Hostname: page-component-76fb5796d-x4r87 Total loading time: 0 Render date: 2024-04-25T16:22:05.833Z Has data issue: false hasContentIssue false
  • English
  • Français

Diffusion approximations for queues with server vacations

Published online by Cambridge University Press:  01 July 2016

Offer Kella  and
Ward Whitt
Offer Kella*
Affiliation:
Yale University
Ward Whitt*
Affiliation:
AT&T Bell Laboratories
*
∗Postal address: Department of Operations Research, Yale University, 84 Trumbull St, New Haven, CT 06520, USA.
∗∗Postal address: Room MH 2C-178, AT&T Bell Laboratories, 600 Mountain Avenue, Murray Hill, NJ 07974, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper studies the standard single-server queue with unlimited waiting space and the first-in first-out service discipline, modified by having the server take random vacations. In the first model, there is a vacation each time the queue becomes empty, as occurs for high-priority customers with a non-preemptive priority service discipline. Approximations for both the transient and steady-state behavior are developed for the case of relatively long vacations by proving a heavy-traffic limit theorem. If the vacation times increase appropriately as the traffic intensity increases, the workload and queue-length processes converge in distribution to Brownian motion with a negative drift, modified to have a random jump up whenever it hits the origin. In the second model, vacations are generated exogenously. In this case, if both the vacation times and the times between vacations increase appropriately as the traffic intensity increases, then the limit process is reflecting Brownian motion, modified by the addition of an exogenous jump process. The steady-state distributions of these two limiting jump-diffusion processes have decomposition properties previously established for vacation queueing models, i.e., in each case the steady-state distribution is the convolution of two distributions, one of which is the exponential steady-state distribution of the reflecting Brownian motion obtained as the heavy-traffic limit without vacations.

Keywords

SERVICE INTERRUPTIONSLIMIT THEOREMSHEAVY TRAFFICSTOCHASTIC DECOMPOSITION
Type
Research Article
Information
Advances in Applied Probability , Volume 22 , Issue 3 , September 1990 , pp. 706 - 729
Copyright
Copyright © Applied Probability Trust 1990 

