Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-22T03:19:36.726Z Has data issue: false hasContentIssue false
  • English
  • Français

Networks of non-homogeneous M/G/∞ systems

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

J. Keilson  and
L. D. Servi
Get access Rights & Permissions [Opens in a new window]

Abstract

For a network of G/∞ service facilities, the transient joint distribution of the facility populations is shown by new simple methods to have a simple Poisson product form with simple explicit formulas for the means. In the network it is assumed that: (a) each facility has an infinite number of servers; (b) the service time distributions are general; (c) external traffic is non-homogeneous in time; (d) arrivals have random or deterministic routes through the network possibly returning to the same facility more than once; (e) arrivals use the facilities on their route sequentially or in parallel (as in the case of a circuitswitched telecommunication network). The results have relevance to communication networks and manufacturing systems.

Keywords

POISSON PRODUCT FORMBCMP NETWORKS

MSC classification

Primary: 60K25: Queueing theory
Type
Part 3 Queueing Theory
Information
Journal of Applied Probability , Volume 31 , Issue A , 1994 , pp. 157 - 168
Copyright
Copyright © Applied Probability Trust 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

Baskett, F., Chandy, M., Muntz, R. and Palacios, J. (1975) Open, closed and mixed networks and queues with different classes of customers. J. Assoc. Comput. Mach. 22, 248–260.Google Scholar
Brown, M. and Ross, S. (1969) Some results for infinite server Poisson queues. J. Appl. Prob. 6, 604–611.Google Scholar
Eick, S. G., Massey, W. A. and Whitt, W. (1993) The physics of the Mt/G/8 queue. Operat. Res. 41, 731–742.Google Scholar
Foley, R. D. (1982) The non-homogeneous M/G/8 queue. Opsearch 19, 40–48.Google Scholar
Harrison, J. M. and Lemoine, A. J. (1981) A note on networks of infinite server queues. J. Appl. Prob. 18, 561–567.CrossRefGoogle Scholar
Jagerman, D. L. (1984) Methods in traffic calculations. AT & T Bell Lab. Tech. J. 63, 1283–1303.Google Scholar
Keilson, J. and Servi, L. D. (1989) Networks of non-homogeneous M/G/8 systems. GTE Technical Report.Google Scholar
Keilson, J. and Steutel, F. W. (1974) Mixtures of distributions, moment inequalities and measures of exponentiality and normality. Ann. Prob. 2, 112–130.Google Scholar
Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, New York.Google Scholar
Khintchine, A. Y. (1955) Mathematical methods in the theory of queueing (in Russian). Trudy Math. Inst. Steklov 49 (English translation published by Charles Griffin and Co. London, 1960).Google Scholar
Massey, W. A. and Whitt, W. (1993a) A probabilistic generalization of Taylor's theorem. Statist. Prob. Letters. 16, 51–54.Google Scholar
Massey, W. A. and Whitt, W. (1993b) Networks of infinite-server queues with non-stationary Poisson input. QUESTA. 13, 183–250.Google Scholar
Massey, W. A. and Whitt, W. (1993C) Stationary-process approximations for the non-stationary Erlang loss model. Operat. Res. To appear.Google Scholar
Palm, C. (1943) Intensity variations in telephone traffic (in German). Ericsson Technics 44, 1–189 (English translation published by North-Holland, 1988).Google Scholar
Prekopa, A. (1958) On secondary processes generated by a random point distribution of Poisson type. Ann. Univ. Sci. Budapest de Eötvös Nom. Sectio Math. 1, 153–170.Google Scholar
Ramakrishnan, C. S. (1980) A note on the M/D/8 queue. Opsearch 17, 118.Google Scholar
Renyi, A. (1967) Remarks on the Poisson process. Studia Sci. Math. Hungar. 2, 119–123.Google Scholar
Serfozo, R. (1990) In Operations Research and Management Science , Vol. 2 Stochastic Models , ed. Heyman, D. P. and Sobel, M. J., North-Holland, Amsterdam.CrossRefGoogle Scholar
Takács, L. (1954) On secondary processes generated by a Poisson process and their applications in physics. Acta. Math. Acad. Sci. Hungar. 5, 203–236.Google Scholar
Takács, L. (1957) On secondary stochastic processes generated by a multidimensional Poisson process. Magyar Tudományos Akad. Mat. Kutató Intézetének Közleményei 2, 71–80.Google Scholar