Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-09T01:36:00.157Z Has data issue: false hasContentIssue false
  • English
  • Français

Work-modulated queues with applications to storage processes

Part of: Special processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Sid Browne  and
Karl Sigman
Sid Browne*
Affiliation:
Columbia University
Karl Sigman*
Affiliation:
Columbia University
*
∗Postal addresses: Graduate School of Business, 402 Uris Hall, and
∗∗Department of Industrial Engineering and Operations Research, Columbia University, Mudd Building, New York, NY 10027, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

We study two FIFO single-server queueing models in which both the arrival and service processes are modulated by the amount of work in the system. In the first model, the nth customer's service time, Sn, depends upon their delay, Dn, in a general Markovian way and the arrival process is a non-stationary Poisson process (NSPP) modulated by work, that is, with an intensity that is a general deterministic function g of work in system V(t). Some examples are provided. In our second model, the arrivals once again form a work-modulated NSPP, but, each customer brings a job consisting of an amount of work to be processed that is i.i.d. and the service rate is a general deterministic function r of work. This model can be viewed as a storage (dam) model (Brockwell et al. (1982)), but, unlike previous related literature, (where the input is assumed work-independent and stationary), we allow a work-modulated NSPP. Our approach involves an elementary use of Foster's criterion (via Tweedie (1976)) and in addition to obtaining new results, we obtain new and simplified proofs of stability for some known models. Using further criteria of Tweedie, we establish sufficient conditions for the steady-state distribution of customer delay and sojourn time to have finite moments.

Keywords

STATIONARY DISTRIBUTIONNON-STATIONARY POISSON PROCESSREGENERATIVEFOSTER'S CRITERIONFINITE MOMENTS

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60G55: Point processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 29 , Issue 3 , September 1992 , pp. 699 - 712
Copyright
Copyright © Applied Probability Trust 1992 

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

Research supported in part by NSF grant DDM 895 7825.

References

[1] Asmussen, S. (1987) Applied Probability and Queues. Wiley, New York.Google Scholar
[2] Bambos, N. and Walrand, J. (1989) On stability of state-dependent queues and acyclic queueing networks. Adv. Appl. Prob. 21, 681–701.CrossRefGoogle Scholar
[3] Brockwell, P. J., Resnick, S. I. and Tweedie, R. L. (1982) Storage processes with general release rule and additive inputs. Adv. Appl. Prob. 14, 392–433.CrossRefGoogle Scholar
[4] Daley, D. J. and Vere-Jones, D. (1988) An Introduction to the Theory of Point Processes. Springer-Verlag, New York.Google Scholar
[5] Jagerman, D. L. (1985) Certain Volterra integral equations arising in queueing. Stochastic Models 1(2).Google Scholar
[6] Knessl, C., Matkowsky, B. J., Schuss, Z. and Tier, C. (1987) Busy period distribution in state-dependent queues. QUESTA 2, 285–305.Google Scholar
[7] Natvig, B. (1974) On the transient state probabilities for a queueing model where potential customers are discouraged by queue length. J. Appl. Prob. 11, 345–354.CrossRefGoogle Scholar
[8] Tweedie, R. L. (1976) Criteria for classifying general Markov chains. Adv. Appl. Prob. 8, 737–771.CrossRefGoogle Scholar
[9] Tweedie, R. L. (1983) The existence of moments for stationary Markov chains. J. Appl. Prob. 20, 191–196.CrossRefGoogle Scholar
[10] Van Doorn, E. A. (1981) The transient state probabilities for a queueing model where potential customers are discouraged by queue length. J. Appl. Prob. 18, 499–506.CrossRefGoogle Scholar
[11] Whitt, W. (1990) Queues with service times and interarrival times depending linearly and randomly upon waiting times. QUESTA, 6, 335–352.Google Scholar
[12] Wolff, R. W. (1989) Stochastic Modeling and the Theory of Queues. Prentice Hall, Englewood Cliffs, NJ.Google Scholar
[13] Wong, E. and Hajek, B. (1985) Stochastic Processes In Engineering Systems. Springer-Verlag, New York.CrossRefGoogle Scholar