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

Abate, J. and Whitt, W. (1987) The transient behavior of regulated Brownian motion, I and II. Adv. Appl. Prob. 19, 560–631.Google Scholar
Asmussen, S. (1987) Applied Probability and Queues. Wiley, New York.Google Scholar
Asmussen, S. (1988) The heavy traffic limit of a class of Markovian queueing models. Operat. Res. Letters 6, 301–306.CrossRefGoogle Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Burman, D. Y. (1987a) Diffusion approximations for queueing systems an analytic approach. AT&T Bell Laboratories, Murray Hill, N.J. Google Scholar
Burman, D. Y. (1987b) Approximations for a service system with interruptions. AT&T Bell Laboratories, Murray Hill, N.J. Google Scholar
Burman, D. Y. and Smith, D. R. (1986) An asymptotic analysis of a queueing system with Markov-modulated arrivals. Operat. Res. 34, 105–119.Google Scholar
Chen, H. and Mandelbaum, A. (1988) Stochastic discrete flow networks: diffusion approximations and bottlenecks. Graduate School of Business, Stanford University.Google Scholar
Chung, K. L. (1974) A course in Probability Theory. 2nd edition. Academic Press, New York.Google Scholar
Chung, K. L. and Williams, R. J. (1983) Introduction to Stochastic Integration. Birkhäuser, Boston.Google Scholar
Cox, D. R. (1972) Renewal Theory. Methuen, London.Google Scholar
Doshi, B. T. (1985) A note on stochastic decomposition in a GI/G/1 queue with vacations or set-up times. J. Appl. Prob. 22, 419–428.Google Scholar
Doshi, B. T. (1986) Queueing systems with vacations—a survey. Queueing Systems 1, 29–66.Google Scholar
Doshi, B. T. (1990a) Generalizations of the stochastic decomposition results for single server queues with vacations. Stochastic Models. To appear.Google Scholar
Doshi, B. T. (1990b) Single server queues with vacations. In Stochastic Analysis of Computer and Communications Systems, ed. Takagi, H., North-Holland, Amsterdam, 217–265.Google Scholar
Federgruen, A. and Green, L. (1986) Queueing systems with service interruptions. Operat. Res. 34, 752–768.Google Scholar
Fendick, K. W., Saksena, V. R. and Whitt, W. (1989) Dependence in packet queues. IEEE Trans. Commun., 37, 1173–1183.CrossRefGoogle Scholar
Fischer, M. J. (1977) An approximation to queueing systems with interruptions. Management Sci. 24, 338–344.Google Scholar
Fuhrmann, S. and Cooper, R. B. (1985) Stochastic decompositions in an M/G/1 queue with generalized vacations. Operat. Res. 33, 1117–1129.Google Scholar
Glynn, P. W. and Whitt, W. (1988) Ordinary CLT and WLLN versions of L = ?W. Math. Operat. Res. 13, 674–692.Google Scholar
Harrison, J. M. (1977) The supremum distribution of a Lévy process with no negative jumps. Adv. Appl. Prob. 9, 417–422.Google Scholar
Harrison, J. M. (1985) Brownian Motion and Stochastic Flow Systems. Wiley, New York.Google Scholar
Harrison, J. M. (1988) Brownian models of queueing networks with heterogeneous customer populations. In Stochastic Differential Systems, Stochastic Control Theory and Applications. ed. Fleming, W. and Lions, P. L., Springer-Verlag, New York, 147–186.Google Scholar
Harrison, J. M. and Lemoine, A. J. (1981) Sticky Brownian motion as the limit of a storage process. J. Appl. Prob. 18, 216–226.CrossRefGoogle Scholar
Harrison, J. M. and Reiman, M. I. (1981) Reflected Brownian motion on the orthant. Ann. Prob. 9, 302–308.Google Scholar
Harrison, J. M. and Williams, R. (1987) Brownian models of open queueing networks with homogeneous customer populations. Stochastics 22, 77–115.Google Scholar
Iglehart, D. L. and Whitt, W. (1970) Multiple channel queues in heavy traffic, I and II. Adv. Appl. Prob. 2, 150–177 and 355–369.Google Scholar
Kella, O. and Whitt, W. (1991) Queues with server vacations and Lévy processes with secondary jump inputs. Ann. Appl. Prob. 1. To appear.Google Scholar
Kingman, J. F. C. (1962) On queues in heavy traffic. J. R. Statist. Soc. B24, 383–392.Google Scholar
Kolmogorov, A. N. (1956) On Skorohod convergence. Theory Prob. Appl. 1, 215–222.Google Scholar
Laslett, G. M., Pollard, D. B. and Tweedie, R. L. (1978) Techniques for establishing ergodic and recurrence properties of continuous-valued Markov chains. Naval Res. Log. Quart. 25, 455–472.Google Scholar
Lucantoni, D., Meier-Hellstern, K. and Neuts, M. F. (1990) A single server queue with server vacations and a class of non-renewal arrival processes. Adv. Appl. Prob. 22, 676–705.Google Scholar
Meyer, P.-A. (1976) Un Cours sur les Intégrales Stochastiques. Lecture Notes in Mathematics 511, Springer-Verlag, New York.Google Scholar
Newell, G. F. (1982) Applications of Queueing Theory, 2nd edn. Chapman and Hall, London.CrossRefGoogle Scholar
Pollard, O. (1984) Convergence of Stochastic Processes. Springer-Verlag, New York.Google Scholar
Pomarede, J. L. (1976) A Unified Approach Via Graphs to Skorohod Topologies on the Function Space D. Ph.D. Dissertation, Yale University.Google Scholar
Reiman, M. I. (1984) Open queueing networks in heavy traffic. Math. Operat. Res. 9, 441–458.CrossRefGoogle Scholar
Reiman, M. I. and Simon, B. (1988) An interpolation approximation for queueing systems with Poisson input. Operat. Res. 36, 454–469.Google Scholar
Skorohod, A. V. (1956) Limit theorems for stochastic processes. Theory Prob. Appl. 1, 261–290.Google Scholar
Takagi, H. (1986) Analysis of Polling Systems. MIT Press, Cambridge, MA.Google Scholar
Whitt, W. (1971) Weak convergence theorems for priority queues: preemptive-resume discipline. J. Appl. Prob. 8, 74–94.Google Scholar
Whitt, W. (1980) Some useful functions for functional limit theorems. Math. Operat. Res. 5, 67–85.Google Scholar
Whitt, W. (1982) Refining diffusion approximations for queues. Operat. Res. Letters 1, 165–169.Google Scholar
Whitt, W. (1989) An interpolation approximation for the mean workload in the GI/G/1 queue. Operat. Res. 37, 936–952.CrossRefGoogle Scholar
Wolff, R. W. (1982) Poisson arrivals see time averages. Operat. Res. 30, 223–231.Google Scholar