# Perturbation analysis of an M/M/1 queue in a diffusion random environment

Abstract : An $M/M/1$ queue whose server rate depends on the state of an independent Ornstein-Uhlenbeck diffusion process is studied in this paper by means of a regular perturbation analysis. Specifically, if $(X(t))$ denotes the modulating Ornstein-Uhlenbeck process, then the server rate at time $t$ is $\phi(X(t))$, where $\phi$ is some given function. After establishing the Fokker-Planck equation characterizing the joint distribution of the occupation process of the $M/M/1$ queue and the state of the modulating Ornstein-Uhlenbeck process, we show, under the assumption that the server rate is weakly perturbed by the diffusion process, that the problem can be solved via a perturbation analysis of a self-adjoint operator defined in an adequate Hilbert space. We then perform a detailed analysis when the perturbation function is linear, namely of the form $\phi(x) = 1-\varepsilonx$. We compute in particular the different terms of the expansion of the solution in power series of $\varepsilon$ and we determine the radius of convergence of the solution. The results are finally applied to study of the integration of elastic and streaming flows in telecommunication network and we show that at the first order the reduced service rate approximation is valid.
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https://hal.inria.fr/inria-00070584
Contributor : Rapport de Recherche Inria <>
Submitted on : Friday, May 19, 2006 - 8:56:39 PM
Last modification on : Friday, May 25, 2018 - 12:02:03 PM
Long-term archiving on : Sunday, April 4, 2010 - 9:31:21 PM

### Identifiers

• HAL Id : inria-00070584, version 1

### Citation

Christine Fricker, Fabrice Guillemin, Philippe Robert. Perturbation analysis of an M/M/1 queue in a diffusion random environment. [Research Report] RR-5422, INRIA. 2005, pp.30. ⟨inria-00070584⟩

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