Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-24T11:54:39.996Z Has data issue: false hasContentIssue false
  • English
  • Français

An M/G/c queue in which the number of servers required is random

Published online by Cambridge University Press:  14 July 2016

Awi Federgruen  and
Linda Green
Awi Federgruen*
Affiliation:
Columbia University
Linda Green*
Affiliation:
Columbia University
*
∗Postal address: Columbia Business School, Columbia University, New York, NY 10027, USA.
∗Postal address: Columbia Business School, Columbia University, New York, NY 10027, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

Many queueing situations such as computer, communications and emergency systems have the feature that customers may require service from several servers at the same time. They may thus be delayed until the required number of servers is available and servers may be idle when customers are waiting. We consider general server-completion-time distributions and derive approximation methods for the computation of the steady-state distribution of the number of customers in queue as well as the moments of the waiting-time distribution. Extensive computational results are reported.

Keywords

MULTISERVER QUEUESRANDOM SERVER REQUIREMENTAPPROXIMATIONS FOR STEADY-STATE DISTRIBUTIONS
Type
Research Papers
Information
Journal of Applied Probability , Volume 21 , Issue 3 , September 1984 , pp. 583 - 601
Copyright
Copyright © Applied Probability Trust 1984 

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

Arthurs, E. and Kaufman, J. S. (1979) Sizing a message store subject to blocking criteria. 4th International Symposium on Modelling and Performance Evaluation of Computer Systems, Vienna.Google Scholar
Chaiken, J. (1971) The number of emergency units busy at alarms which require multiple servers. R-531-NYC/HUD, The Rand Corporation, Santa Monica, CA.Google Scholar
Chaiken, J. and Dormont, P. (1978) A patrol car allocation model: background and capabilities and algorithms. Management Sci. 24, 1280–1300.Google Scholar
Chelst, K. R. and Barlach, Z. (1981) Multiple unit dispatches in emergency services. Management Sci. 27, 1390–1409.CrossRefGoogle Scholar
Federgruen, A. and Tijms, H. C. (1980) The computation of the stationary distribution of the queue size in an M/G/1 queuing system with variable service rate. J. Appl. Prob. 17, 515–522.Google Scholar
Gimpelson, L. A. (1965) Analysis of mixtures of wide- and narrow-band traffic. IEEE Trans. Commun. Technol. 13, 258–266.Google Scholar
Green, L. (1980) A queuing system in which customers require a random number of servers. Operat. Res. 28, 1335–1346.Google Scholar
Green, L. (1982) A limit theorem on subintervals of interrenewal times. Operat. Res. 30, 210–215.CrossRefGoogle Scholar
Green, L. (1984) A multiple dispatch queueing model of police patrol operations. Management Sci. 30.Google Scholar
Green, L. and Kolesar, P. (1983) Testing the validity of a queueing model of police patrol operations. Columbia Univ. Research Working Paper, No. 521A.Google Scholar
Green, L. and Kolesar, P. (1984) The feasibility of one-officer patrol in New York City. Management Sci. 30.CrossRefGoogle Scholar
Gross, D. and Harris, C. M. (1974) Fundamentals of Queuing Theory. Wiley, New York.Google Scholar
Heller, N. (1967) Service time histograms for police patrol activities in St. Louis. St. Louis Police Department.Google Scholar
Heyman, D. P. and Sobel, M. J. (1982) Stochastic Models in Operations Research, Vol. I. McGraw-Hill, New York.Google Scholar
Hordijk, A. and Tijms, H. C. (1976) A simple proof of the equivalence of the limiting distributions of the continuous-time and the embedded process of the queue size in the M/G/1 queue. Statist. Neerlandica 30, 97–100.Google Scholar
Kemeny, J. G. and Snell, J. C. (1960) Finite Markov Chains. Van Nostrand, Princeton, N.J.Google Scholar
Larson, R. (1972) Urban Police Patrol Analysis. MIT Press, Cambridge, Mass.Google Scholar
Neuts, M. F. (1981) Matrix-Geometric Solutions in Stochastic Models. Johns Hopkins University Press, Baltimore.Google Scholar
Nigam, A. K. (1975) Analysis for a satellite communications system. Interfaces 5, 37–47.Google Scholar
Nozaki, S. and Ross, S. (1978) Approximations in finite-capacity multiserver queues with Poisson arrivals. J. Appl. Prob. 15, 826–834.CrossRefGoogle Scholar
Omahen, K. J. (1977) Capacity bounds for multi-resource queues. J. Assoc. Comput. Mach. 24, 646–663.CrossRefGoogle Scholar
Ross, S. M. (1970) Applied Probability Models with Optimization Applications. Holden-Day, San Francisco.Google Scholar
Stidham, S. Jr (1972) Regenerative processes in the theory of queues, with applications to the alternating-priority queue. Adv. Appl. Prob. 4, 542–577.Google Scholar
Takács, L. (1962) Introduction to the Theory of Queues. Oxford University Press, New York.Google Scholar
Tijms, H. C., Van Hoorn, M. H. and Federgruen, A. (1981) Approximations for the steady-state probabilities in the M/G/c queue. Adv. Appl. Prob. 13, 186–206.Google Scholar
Van Hoorn, M. H. (1983) Algorithms and Approximations for Queueing Systems. Mathematical Centre Tract, Mathematical Centre, Amsterdam.Google Scholar
Van Hoorn, M. H. and Tijms, H. C. (1982) Approximations for the waiting time distribution of the M/G/c queue. Performance Analysis 2, 22–28.Google Scholar
Whitt, W. (1982) Approximating a point process in a renewal process I: two basic methods. Operat. Res. 30, 125–147.Google Scholar
Wolman, E. (1972) The camp-on problem for multiple-address traffic. Bell System Tech. J., 1363–1422.Google Scholar