Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-23T18:21:44.304Z Has data issue: false hasContentIssue false
  • English
  • Français

Oscillating random walk models for GI/G/1 vacation systems with Bernoulli schedules

Published online by Cambridge University Press:  14 July 2016

J. Keilson  and
L. D. Servi
J. Keilson*
Affiliation:
University of Rochester
L. D. Servi*
Affiliation:
GTE Laboratories
*
∗Postal address: Graduate School of Management, University of Rochester, Rochester, NY 14627, USA.
∗∗Postal address: GTE Laboratories Inc., 40 Sylvan Road, Waltham, MA 02254, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

Processors handling multi-class traffic typically alternate between serving a particular class of traffic and performing other tasks, e.g., secondary service tasks or routine maintenance. The stochastic behavior of such systems is modeled by a newly introduced class of Bernoulli GI/G/1 vacation models. For this model, when a vacation is completed and customers are present, a customer is served. When a customer has just been served and other customers are present, the server accepts a customer with fixed probability p or commences a vacation of prespecified random duration with probability 1 – p. Whenever no customers are present, a vacation is taken. When p = 0 or p = 1 this schedule reduces to the previously introduced single service schedule and the exhaustive service schedule, respectively. An analysis of all three schedules on a state space incorporating server vacations is presented using simple methods in the complex plane. It is shown that the recent decomposition results for exhaustive service extend to the more general class of Bernoulli schedules.

Keywords

PROCESSOR DISCIPLINESSTOCHASTIC SCHEDULING
Type
Research Paper
Information
Journal of Applied Probability , Volume 23 , Issue 3 , September 1986 , pp. 790 - 802
Copyright
Copyright © Applied Probability Trust 1986 

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] Copson, E. T. (1935) An introduction to Theory of Functions of a Complex Variable. Clarendon Press, Oxford.Google Scholar
[2] 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.CrossRefGoogle Scholar
[3] Fuhrman, S. W. (1984) A note on the M/G/l queue with server vacations. Operat. Res. 32,Google Scholar
[4] Gelenbe, E. and Iasnogorodski, R. (1980) A queue with server of walking type. Ann. Inst. H. Poincaré 16, 63–73.Google Scholar
[5] Hashida, O. (1972) Analysis of a multiqueue. Rev. Elect. Commun. Lab, NTT 20, 189–199.Google Scholar
[6] Hashida, O. and Ohara, K. (1972) Line accommodation capacity of communication control unit. Rev. Elect. Commun. Lab., NTT 20, 231–239.Google Scholar
[7] Keilson, J. (1961) Inst. Internat. Statist., 33rd Session, Paris.Google Scholar
[8] Keilson, J. (1965) Green's Function Methods in Probability Theory. Griffin, London.Google Scholar
[9] Kemperman, J. H. B. (1974) Oscillating random walk. Stoch. Proc. Appl. 2, 1–29.CrossRefGoogle Scholar
[10] Kingman, J. F. C. (1965) The heavy traffic approximation in the theory of queues (Discussion by J. Keilson and J. Th. Runnenberg). In Proc. Symp. Congestion Theory, University of North Carolina Press, Chapel Hill, 162–165.Google Scholar
[11] Kleinrock, L. (1975) Queueing Systems, Vol. 1. Wiley, New York.Google Scholar
[12] Kuehn, P. J. (1979) Multiqueue systems with non-exhaustive cyclic service. Bell System Tech. J. 58, 670–678.Google Scholar
[13] Lee, T. T. (1984) M/G/1/N queue with vacation and exhaustive service discipline. Operat. Res. 32,CrossRefGoogle Scholar
[14] Lee, T. T. (1985) Numerical approximations of multi-queue systems served in cyclic order. In preparation.Google Scholar
[15] Leibowitz, M. A. (1961) An approximation method for treating a class of multi-queue problems. IBM Journal 5, 204–209.Google Scholar
[16] Muskhelishvili, N. I. (1953) Singular Integral Equations. Noordhoff, Groningen.Google Scholar
[17] Scholl, M. and Kleinrock, L. (1983) On the M/G/l queue with rest periods and certain service-dependent queueing disciplines. Operat. Res. 31,Google Scholar
[18] Servi, L. D. (1986) D/G/1 queue with vacation. Operat. Res. To appear.Google Scholar
[19] Servi, L. D. (1986) Average delay approximation of M/G/l cyclic service queues with Bernoulli schedules. IEEE. J. Sel. Areas Comm. To appear.CrossRefGoogle Scholar