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Asymptotic exponentiality of the tail of the waiting-time distribution in a Ph/Ph/C queue

Published online by Cambridge University Press:  01 July 2016

Yukio Takahashi
Yukio Takahashi*
Affiliation:
Tohoku University
*
∗Postal address: Tohoku University, Faculty of Economics, Kawauchi, Sendai 980, Japan.
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Abstract

It is shown that, in a multiserver queue with interarrival and service-time distributions of phase type (PH/PH/c), the waiting-time distribution W(x) has an asymptotically exponential tail, i.e., 1 – W(x) ∽ Ke–ckx. The parameter k is the unique positive number satisfying T*(ck) S*(–k) = 1, where T*(s) and S*(s) are the Laplace–Stieltjes transforms of the interarrival and the service-time distributions. It is also shown that the queue-length distribution has an asymptotically geometric tail with the rate of decay η = T*(ck). The proofs of these results are based on the matrix-geometric form of the state probabilities of the system in the steady state.

The equation for k shows interesting relations between single- and multiserver queues in the rates of decay of the tails of the waiting-time and the queue-length distributions.

The parameters k and η can be easily computed by solving an algebraic equation. The multiplicative constant K is not so easy to compute. In order to obtain its numerical value we have to solve the balance equations or estimate it from simulation.

Keywords

PH/PH/c QUEUE; WAITING-TIME DISTRIBUTION; APPROXIMATIONS IN QUEUES; MARKOV CHAINS
Type
Research Article
Information
Advances in Applied Probability , Volume 13 , Issue 3 , September 1981 , pp. 619 - 630
Copyright
Copyright © Applied Probability Trust 1981 

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References

Bellman, R. (1960) Introduction to Matrix Analysis. McGraw-Hill, New York.Google Scholar
Borovkov, A. A. (1976) Stochastic Processes in Queueing Theory. Springer-Verlag, New York.Google Scholar
Chung, K. L. (1967) Markov Chains with Stationary Transition Probabilities, 2nd edn. Springer-Verlag, New York.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol II, 2nd edn. Wiley, New York.Google Scholar
Gantmacher, F. R. (1959) The Theory of Matrices, Vol. II. Chelsea, New York.Google Scholar
Kingman, J. F. C. (1966) On the algebra of queues. J. Appl. Prob. 3, 285–326.Google Scholar
Neuts, M. F. (1975) Probability distributions of phase type. In Liber Amicorum Professor Emeritus H. Florin, Dept. of Mathematics, University of Louvain, Belgium, 173–206.Google Scholar
Neuts, M. F. (1978a) Markov chains with applications in queueing theory, which have a matrix-geometric invariant vector. Adv. Appl. Prob. 10, 185–212.CrossRefGoogle Scholar
Neuts, M. F. (1978b) Renewal processes of phase type. Naval Res. Logist Quart. 25, 445–454.Google Scholar
Neuts, M. F. (1980) The probabilistic significance of the rate matrix in matrix-geometric invariant vectors. J. Appl. Prob. 17, 291–296.Google Scholar
Neuts, M. F. (1981) Matrix-Geometric Solutions in Stochastic Models—An Algorithmic Approach. The Johns Hopkins University Press, Baltimore, MD.Google Scholar
Takahashi, Y. and Takami, Y. (1976) A numerical method for the steady-state probabilities of a GI/G/c queueing system in a general class. J. Operat. Res. Soc. Japan 19, 147–157.Google Scholar