Hostname: page-component-8448b6f56d-jr42d Total loading time: 0 Render date: 2024-04-23T13:11:01.741Z Has data issue: false hasContentIssue false
  • English
  • Français

Analysis of a two-queue model with Bernoulli schedules

Part of: Operations research and management science Miscellaneous topics of analysis in the complex domain Fredholm integral equations

Published online by Cambridge University Press:  14 July 2016

Duan-Shin Lee
Duan-Shin Lee*
Affiliation:
C&C Research Laboratories
*
∗Postal address: C&C Research Laboratories, NEC USA Inc., 4 Independence Way, Princeton, NJ 08540, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we analyze a single server two-queue model with Bernoulli schedules. This discipline is very flexible and contains the exhaustive and 1-limited disciplines as special cases. We formulate the queueing system as a Riemann boundary value problem with shift. The boundary value problem is solved by exploring a Fredholm integral equation around the unit circle. Some numerical examples are presented at the end of the paper.

Keywords

CYCLIC SERVER QUEUEBERNOULLI SCHEDULERIEMANN BOUNDARY VALUE PROBLEMFREDHOLM INTEGRAL EQUATION

MSC classification

Primary: 90B22: Queues and service
Secondary: 30E25: Boundary value problems 45B05: Fredholm integral equations
Type
Research Papers
Information
Journal of Applied Probability , Volume 34 , Issue 1 , March 1997 , pp. 176 - 191
Copyright
Copyright © Applied Probability Trust 1997 

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] Boxma, O. J. (1986) Methods of two queues: a few new views. Teletraffic Analysis and Computer Performance Evaluation. ed. Boxma, O. J., Cohen, J. W. and Hijms, H. C. North-Holland, Amsterdam. pp. 75–98.Google Scholar
[2] Boxma, O. J. and Groenenduk, W. P. (1987) Pseudo-conservation laws in cyclic-service systems. J. Appl. Prob. 24, 949–964.CrossRefGoogle Scholar
[3] Boxma, O. J. and Groenenduk, W. P. (1988) Two queues with alternating service and switching times. In Queueing Theory and its Applications – Liber Amicorum for J. W. Cohen. ed. Boxma, O. J. and Syski, R. North-Holland, Amsterdam. pp. 261–282.Google Scholar
[4] Boxma, O. J., Koole, G. and Mitrani, I. (1995) A two-queue polling model with a threshold service policy. Proc. MASCOTS '95. ed. Dowd, P. and Gelenbe, E. IEEE Computer Society Press, pp. 84–89.Google Scholar
[5] Cohen, J. W. (1984) On a functional relation in three complex variables: three coupled processors: Preprint NR. 359. Department of Mathematics, University of Utrecht.Google Scholar
[6] Cohen, J. W. and Boxma, O. (1981) The M/G/1 queue with alternating service formulated as a Riemann-Hilbert problem. Performance 81. pp. 181–199.Google Scholar
[7] Cohen, J. W. and Boxma, O. J. (1983) Boundary Value Problems in Queueing System Analysis. North-Holland, Amsterdam.Google Scholar
[8] Eisenberg, M. (1972) Queues with periodic service and changeover time. Operat. Res. 20, 440–451.Google Scholar
[9] Eisenberg, M. (1979) Two queues with alternating service. SIAM J. Appl. Math. 36, 287–303.Google Scholar
[10] Evgrafov, M. A. (1966) Analytic Functions. Dover, New York.Google Scholar
[11] Feller, W. (1957) An Introduction to Probability Theory and Its Applications. Vol. 1. Wiley, New York.Google Scholar
[12] Gakhov, F. D. (1966) Boundary Value Problems. Pergamon, Oxford.Google Scholar
[13] Keilson, J. and Servi, L. D. (1986) Oscillating random walk models for GI/GI/1 vacation systems with Bernoulli schedules. J. Appl. Prob. 20, 813–822.Google Scholar
[14] Kolmogorov, A. N. and Fomin, S. V. (1970) Introductory Real Analysis. Dover, New York.Google Scholar
[15] Lee, D.-S. (1996) A two-queue model with exhaustive and limited service disciplines. Stoch. Models 12, 285–305.CrossRefGoogle Scholar
[16] Lee, D.-S. and Sengupta, B. (1993) Queueing analysis of a threshold based priority scheme for ATM networks. IEEE Trans. Networking 1, 709–717.Google Scholar
[17] Mikhlin, S. G. (1957) Integral Equations and Their Applications to Certain Problems in Mechanics, Mathematical Physics and Technology. Pergamon, New York.Google Scholar
[18] Nehari, Z. (1957) Conformal Mapping. Dover, New York.Google Scholar
[19] Ramaswamy, R. and Servi, L. D. (1988) The busy period of the M/G/l vacation model with a Bernoulli Schedule. Commun. Stat. – Stoch. Models. 4, 507–521.Google Scholar
[20] Servi, L. D. (1986) Average delay approximation of M/G/1 cyclic service queues with Bernoulli Schedules. IEEE J. Selected Areas in Commun. SAC-4, 813–822.CrossRefGoogle Scholar
[21] Takács, L. (1962) Introduction to the Theory of Queues. Oxford University Press, New York.Google Scholar
[22] Takács, L. (1968) Two queues attended by a single server. Operat. Res. 16, 639–650.Google Scholar
[23] Teduanto, (1990) Exact results for the cyclic-service queue with a Bernoulli schedule. Perf. Eval. 11, 107–115.Google Scholar
[24] Weststrate, J. A. and Van Der Mei, R. D. (1994) Waiting time in a two-queue model with exhaustive and Bernoulli service. Z.O.R. – Math. Meth. Operat. Res. 40, 289–303.Google Scholar