Hostname: page-component-7479d7b7d-8zxtt Total loading time: 0 Render date: 2024-07-08T12:37:35.483Z Has data issue: false hasContentIssue false
  • English
  • Français

On the stationary waiting-time distribution in the GI/G/1 queue, I: Transform methods and almost-phase-type distributions

Published online by Cambridge University Press:  01 July 2016

Teunis J. Ott
Teunis J. Ott*
Affiliation:
Bell Communications Research
*
∗Postal address: Bell Communications Research, 435 South Street, Morristown, NJ 07960-1961, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper establishes a link between the Wiener–Hopf factorization and the phase-type method for studying the GI/G/1 queue. Using the Wiener–Hopf factorization, infinite-matrix type results are established for the GI/G/1 queue. An iterative numerical procedure (‘Levinson&s method’) based on these results is described. This method does not always converge. For the situation where either the interarrival times or the service times are of the so-called almost phase type (APH) an alternative, probabilistic derivation of the same results is given. This alternative derivation shows that in the APH situation Levinson&s method converges, converges essentially monotonically, and converges to the correct values.

The algorithm has been coded and examples of numerical results are included.

Keywords

SINGLE-SERVER QUEUENUMERICAL METHODSPHASE-TYPE METHODSWIENER-HOPFALMOST PHASE TYPEGENERALIZED PHASE TYPE
Type
Research Article
Information
Advances in Applied Probability , Volume 19 , Issue 1 , March 1987 , pp. 240 - 265
Copyright
Copyright © Applied Probability Trust 1987 

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

Ackroyd, M. H. (1984) Efficient computation of stationary and cyclostationary finite buffer behavior. AT & T Bell Lab. Tech. J. 63, 2159–2170.CrossRefGoogle Scholar
De Smit, J. H. A. (1983) A numerical solution for the multiserver queue with hyperexponential service times. Operat. Res. Letters 2, 217–224.CrossRefGoogle Scholar
Grassman, W. K. and Jain, J. L. (1985) Numerical solutions of the waiting time of the GI/G/1 queue. Preprint, University of Saskatchewan.Google Scholar
Keilson, J. (1965) Green&s Function Methods in Probability Theory. Hafner, New York.Google Scholar
Keilson, J. and Sumita, U. (1983) The depletion times for M/G/1 Systems. Adv. Appl. Prob. 15, 420–443.CrossRefGoogle Scholar
Kingman, J. F. C. (1964) The heavy traffic approximation in the theory of queues. Proc. Symp. Congestion Theory, University of North Carolina Press, Chapel Hill, 137–169.Google Scholar
Konheim, A. G. (1975) An elementary solution of the queueing system G/G/1. SIAM J. Comput. 4, 540–545.CrossRefGoogle Scholar
Lindley, D. V. (1952) Theory of queues with a single server. Proc. Camb. Phil. Soc. 48, 277–289.CrossRefGoogle Scholar
Lucantoni, D. M. and Ramaswami, V. (1985) Efficient algorithms for solving the non-linear matrix equations arising in phase type queues. Stoch. Models 1, 29–52.CrossRefGoogle Scholar
Luchak, G. (1956) The solution of the single-channel queueing equations characterized by a time-dependent Poisson-distributed arrival rate and a general class of holding times. Operat. Res. 4, 711–732.CrossRefGoogle Scholar
Luchak, G. (1957) The distribution of the time required to reduce to some preassigned level a single-channel queue characterized by a time-dependent Poisson-distributed arrival rate and a general class of holding times. Operat. Res. 5, 205–209.CrossRefGoogle Scholar
Luchak, G. (1958) The continuous time solution of the equations of the single channel queue with a general class of service-time distributions by the method of generating functions. J. R. Statist. Soc. B. 20, 176–181.Google Scholar
Luracs, E. (1970) Characteristic Functions, 2nd edn. Hafner, New York.Google Scholar
Neuts, M. F. (1981) Matrix-Geometric Solutions in Stochastic Models–An Algorithmic Approach. The Johns Hopkins University Press, Baltimore.Google Scholar
Ott, T. J. (1987) On the stationary waiting time distribution in the GI/G/1 queue, II: Real domain methods. Work in progress.CrossRefGoogle Scholar
Ponstein, J. (1974) Theory and numerical solution of a discrete queueing problem. Statist. Neerlandica 20, 139–152.CrossRefGoogle Scholar
Powell, W. B. (1985) Analysis of vehicle holding and cancellation strategies in bulk arrival, bulk service queues. Transportation Sci. 19, 352–377.CrossRefGoogle Scholar
Powell, W. B. (1986) Iterative algorithms for bulk arrival, bulk service queues with Poisson and non-Poisson arrivals. Transportation Sci. 20, 65–79.CrossRefGoogle Scholar
Powell, W. B. and Humblet, P. (1986) The bulk service queue with a general control strategy: theoretical analysis and a new computational procedure. Operat. Res. 34, 267–275.CrossRefGoogle Scholar
Ramaswami, V. and Lucantoni, D. M. (1985) Stationary waiting time distribution in queues with phase type service and in quasi-birth-and-death processes. Stoch. Models 1, 125–136.CrossRefGoogle Scholar
Schassberger, R. A. (1970) On the waiting time in the system GI/G/1. Ann. Math. Statist. 41, 182–187.CrossRefGoogle Scholar
Schassberger, R. A. (1973) Investigation of queueing and related systems with the phase method. Proc. Internat. Teletraffic Congr., Paper no. 324.Google Scholar
Schassberger, R. A. (1974) A broad analysis of single server priority queues with two independent input streams, one of them Poisson. Adv. Appl. Prob. 6, 666–688.CrossRefGoogle Scholar
Seelen, L. P. (1986) An algorithm for PH/PH/c queues. European J. Operat. Res. 23, 118–127.CrossRefGoogle Scholar
Seelen, L. P., Tijms, H. C., and Van Hoorn, M. H. (1985) Tables for Multi-Server Queues. North-Holland, Amsterdam.Google Scholar
Shanthikumar, J. G. (1985) Bilateral phase-type distributions. Naval Res. Logist. Quart. 32, 119–136.CrossRefGoogle Scholar
Smith, W. L. (1953) On the distribution of queueing times. Proc. Camb. Phil. Soc. 49, 449–461.CrossRefGoogle Scholar
Sumita, U. (1981) Development of the Laguerre Transform Method For Numerical Exploration of Applied Probability Models. Ph.D. Dissertation, Graduate School of Management, University of Rochester.Google Scholar
Tijms, H. C. and Van Hoorn, M. H. (1981) Approximations for the steady state probabilities in the M/G/c queue. Adv. Appl. Prob. 13, 186–206.CrossRefGoogle Scholar
Tijms, H. C. and Van Hoorn, M. H. (1982) Computational methods for single server queues with Markovian input and general service times. In Applied Probability–Computer Science, the Interface Vol. II, ed. Disney, R. L. and Ott, T. J., Birkhauser, Boston, 71–102.CrossRefGoogle Scholar
Van Hoorn, M. H. (1984) Algorithms and Approximations For Queueing Systems. CWI Tract No. 8, Mathematical Center, Amsterdam.Google Scholar