Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-26T14:59:22.242Z Has data issue: false hasContentIssue false
  • English
  • Français

On the time-dependent occupancy and backlog distributions for the GI/G/∞ queue

Part of: Stochastic processes Special processes

Published online by Cambridge University Press:  14 July 2016

H. Ayhan ,
J. Limon-Robles  and
M. A. Wortman
H. Ayhan*
Affiliation:
Georgia Institute of Technology
J. Limon-Robles*
Affiliation:
ITESM, Monterrey
M. A. Wortman*
Affiliation:
Texas A&M University
*
∗Postal address: School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA. Email address: [email protected]
∗∗Postal address: Department of Control Engineering, ITESM, Monterrey, Mexico.
∗∗∗Postal address: Department of Industrial Engineering, Texas A&M University, College Station, TX 77843, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider an infinite server queueing system. An examination of sample path dynamics allows a straightforward development of integral equations having solutions that give time-dependent occupancy (number of customers) and backlog (unfinished work) distributions (conditioned on the time of the first arrival) for the GI/G/∞ queue. These integral equations are amenable to numerical evaluation and can be generalized to characterize GIX/G/∞ queue. Two examples are given to illustrate the results.

Keywords

Infinite serverintegral equationsoccupancybacklog

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60G35: Signal detection and filtering
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 2 , June 1999 , pp. 558 - 569
Copyright
Copyright © Applied Probability Trust 1999 

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.)

Footnotes

This research has been supported by National Science Foundation Grants DMI-9622138 and DMI-9215662.

References

Baccelli, F. and Brémaud, P. (1994). Elements of Queueing Theory: Palm Martingale Calculus and Stocahstic Recurrences. Springer, Berlin.Google Scholar
Billingsley, F. (1995). Probability and Measure. Wiley, New York.Google Scholar
Brandt, A., Franken, P. and Lisek, B. (1990). Stationary Stochastic Models. Akademic-Verlag, Berlin.Google Scholar
Gakis, K. G., and Sivazlian, B. D. (1994). A Voltera type integral equation for the characteristic function of the extended compound renewal process with applications. Stoch. Anal. Appl. 12, 159–174.Google Scholar
Gakis, K. G., and Sivazlian, B. D. (1994). The moments of the extended compound point processes with applications. Stoch. Anal. Appl. 12, 175–191.Google Scholar
Gross, D., and Harris, C. M. (1985). Fundamentals of Queueing Theory. Wiley, New York.Google Scholar
Holman, D. F., Chaudry, M. L., and Kashyap, B. R. K. (1982). On the number in the system GI X /G/∞. Sankhyā A44, 294–297.Google Scholar
Liu, L., Kashyap, B. R. K., and Templeton, J. G. C. (1990). On the GI X /G/∞ system. J. Appl. Prob. 27, 671–683.Google Scholar
Liu, L., and Templeton, J. G. C. (1991). The GX n /Gn /∞ system: system size. Queueing Systems 8, 323–356.Google Scholar
Liu, L., and Templeton, J. G. C. (1995). Departures in GX n /Gn /∞. Queueing Systems 19, 399–419.Google Scholar
Liu, L., and Shi, D-H. (1996). Busy period in G X /G/∞. J. Appl. Prob. 33, 815–829.CrossRefGoogle Scholar
Ramachandran, S. (1994). Backlog and occupancy in the GI/GI/∞ queue. Ph.D. Dissertation, Texas A&M University, College Station, TX.Google Scholar
Shanbhag, D. N. (1966). On infinite server queues with batch arrivals. J. Appl. Prob. 3, 274–279.Google Scholar
Shirayev, A. N. (1984). Probability. Springer, New York.Google Scholar
Sigman, K. (1995). Stationary Marked Point Processes : An Intuitive Approach. Chapman & Hall, New York.Google Scholar
Takács, L. (1958). On a coincidence problem in telephone traffic. Acta Math. Acad. Sci. Hung. 9, 45–80.Google Scholar
Takács, L. (1980). Queues with infinitely many servers, R.A.I.R.O Rech. Opérat. 14, 109–113.Google Scholar