Hostname: page-component-76fb5796d-9pm4c Total loading time: 0 Render date: 2024-04-28T08:03:15.776Z Has data issue: false hasContentIssue false
  • English
  • Français

Multiple channel queues in heavy traffic. I

Published online by Cambridge University Press:  01 July 2016

Donald L. Iglehart  and
Ward Whitt
Donald L. Iglehart
Affiliation:
Stanford University
Ward Whitt
Affiliation:
Yale University
Get access Rights & Permissions [Opens in a new window]

Extract

The queueing systems considered in this paper consist of r independent arrival channels and s independent service channels, where as usual the arrival and service channels are independent. Arriving customers form a single queue and are served in the order of their arrival without defections. We shall treat two distinct modes of operation for the service channels. In the standard system a waiting customer is assigned to the first available service channel and the servers (servers ≡ service channels) are shut off when they are idle. Thus the classical GI/G/s system is a special case of our standard system. In the modified system a waiting customer is assigned to the service channel that can complete his service first and the servers are not shut off when they are idle. While the modified system is of some interest in its own right, we introduce it primarily as an analytical tool. Let λi denote the arrival rate (reciprocal of the mean interarrival time) in the ith arrival channel and μj the service rate (reciprocal of the mean service time) in the jth service channel. Then is the total arrival rate to the system and is the maximum service rate of the system. As a measure of congestion we define the traffic intensity ρ = λ/μ.

Type
Research Article
Information
Advances in Applied Probability , Volume 2 , Issue 1 , Spring 1970 , pp. 150 - 177
Copyright
Copyright © Applied Probability Trust 

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

[1] Billingsley, P. (1968) Convergence of Probability Measures. John Wiley and Sons, New York.Google Scholar
[2] Borovkov, A. A. (1965) Some limit theorems in the theory of mass service, II. Theor. Probability Appl. 10, 375–400.CrossRefGoogle Scholar
[3] Champernowne, D. (1956) An elementary method of solution of the queueing problem with a single server and constant parameters. J. R. Statist. Soc. B 18, 125–128.Google Scholar
[4] Doob, J. (1948) Renewal theory from the point of view of the theory of probability. Trans. Amer. Math. Soc. 63, 422–438.CrossRefGoogle Scholar
[5] Itô, K. and McKean, H. Jr. (1965) Diffusion Processes and their Sample Paths. Springer-Verlag, Berlin.Google Scholar
[6] Kingman, J. F. C. (1961) The single server queue in heavy traffic. Proc. Camb. Phil. Soc. 57, 902–904.Google Scholar
[7] Kingman, J. F. C. (1962) On queues in heavy traffic. J. R. Statist. Soc. B 24, 383–392.Google Scholar
[8] Kingman, J. F. C. (1965) The heavy traffic approximation in the theory of queues. Smith, W. and Wilkinson, W. (eds.) Proc. Symposium on Congestion Theory. Univ. of North Carolina Press, Chapel Hill, 137–159.Google Scholar
[9] Prabhu, N. U. (1965) Queues and Inventories. John Wiley and Sons, New York.Google Scholar
[10] Presman, E. (1965) On the waiting time for many-server queueing systems. Theor. Probability Appl. 10, 63–73.CrossRefGoogle Scholar
[11] Prohorov, Yu. V. (1963) Transient phenomena in processes of mass service (in Russian). Litovsk. Mat. Sb. 3, 199–205.Google Scholar
[12] Whitt, W. (1968) Weak Convergence Theorems for Queues in Heavy Traffic. . Cornell University. (Technical Report No. 2, Department of Operations Research, Stanford University.).Google Scholar