Towards an Erlang formula for multiclass networks |
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Authors: | Matthieu Jonckheere Jean Mairesse |
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Institution: | (1) Statistical Science & Operations Research, Southern Methodist University, Dallas, TX 75275-0332, USA |
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Abstract: | Consider a multiclass stochastic network with state-dependent service rates and arrival rates describing bandwidth-sharing
mechanisms as well as admission control and/or load balancing schemes. Given Poisson arrival and exponential service requirements,
the number of customers in the network evolves as a multi-dimensional birth-and-death process on a finite subset of ℕ
k
. We assume that the death (i.e., service) rates and the birth rates depending on the whole state of the system satisfy a
local balance condition. This makes the resulting network a Whittle network, and the stochastic process describing the state
of the network is reversible with an explicit stationary distribution that is in fact insensitive to the service time distribution.
Given routing constraints, we are interested in the optimal performance of such networks. We first construct bounds for generic
performance criteria that can be evaluated using recursive procedures, these bounds being attained in the case of a unique
arrival process. We then study the case of several arrival processes, focusing in particular on the instance with admission
control only. Building on convexity properties, we characterize the optimal policy, and give criteria on the service rates
for which our bounds are again attained. |
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Keywords: | |
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