Hostname: page-component-848d4c4894-tn8tq Total loading time: 0 Render date: 2024-06-30T04:09:57.306Z Has data issue: false hasContentIssue false
  • English
  • Français

Multiple channel queues in heavy traffic. II: sequences, networks, and batches

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

This paper is a sequel to [7], in which heavy traffic limit theorems were proved for various stochastic processes arising in a single queueing facility with r arrival channels and s service channels. Here we prove similar theorems for sequences of such queueing facilities. The same heavy traffic behavior prevails in many cases in this more general setting, but new heavy traffic behavior is observed when the sequence of traffic intensities associated with the sequence of queueing facilities approaches the critical value (ρ = 1) at appropriate rates.

Type
Research Article
Information
Advances in Applied Probability , Volume 2 , Issue 2 , Autumn 1970 , pp. 355 - 369
Copyright
Copyright © Applied Probability Trust 1970 

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.Google Scholar
[3] Borovkov, A. A. (1967) On limit laws for service processes in multi-channel systems. Siberian Math, J. 8, 746–763.CrossRefGoogle Scholar
[4] Cox, D. R. and Miller, H. D. (1965) The Theory of Stochastic Processes. John Wiley and Sons, New York.Google Scholar
[5] Gaver, D. P. (1968) Diffusion approximations and models for certain congestion problems. J. Appl. Prob. 5, 607–623.Google Scholar
[6] Iglehart, D. L. and Kennedy, D. P. (1970) Weak convergence of the average of flag processes. J. Appl. Prob. (To appear).Google Scholar
[7] Iglehart, D. L. and Whitt, W. (1969) Multiple channel queues in heavy traffic. I. Adv. Appl Prob. 2. 150–177.Google Scholar
[8] Iglehart, D. L. and Whitt, W. (1969) The equivalence of functional central limit theorems for counting processes and associated partial sums. Technical Report No. 5, Department of Operations Research, Stanford University.Google Scholar
[9] Kingman, J. F. C. (1961) The single server queue in heavy traffic. Proc. Camb. Phil. Soc. 57, 902–904.Google Scholar
[10] 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
[11] Parthasarathy, K. R. (1967) Probability Measures on Metric Spaces. Academic Press, New York.Google Scholar
[12] Prabhu, N. U. (1970) Limit theorems for the single server queue with traffic intensity one. J. Appl. Prob. 7, 227–233.Google Scholar
[13] Prohorov, Yu. V. (1956) Convergence of random processes and limit theorems in probability theory. Theor. Probability Appl. 1, 157–214.CrossRefGoogle Scholar
[14] Prohorov, Yu. V. (1963) Transient phenomena in processes of mass service (in Russian). Litovsk. Mat. Sb. 3, 199–205.Google Scholar
[15] Takács, L. (1967). Combinatorial Methods in the Theory of Stochastic Processes. John Wiley and Sons, New York.Google Scholar
[16] 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
[17] Whitt, W. (1970) Multiple channel queues in heavy traffic. III: random server selection. Adv. Appl. Prob. 2, 370–375.Google Scholar