Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-11T01:58:12.957Z Has data issue: false hasContentIssue false
  • English
  • Français

A multiclass feedback queue in heavy traffic

Published online by Cambridge University Press:  01 July 2016

Martin I. Reiman
Martin I. Reiman*
Affiliation:
AT&T Bell Laboratories
*
∗Postal address: AT&T Bell Laboratories, 600 Mountain Avenue, Murray Hill, NJ 07974, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a single station queueing system with several customer classes. Each customer class has its own arrival process. The total service requirement of each customer is divided into a (possibly) random number of service quanta, where the distribution of each quantum may depend on the customer's class and the other quanta of that customer. The service discipline is round-robin, with random quanta.

We prove a heavy traffic limit theorem for the above system which states that as the traffic intensity approaches unity, properly normalized sequences of queue length and sojourn time processes converge weakly to one-dimensional reflected Brownian motion.

Keywords

BROWNIAN MOTIONDIFFUSION APPROXIMATIONWEAK CONVERGENCE
Type
Research Article
Information
Advances in Applied Probability , Volume 20 , Issue 1 , March 1988 , pp. 179 - 207
Copyright
Copyright © Applied Probability Trust 1988 

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. M., Muntz, R. R. and Palacios, F. G. (1975) Open, closed and mixed networks of queues with different classes of customers. J. Assoc. Comput. Mach. 22, 248–260.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Burman, D. Y. and Smith, D. R. (1986) An asymptotic analysis of queueing systems with Markov modulated arrivals. Operat. Res. 34, 105–119.Google Scholar
Cox, D. R. and Miller, H. D. (1965) The Theory of Stochastic Processes. Wiley, New York.Google Scholar
Harrison, J. M. and Reiman, M. I. (1981) Reflected Brownian motion on an orthant. Ann. Prob. 9, 302–308.Google Scholar
Iglehart, D. L. (1973) Weak convergence in queueing theory. Adv. Appl. Prob. 5, 570–594.Google Scholar
Iglehart, D. L. and Whitt, W. (1970) Multiple channel queues in heavy traffic, I and II. Adv. Appl. Prob. 2, 150–177; 355-364.Google Scholar
Johnson, D. P. (1983) Diffusion Approximations for Optimal Filtering of Jump Processes and for Queueing Networks. , University of Wisconsin.Google Scholar
Kelly, F. P. (1975) Networks of queues with customers of different classes. J. Appl. Prob. 12, 542–554.Google Scholar
Kelly, F. P. (1978) Reversibility and Stochastic Networks. Wiley, New York.Google Scholar
Lam, S. S. and Shankar, A. U. (1981) A derivation of response time distributions for a multi-class feedback queueing system. Perf. Eval. 1, 48–61.Google Scholar
Massey, W. A. (1984) Open networks of queues: their algebraic structure and estimating their transient behavior. Adv. Appl. Prob. 16, 176–201.Google Scholar
Peterson, W. P. (1985) Diffusion Approximations for Networks of Queues with Multiple Customer Types. , Stanford University.Google Scholar
Prohorov, Yu. V. (1956) Convergence of random processes and limit theorems in probability theory. Theory Prob. Appl. 1, 157–214.Google Scholar
Reiman, M. I. (1983) Some diffusion approximations with state-space collapse. Proc. Internat. Seminar on Modeling and Performance Evaluation Methodology. Springer-Verlag, Berlin.Google Scholar
Reiman, M. I. (1984) Open queueing networks in heavy traffic. Math. Operat. Res. 9, 441–458.Google Scholar
Reiman, M. I. and Simon, B. (1988a) Open queueing systems in light traffic. Math. Operat. Res. to appear.Google Scholar
Reiman, M. I. and Simon, B. (1988b) An interpolation approximation for queueing systems with Poisson input. Operat. Res. 36.Google Scholar
Simon, B. (1984) Priority queues with feedback. J. Assoc. Comput. Mach. 31, 134–149.Google Scholar
Whitt, W. (1980) Some useful functions for functional limit theorems. Math. Operat. Res. 5, 67–85.Google Scholar