Queueing Systems

, Volume 61, Issue 1, pp 1–35 | Cite as

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

  • Christine Fricker
  • Fabrice Guillemin
  • Philippe Robert


We study in this paper an M/M/1 queue whose server rate depends upon the state of an independent Ornstein–Uhlenbeck diffusion process (X(t)) so that its value at time t is μ φ(X(t)), where φ(x) is some bounded function and μ>0. We first establish the differential system for the conditional probability density functions of the couple (L(t),X(t)) in the stationary regime, where L(t) is the number of customers in the system at time t. By assuming that φ(x) is defined by φ(x)=1−ε((x a/ε)(−b/ε)) for some positive real numbers a, b and ε, we show that the above differential system has a unique solution under some condition on a and b. We then show that this solution is close, in some appropriate sense, to the solution to the differential system obtained when φ is replaced with Φ(x)=1−ε x for sufficiently small ε. We finally perform a perturbation analysis of this latter solution for small ε. This allows us to check at the first order the validity of the so-called reduced service rate approximation, stating that everything happens as if the server rate were constant and equal to \(\mu(1-\varepsilon {\mathbb{E}}(X(t)))\) .


M/M/1 queue Self-adjoint operators Perturbation analysis Power series expansion Reduced service rate 

Mathematics Subject Classification (2000)



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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Christine Fricker
    • 1
  • Fabrice Guillemin
    • 2
  • Philippe Robert
    • 1
  1. 1.INRIA Paris—RocquencourtLe ChesnayFrance
  2. 2.Orange LabsLannionFrance

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