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Some explicit formulas and computational methods for infinite-server queues with phase-type arrivals

  • V. Ramaswami (a1) and Marcel F. Neuts (a2)
Abstract

This paper discusses infinite-server queues with phase-type input. The problems of obtaining the transient and steady-state distributions and moments of the queue length are reduced to the solution of certain well-behaved systems of linear differential equations. Sample computations, provided with as many as ten phases, show that although (even the time-dependent) mean queue length is very insensitive to substantial random variability in the arrival process, the higher moments of the queue length are highly sensitive. These examples indicate that considerable caution should be exercised in using robustness results for such stochastic models.

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Corresponding author
Postal address: Department of Mathematical Sciences, Drexel University, Philadelphia, PA 19104, U.S.A.
∗∗ Postal address: Department of Mathematics, University of Delaware, 501 Kirkbride Office Building, Newark, DE 19711, U.S.A.
Footnotes
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Research supported by AFOSR–72–2350C at the Department of Statistics and Computer Science, University of Delaware.

This paper is based on Part II of the author's Ph. D. Thesis.

Research supported by AFOSR–77–3236 at the Department of Statistics and Computer Science, University of Delaware.

Footnotes
References
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[1] Bellman, R. (1960) Introduction to Matrix Analysis. McGraw-Hill, New York.
[2] Cesari, L. (1963) Asymptotic Behavior and Stability Problems in Ordinary Differential Equations. Springer-Verlag, Berlin.
[3] Kingman, J. F. C. (1969) Markov population processes. J. Appl. Prob. 6, 118.
[4] Levinson, N. (1948) The asymptotic nature of solutions of linear systems of differential equations. Duke Math. J. 15, 111126.
[5] Loève, M. (1963) Probability Theory. Van Nostrand, Princeton, N.J.
[6] Neuts, M. F. (1976) Probability distributions of phase type. In Liber Amicorum Professor Emeritus H. Florin, Department of Mathematics, University of Louvain, 173206.
[7] Neuts, M. F. (1978) Renewal processes of phase type. Naval Res. Log. Quart. 25, 445454.
[8] Neuts, M. F. (1976) Markov chains with applications in queueing theory which have a matrix-geometric invariant probability vector. Adv. Appl. Prob. 10, 185212.
[9] Ramaswami, V. and Neuts, M. F. (1978) Some explicit formulas and computational methods for infinite server queues with phase type arrivals. Technical Report 78/5, Department of Statistics and Computer Science, University of Delaware.
[10] Takács, L. (1962) Introduction to the Theory of Queues. Oxford University Press, New York.
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Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
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