Hostname: page-component-77c89778f8-m8s7h Total loading time: 0 Render date: 2024-07-19T23:52:38.496Z Has data issue: false hasContentIssue false
  • English
  • Français

A state-dependent GI/G/1 queue

Published online by Cambridge University Press:  26 September 2008

Charles Knessl ,
Charles Tier ,
B. J. Matkowsky  and
Z. Schuss
Charles Knessl
Affiliation:
Department of Mathematics, Statistics and Computer Science, University of Illinoisat Chicago, Chicago, IL 60607, USA
Charles Tier
Affiliation:
Department of Mathematics, Statistics and Computer Science, University of Illinoisat Chicago, Chicago, IL 60607, USA
B. J. Matkowsky
Affiliation:
Department of Engineering Sciences and Applied Mathematics, The Technological Institute, Northwestern University, Evanston, IL 60201, USA
Z. Schuss
Affiliation:
Department of Mathematics, Tel Aviv University, Tel Aviv, Israel
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a state-dependent GI/G/1 queueing system characterized by the unfinished work U(t) in the system at time t. We introduce state-dependence by allowing (i) the arrival process to depend on the instantaneous value of U(t), (ii) the service rate, that is, the rate at which U(t) decreases in the absence of arrivals, to depend on U(t), and (iii) the customer's service requirement to depend on U(t*) where t* denotes the instant in which that customer entered the system. We consider the limit of short inter-arrival times and small service requests and compute asymptotic approximations to the stationary density of the unfinished work, including the stationary probability of finding the system empty, using the WKB method and the method of matched asymptotic expansions.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 5 , Issue 2 , June 1994 , pp. 217 - 241
Copyright
Copyright © Cambridge University Press 1994

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]King, P. J. B. 1990 Computer and Communication Systems Performance Modelling, Prentice Hall, Hemel Hempstead.Google Scholar
[2]Kleinrock, L. 1976 Queueing Systems, vols. 1, II. Wiley, New York.Google Scholar
[3]Moran, P. A. P. 1957 Theory of Storage. Methuen, London.Google Scholar
[4]Prabhu, N. U. 1980 Stochastic Storage Processes: Queues, Insurance, Risk and Dams. Springer-Verlag.CrossRefGoogle Scholar
[5]Cohen, J. W. 1982 The Single Server Queue. North-Holland, Amsterdam.Google Scholar
[6]Courtois, P. J. 1985 On time and space decomposition of complex systems. Comm. ACM 28, 590–603.CrossRefGoogle Scholar
[7]Neuts, M. F. 1989 Structured Stochastic Matrices of the M/G/I Type and Their Application. Marcel Dekker, New York.Google Scholar
[8]Agrawal, S. C. 1985 Metamodeling: A Study of Approximations in Queueing Models. MIT Press, Cambridge, MA.CrossRefGoogle Scholar
[9]Eager, D. L. & Sevcik, K. C. 1986 Performance hierarchies for multiple-class queueing networks. J. ACM 33, 179–206.CrossRefGoogle Scholar
[10]Mitra, D. & McKenna, J. 1986 Asymptotic expansions for closed Markovian networks with state-dependent service rates. J. ACM 33, 568–592.CrossRefGoogle Scholar
[11]Lindley, D. V. 1952 The theory of queues with a single server. Proc. Camb. Phil. Soc. 48, 277–289.CrossRefGoogle Scholar
[12]Feller, W. 1966 Probability Theory and its Applications. Vol. II. Wiley, New York.Google Scholar
[13]Bender, C. M. & Orszag, S. A. 1978 Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill, New York.Google Scholar
[14]Kevorkian, J. & Cole, J. D. 1981 Perturbation Methods in Applied Mathematics. Springer-Verlag, New York.CrossRefGoogle Scholar
[15]Knessl, C., Matkowsky, B. J., Schuss, Z. & Tier, C. 1986 Asymptotic analysis of a state-dependent M/G/l queueing system. SI AM J. Appl. Math. 46, 483–505.CrossRefGoogle Scholar
[16]Knessl, C., Matkowsky, B. J., Schuss, Z. & Tier, C. 1986 A finite capacity single-server queue with customer loss. Commun. Statist.: Stochastic Models 2, 97–121.Google Scholar
[17]Knessl, C., Matkowsky, B. J., Schuss, Z. & Tier, C. 1986 System crash in a finite capacity M/G/l queue. Commun. Statist.: Stochastic Models 2, 171–201.Google Scholar
[18]Knessl, C., Matkowsky, B. J., Schuss, Z. & Tier, C. 1986 On the performance of state-dependent single server queues. SIAM J. Appl. Math. 46, 657–697.CrossRefGoogle Scholar
[19]Knessl, C., Matkowsky, B. J., Schuss, Z. & Tier, C. 1987 Asymptotic expansions for a closed multiple access system. SIAM J. Comp. 16, 378–398.CrossRefGoogle Scholar
[20]Knessl, C., Matkowsky, B. J., Schuss, Z. & Tier, C. 1987 The two repairmen problem: A finite source M/G/2 queue. SIAM J. Appl. Math. 47, 367–397.CrossRefGoogle Scholar
[21]Suri, R. 1989 Perturbation analysis: the state of the art and research issues explained via the GI/G/1 queue. Proc. IEEE 11, 114–137.CrossRefGoogle Scholar
[22]Cao, X. 1990 Realization factors and sensitivity analysis of queueing networks with state-dependent service rates. Adv. Appl. Prob. 22, 178–210.CrossRefGoogle Scholar
[23]Knessl, C., Matkowsky, B. J., Schuss, Z. & Tier, C. 1988 A state dependent M/M/l queue: a production-inventory model. Appl. Math. Lett 1, 235–239.CrossRefGoogle Scholar
[24]Knessl, C. & Tier, C. 1990 Approximations to the moments of the sojourn time in a tandem queue with overtaking. Comm. Stat.: Stochastic Models 6, 499–524.Google Scholar
[25]Knessl, C. & Tier, C. 1990 Asymptotic expansions for large closed queueing networks. J. ACM 31, 144–174.CrossRefGoogle Scholar