Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-06-09T01:33:34.046Z Has data issue: false hasContentIssue false
  • English
  • Français

Geometric product form queueing networks with concurrent batch movements

Part of: Markov processes Special processes Operations research and management science

Published online by Cambridge University Press:  01 July 2016

Hideaki Yamashita  and
Masakiyo Miyazawa
Hideaki Yamashita*
Affiliation:
Tohoku University
Masakiyo Miyazawa*
Affiliation:
Science University of Tokyo
*
∗ Postal address: Faculty of Economics, Tohoku University, Kawauchi, Sendai 980-8576, Japan. Email address: hideaki@econ.tohoku.ac.jp
∗∗ Postal address: Department of Information Sciences, Science University of Tokyo, Noda, Chiba 278-8510, Japan.
Get access Rights & Permissions [Opens in a new window]

Abstract

Queueing networks have been rather restricted in order to have product form distributions for network states. Recently, several new models have appeared and enlarged this class of product form networks. In this paper, we consider another new type of queueing network with concurrent batch movements in terms of such product form results. A joint distribution of the requested batch sizes for departures and the batch sizes of the corresponding arrivals may be arbitrary. Under a certain modification of the network and mild regularity conditions, we give necessary and sufficient conditions for the network state to have the product form distribution, which is shown to provide an upper bound for the one in the original network. It is shown that two special settings satisfy these conditions. Algorithms to calculate their stationary distributions are considered, with numerical examples.

Keywords

Markovian queueing networkconcurrent batch movementgeometric distributionproduct form stationary distributioncharacterizationstochastic order

MSC classification

Primary: 60K25: Queueing theory 90B22: Queues and service
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 30 , Issue 4 , December 1998 , pp. 1111 - 1129
Copyright
Copyright © Applied Probability Trust 1998 

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

Boucherie, R. J. and van Dijk, N. M. (1991). Product forms for queueing networks with state-dependent multiple job transitions. Adv. Appl. Prob. 23, 152–187.CrossRefGoogle Scholar
Chao, X. (1995). Networks of queues with customers, signals and arbitrary service time distributions. Operat. Res. 43, 537–544.CrossRefGoogle Scholar
Chao, X. and Miyazawa, M. (1998). Queueing networks with instantaneous movements: A coupling approach by quasi-reversibility. To appear in Operat. Res. 46.Google Scholar
Chao, X. and Pinedo, M. (1995). Networks of queues with batch services, signals, and product form solutions. Operat. Res. Lett. 17, 237–242.CrossRefGoogle Scholar
Gelenbe, E. (1991). Product-form queueing networks with negative and positive customers. J. Appl. Prob. 28, 656–663.CrossRefGoogle Scholar
Gelenbe, E. (1993). G-networks with triggered customer movement. J. Appl. Prob. 30, 742–748.CrossRefGoogle Scholar
Henderson, W. and Taylor, P. G. (1990). Product form in networks of queues with batch arrivals and batch services. Queueing Systems 6, 71–88.CrossRefGoogle Scholar
Henderson, W. (1993) Queueing networks with negative customers and negative queueing lengths, Journal of Applied Probability, 30, 931–942.CrossRefGoogle Scholar
Kelly, F. P. (1979). Reversibility and Stochastic Networks. Wiley, New York.Google Scholar
Miyazawa, M. (1996a). Stochastic bound and stability of discrete-time Jackson networks with batch movements. In Stochastic networks: Stability and rare events, ed. Glasserman, P., Sigman, K. and Yao, D. D. (Lecture Notes in Statist. 117). Springer, pp. 75–94.CrossRefGoogle Scholar
Miyazawa, M. (1996b). Structure-reversibility and departure functions of queueing networks with batch movements and state dependent routing. Queueing Systems 25, 45–75.CrossRefGoogle Scholar
Miyazawa, M. and Taylor, P. G. (1997). A geometric product-form distribution for a queueing network with nonstandard batch arrivals and batch transfers. Adv. Appl. Prob. 29, 523–544.CrossRefGoogle Scholar
Serfozo, R. F. (1993). Queueing networks with dependent nodes and concurrent movements. Queueing Systems 13, 143–182.CrossRefGoogle Scholar
Walrand, J. (1988). An Introduction to Queueing Networks. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar