Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-16T09:12:49.680Z Has data issue: false hasContentIssue false
  • English
  • Français

One-node closed multichannel service system: several types of customers and service rates, and random pick-up from the waiting line

Published online by Cambridge University Press:  01 July 2016

O. Bronshtein ,
I. Gertsbakh ,
B. Pittel  and
S. Shahaf
O. Bronshtein*
Affiliation:
Israel Aircraft Industries, Lod
I. Gertsbakh*
Affiliation:
Ben-Gurion University
B. Pittel*
Affiliation:
The Ohio State University
S. Shahaf*
Affiliation:
Israel Aircraft Industries, Lod
*
∗Postal address: Israel Aircraft Industries Ltd, Department 4616, Ben Gurion International Airport, Lod, Israel.
∗∗ Postal address: Department of Mathematics, Ben Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84 105, Israel.
∗∗∗ Postal address: Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, OH 43210-1174, USA.
∗Postal address: Israel Aircraft Industries Ltd, Department 4616, Ben Gurion International Airport, Lod, Israel.
Get access Rights & Permissions [Opens in a new window]

Abstract

A closed queueing system has Q service channels and a waiting line. There are Ni customers of type i in the system, i = 1, ···, m, ∑mi=1Ni = N > Q. Q customers are served and Q0 = N – Q stay in the waiting line. Q channels are partitioned into n groups with Qj channels in the jth group, j = 1, ···, n. The service time of the ith type customer by a channel of the jth group is τij ~ Exp (λij). When a customer leaves the channel, it is immediately replaced by another one picked up randomly from the waiting time. The customer which has cleared service joins the waiting line without delay. Let Xij be the number of ith type customers served by jth group channels in equilibrium. An explicit formula for P(Xij = kij, i = 1, ···, m; j = 1, ···, n) is found. It is shown in a form of a local limit theorem that the asymptotic distribution of {Xij} is a multidimensional normal, if Qj/N and Qj/N have positive limits as N → ∞. Formulas for mean values and covariances are given. It turns out that the means of Xij and covariances between Xij and Xrk can be found, using an efficient iterative algorithm, from the deterministic version of the system. A numerical example demonstrates that the normal approximation is rather accurate.

Keywords

CLOSED NETWORKMULTIVARIATE NORMAL APPROXIMATION
Type
Research Article
Information
Advances in Applied Probability , Volume 19 , Issue 2 , June 1987 , pp. 487 - 504
Copyright
Copyright © Applied Probability Trust 1987 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Baskett, F., Chandy, K., Muntz, R. and Palacios, F. (1975) Open, closed and mixed networks of queues with different classes of customers. J. Assoc. Comput. Mach. 22, 248–260.CrossRefGoogle Scholar
Bregman, L. M. (1967) Proof of the convergence of Sheleikhovskii&s method for a problem with transportation constraints. USSR Comp. Math. Math. Physics 7, 191–204.CrossRefGoogle Scholar
D&Esopo, D. A. and Lefkowitz, B. (1963) An algorithm for computing interzonal transfers using the gravity model. Operat. Res. 11, 901–907.CrossRefGoogle Scholar
Gordon, W. F. and Newell, G. F. (1967) Closed queueing systems with exponential servers. Operat. Res. 15, 254–265.CrossRefGoogle Scholar
Iglehart, D. L. and Lemoine, A. F. (1973) Approximations for the repairman problem with two repair facilities. I: no spares. Adv. Appl. Prob. 5, 595–613.CrossRefGoogle Scholar
Iglehart, D. L. and Lemoine, A. F. (1974) Approximations for the repairman problem with two repair facilities. II: spares. Adv. Appl. Prob. 6, 147–158.CrossRefGoogle Scholar
Karlin, S. (1959) Mathematical Methods and Theory in Games, Programming and Economics. Pergamon Press, New York.Google Scholar
Kelly, F. P. (1975) Networks of queues with customers of different types. J. Appl. Prob. 12, 542–554.CrossRefGoogle Scholar
Kelly, F. P. (1976) Networks of queues. Adv. Appl. Prob. 12, 416–432.CrossRefGoogle Scholar
Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, New York.Google Scholar
Kurtz, T. G. (1970) Solutions of ordinary differential equations as limits of pure jump Markov processes. J. Appl. Prob. 7, 49–58.CrossRefGoogle Scholar
Pittel, B. (1967) A simple model of collective behaviour. Probl. Informat. Trans. 3 (3), 30–44.Google Scholar
Pittel, B. (1979) Closed exponential network of queues with saturation: Jackson-type stationary distribution and its asymptotic analysis. Math. Operat. Res. 4, 357–378.CrossRefGoogle Scholar
Sinkhorn, R. (1964) A relationship between arbitrary positive matrices and doubly stochastic matrices. Ann. Math. Statist. 35, 876–879.CrossRefGoogle Scholar