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Semi-linear Stochastic Difference Equations

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Abstract

We consider in this paper a class of vector valued processes that have the form Y n + 1 = A n ( Y n ) + B n . B n is assumed to be stationary ergodic and A n is assumed to have a divisibility property. This class includes linear stochastic difference equations as well as multi-type branching processes (with a discrete or with a continuous state space). We derive explicit expressions for the probability distribution as well as for the two first moments of state vectors at the stationary regime. We then apply this approach to derive two formalisms to describe the infinite server queue. The first is based on a branching process approach adapted to phase type service time distributions. The second is based on a linear stochastic difference equation and is adapted to independent and generally distributed service times with bounded support. In both cases we allow for generally distributed arrival process (not necessarily i.i.d. nor Markovian).

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Notes

  1. A random filed is an extension of a stochastic process where the “time” parameter is not a scalar but a vector in ℝ+ m.

  2. Note that \( ( J - b_i {\bf 1}^T ) \widehat G^i \) is a row vector of dimension |Θ| whose lth entry equals \(E[ B_0^i - b_i | \theta_0 = l ] \pi( l ) \).

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Acknowledgement

This work was supported by the EuroNF network of excellence.

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  1. INRIA, BP93, 06902, Sophia Antipolis, France

    Eitan Altman

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  1. Eitan Altman
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Correspondence to Eitan Altman.

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Altman, E. Semi-linear Stochastic Difference Equations. Discrete Event Dyn Syst 19, 115–136 (2009). https://doi.org/10.1007/s10626-008-0053-4

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