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An urn model arising from all-optical networks

Part of: Operations research and management science Special processes Combinatorics

Published online by Cambridge University Press:  14 July 2016

John A. Morrison
John A. Morrison*
Affiliation:
Bell Laboratories
*
∗Postal address: Bell Laboratories, Lucent Technologies, 600 Mountain Avenue, Murray Hill, NJ 07974, USA. Email address: [email protected]
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Abstract

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An occupancy model that has arisen in the investigation of randomized distributed schedules in all-optical networks is considered. The model consists of B initially empty urns, and at stage j of the process dj ≤ B balls are placed in distinct urns with uniform probability. Let Mi(j) denote the number of urns containing i balls at the end of stage j. An explicit expression for the joint factorial moments of M0(j) and M1(j) is obtained. A multivariate generating function for the joint factorial moments of Mi(j), 0 ≤ i ≤ I, is derived (where I is a positive integer). Finally, the case in which the dj, j ≥ 1, are independent, identically distributed random variables is investigated.

Keywords

Generating functionjoint factorial momentoccupancy modeloptical networkrandomized schedule

MSC classification

Primary: 05A15: Exact enumeration problems, generating functions
Secondary: 60K30: Applications (congestion, allocation, storage, traffic, etc.) 90B15: Network models, stochastic
Type
Research Papers
Information
Journal of Applied Probability , Volume 43 , Issue 4 , December 2006 , pp. 952 - 966
Copyright
© Applied Probability Trust 2006 

References

Johnson, N. L. and Kotz, S. (1977). Urn Models and Their Application. An Approach to Modern Discrete Probability Theory. John Wiley, New York.Google Scholar
Kolchin, V. F., Sevastyanov, B. A. and Christiakov, V. P. (1978). Random Allocations. V. H. Winston, Washington, DC.Google Scholar
Morrison, J. A., Saniee, I. and Widjaja, I. (2006). Design and performance of randomized network schedules for time-domain wavelength interleaved networks. Preprint.Google Scholar