Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-13T08:37:29.302Z Has data issue: false hasContentIssue false
  • English
  • Français

Busy periods in time-dependent M/G/1 queues

Published online by Cambridge University Press:  01 July 2016

Stig. I. Rosenlund
Stig. I. Rosenlund*
Affiliation:
University of Göteborg
Get access Rights & Permissions [Opens in a new window]

Abstract

A single-server queue with batch arrivals in a non-homogeneous Poisson process and with balking is studied with respect to the busy period, using supplementary variables. A system of integral equations is obtained on the base of which the transforms are expressed in series. For the homogeneous case, assuming finite waiting room, the solutions are obtained via Cramer's rule. This gives asymptotic expressions for the expectations for large arrival intensity. An efficiency measure giving the long run loss probability is given. For a special case contour integral representations are given as solutions.

Keywords

BUSY PERIODBALKINGBATCH ARRIVALSTIME-DEPENDENT ARRIVAL INTENSITYLINEAR EQUATIONS IN TRANSFORMSFINITE WAITING ROOMDETERMINANT CALCULUS
Type
Research Article
Information
Advances in Applied Probability , Volume 8 , Issue 1 , March 1976 , pp. 195 - 208
Copyright
Copyright © Applied Probability Trust 1976 

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] Cohen, J. W. (1971) On the busy periods for the M/G/1 queue with finite and with infinite waiting room. J. Appl. Prob. 8, 821–827.Google Scholar
[2] Hasofer, A. M. (1964) On the single-server queue with non-homogeneous Poisson input and general service times. J. Appl. Prob. 1, 369–384.Google Scholar
[3] Newell, G. F. (1968) Queues with time-dependent arrival rates I — The transition through saturation. J. Appl. Prob. 5, 436–451.Google Scholar
[4] Newell, G. F. (1968) Queues with time-dependent arrival rates II — The maximum queue and the return to equilibrium. J. Appl. Prob. 5, 579–590.Google Scholar
[5] Newell, G. F. (1968) Queues with time dependent arrival rates III — A mild rush hour. J. Appl. Prob. 5, 591–606.Google Scholar
[6] Rosenlund, S. I. (1973) On the length and number of served customers of the busy period of a generalised M/G/1 queue with finite waiting room. Adv. Appl. Prob. 5, 379–389.Google Scholar
[7] Rosenlund, S. I. (1973) An M/G/1 model with finite waiting room in which a customer remains during part of service. J. Appl. Prob. 10, 778–785.Google Scholar
[8] Rosenlund, S. I. (1975) Busy period of a finite queue with phase type service. J. Appl. Prob. 12, 201–204.Google Scholar
[9] Tomko, J. (1967) A limit theorem for a queue when the input rate increases indefinitely. (In Russian) Studia Sci. Math. Hung. 2, 447–454.Google Scholar