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On the convergence to stationarity of the many-server Poisson queue

Published online by Cambridge University Press:  14 July 2016

Wolfgang Stadje*
Affiliation:
University of Osnabrück
P. R. Parthasarathy*
Affiliation:
Indian Institute of Technology, Madras
*
Postal address: Fachbereich Mathematik/Informatik, University of Osnabrück, 49069 Osnabrück, Germany. Email address: [email protected].
∗∗Postal address: Department of Mathematics, Indian Institute of Technology, Madras, Chennai-600036, India.

Abstract

We consider the many-server Poisson queue M/M/c with arrival intensity λ, mean service time 1 and λ/c < 1. Let X(t) be the number of customers in the system at time t and assume that the system is initially empty. Then pn(t) = P(X(t) = n) converges to the stationary probability πn = P(X = n). The integrals ∫0[E(X)-E(X(t))]dt and ∫0[P(Xn) − P(X(t)≤n)]dt are a measure of the speed of convergence towards stationarity. We express these integrals in terms of λ and c.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 1999 

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References

Abate, J., and Whitt, W. (1987). Transient behavior of the M/M/1 queue starting at the origin. Queueing Systems 2, 4165.CrossRefGoogle Scholar
Abate, J., and Whitt, W. (1989). Calculating time-dependent performance measures for the M/M/1 queue. IEEE Trans. Commun. 37, 11021104.CrossRefGoogle Scholar
Ackroyd, M. H. (1982). M/M/1 transient state occupancy probabilities via the discrete Fourier transform. IEEE Trans. Commun. 30, 357559.CrossRefGoogle Scholar
Baccelli, F., and Massey, W. A. (1989). A sample path analysis of the M/M/1 queue. J. Appl. Prob. 26, 418422.CrossRefGoogle Scholar
Boxma, O. J., and Syski, R. (Eds.) (1988). Queueing Theory and Its Applications. North-Holland, Amsterdam.Google Scholar
Buzacott, J. A., and Shantikumar, J. G. (1993). Stochastic Models of Manufacturing Systems. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Cantrell, P. E. (1986). Computation of the transient M/M/1 queue cdf, pdf, and mean with generalized Q-functions. IEEE Trans. Commun. 34, 814817.CrossRefGoogle Scholar
Cantrell, P. E., and Beall, G. R. (1988). Transient M/M/1 queue variance computation using generalized Q-functions. IEEE Trans. Commun. 36, 756758.CrossRefGoogle Scholar
Van de Coevering, M. C. T. (1995). Computing transient performance measures for the M/M/1 queue. OR-Spektrum 17, 1922.CrossRefGoogle Scholar
Conolly, B. W., and Langaris, C. (1993). On a new formula for the transient state probabilities for M/M/1 queues and computational implications. J. Appl. Prob. 30, 237246.CrossRefGoogle Scholar
Guillemin, F., and Majumdar, R. R. (1994). On pathwise behaviour of multiserver queues. Queueing Systems 15, 279288.CrossRefGoogle Scholar
Kijima, M. (1997). Markov Processes for Stochastic Modeling. Chapman and Hall, London.CrossRefGoogle Scholar
Kimura, T. (1994). Approximations for multi-server queues: system interpolations. Queueing Systems 17, 347382.CrossRefGoogle Scholar
Leguesdron, P., Pellaumail, J., Rubino, G., and Sericola, B. (1993). Transient analysis of the M/M/1 queue. Adv. Appl. Prob. 25, 702713,CrossRefGoogle Scholar
Parthasarathy, P. R. (1987). A transient solution to an M/M/1 queue: a simple approach. Adv. Appl. Prob. 19, 997998.CrossRefGoogle Scholar
Parthasarathy, P. R., and Sharafali, M. (1989). Transient solution of the many-server Poisson queue – a simple approach. J. Appl. Prob. 26, 584596.CrossRefGoogle Scholar
Raju, S. N., and Bhat, U. N. (1982). A computationally oriented analysis of the GI/M/1 queue. Opsearch 19, 6783.Google Scholar