Processes Research Project.

References

Brown, M. (1981) Further monotonicity properties for specialized renewal processes. Ann. Prob. 9, 891895.Google Scholar
Brown, M. and Solomon, H. (1974) Some results for secondary processes generated by a Poisson process. Stoch. Proc. Appl. 2, 337348.CrossRefGoogle Scholar
Conolly, B. W. (1971) The busy period for the infinite capacity service system M/G/8 In Studii di Probabilità, Statistica e Ricerca Operativa in Onore di G. Pompilj, Istituto di Calcolo delle Probabilità, Università di Roma, Oderisi Gubbio, Roma, 128130.Google Scholar
Cox, D. R. (1962) Renewal Theory. Methuen, London.Google Scholar
Cox, D. R. and Isham, V. (1980) Point Processes. Chapman and Hall, London.Google Scholar
Ramalhoto, M. F. (1983) A note on the variance of the busy period of the M/G/8 system. Technical Report, Linha-3, Processos Estocásticos Aplicados, do CEAUL do INIC.Google Scholar
Ross, S. (1983) Stochastic Processes. Wiley, New York.Google Scholar
Shanbhag, D. N. (1966) On infinite server queues with batch arrivals. J. Appl. Prob. 3, 274279.Google Scholar
Takács, L. (1954) On secondary processes generated by a Poisson process and their applications in physics. Acta Math. Acad. Sci. Hungar. 5, 203236.Google Scholar
Takács, L. (1962) An Introduction to Queueing Theory. Oxford University Press, New York.Google Scholar