Hostname: page-component-7b7567c8c-2n9c9 Total loading time: 0 Render date: 2026-11-15T17:21:33.078Z Has data issue: false hasContentIssue false
  • English
  • Français

The stationary work in system of a G/G/1 gradual input queue

Part of:Special processes

Published online by Cambridge University Press:  14 July 2016

Issei Kino and
Masakiyo Miyazawa
Issei Kino*
Affiliation:
NEC Corporation
Masakiyo Miyazawa*
Affiliation:
Science University of Tokyo
*
∗ Postal address: C&C Information Technology Research Laboratories, NEC Corporation, Kawasaki, Kanagawa 216, Japan.
∗∗ Postal address: Department of Information Science, Science University of Tokyo, Noda, Chiba 278, Japan.
Get accessRights & Permissions [Opens in a new window]

Abstract

This paper is devoted to the study of a stationary G/G/1 queue in which work input is gradually injected into the system and the work load is processed at unit rate. First, assuming that the input process is stationary, the key formula for the stationary distribution of work in system is derived by appeal to the waiting time of the associated regular G/G/1 queue. The main contribution of this paper is the derivation of Laplace-Stieltjes transforms (LSTs) for the stationary distributions of work in system and related random variables in terms of integrations with respect to a waiting time distribution of the associated regular queue. The results are exemplified by giving explicit formulas for the LST of total work for M/Ek /1 and M/H 2/1. The results generalize the results of Pan et al. (1991) for the M/M/1 gradual input queue.

Keywords

G/G/1 QUEUEGRADUAL INPUTBURST ARRIVALWORK IN SYSTEMATM

MSC classification

Primary:60K25: Queueing theory

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 30 , Issue 1 , March 1993, pp. 207 - 222
Copyright
Copyright © Applied Probability Trust 1993 

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.)

Article purchase

Temporarily unavailable

References

Anick, D., Mitra, D. and Sondhi, M. (1982) Stochastic theory of a data-handling system with multiple sources. Bell System Tech. J. 61, 1871–1894.Google Scholar
Brandt, A., Brandt, M. and Sulanke, H. (1990) A single server model for packetwise transmission of messages. QUESTA 6, 287–310.Google Scholar
Cohen, J. W. (1974) Superimposed renewal processes and storage with gradual input. Stoch. Proc. Appl. 2, 31–58.Google Scholar
Franken, P., König, D., Arndt, U. and Schmidt, V. (1982) Queues and Point Processes. Wiley, New York.Google Scholar
Hashida, O. and Fujiki, M. (1973) Queueing models for buffer memory in store-and-forward systems. Proc. ITC 7, Stockholm 323/1–323/7.Google Scholar
Kosten, L. (1974a) Stochastic theory of a multi-entry buffer (1). Delft Progress Report 1, 10–18.Google Scholar
Kosten, L. (1974b) Stochastic theory of a multi-entry buffer (2). Delft Progress Report 1, 44–50.Google Scholar
Kosten, L. and Vrieze, O. (1975) Stochastic theory of a multi-entry buffer (3). Delft Progress Report 1, 103–115.Google Scholar
Neuts, M. F. (1990) On Viterbi's formula for mean delay in a queue of data packets. Commun. Statist.-Stoch. Models 6, 87–98.Google Scholar
Pan, H., Okazaki, H. and Kino, I. (1991) Analysis of a gradual input model for bursty traffic in ATM, Proc. ITC 13, Copenhagen, 795–800.Google Scholar
Sigman, K. and Yamazaki, G. (to appear) Fluid models with burst arrivals: a sample-path analysis. Prob. Eng. Inf. Sci.Google Scholar
Wolff, R. W. (1989) Stochastic Modeling and the Theory of Queues, Prentice-Hall, Englewood Cliffs, NJ.Google Scholar