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Markov random field models of multicasting in tree networks

Part of: Stochastic processes Special processes Operations research and management science Computer system organization

Published online by Cambridge University Press:  19 February 2016

Kavita Ramanan ,
Anirvan Sengupta ,
Ilze Ziedins  and
Partha Mitra
Kavita Ramanan*
Affiliation:
Bell Laboratories, Lucent Technologies
Anirvan Sengupta*
Affiliation:
Bell Laboratories, Lucent Technologies
Ilze Ziedins*
Affiliation:
University of Auckland
Partha Mitra*
Affiliation:
Bell Laboratories, Lucent Technologies
*
∗ Postal address: Bell Laboratories, Lucent Technologies, 600 Mountain Avenue, Murray Hill, NJ 07974, USA.
∗ Postal address: Bell Laboratories, Lucent Technologies, 600 Mountain Avenue, Murray Hill, NJ 07974, USA.
∗∗∗ Postal address: Department of Statistics, University of Auckland, Private Bag 92019, Auckland, New Zealand.
∗ Postal address: Bell Laboratories, Lucent Technologies, 600 Mountain Avenue, Murray Hill, NJ 07974, USA.
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Abstract

In this paper, we analyse a model of a regular tree loss network that supports two types of calls: unicast calls that require unit capacity on a single link, and multicast calls that require unit capacity on every link emanating from a node. We study the behaviour of the distribution of calls in the core of a large network that has uniform unicast and multicast arrival rates. At sufficiently high multicast call arrival rates the network exhibits a ‘phase transition’, leading to unfairness due to spatial variation in the multicast blocking probabilities. We study the dependence of the phase transition on unicast arrival rates, the coordination number of the network, and the parity of the capacity of edges in the network. Numerical results suggest that the nature of phase transitions is qualitatively different when there are odd and even capacities on the links. These phenomena are seen to persist even with the introduction of nonuniform arrival rates and multihop multicast calls into the network. Finally, we also show the inadequacy of approximations such as the Erlang fixed-point approximations when multicasting is present.

Keywords

MulticastingunicastingbroadcastingCayley treesBethe latticeloss networksblocking probabilitiesphase transitionsErlang fixed-point approximationsreduced load approximationsstatistical mechanicsspin modelshard core modelssymmetry breaking

MSC classification

Primary: 60G60: Random fields 90B15: Network models, stochastic
Secondary: 68M20: Performance evaluation; queueing; scheduling 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 34 , Issue 1 , March 2002 , pp. 58 - 84
Copyright
Copyright © Applied Probability Trust 2002 

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