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- Title
Perturbation analysis of an M/ M/1 queue in a diffusion random environment.
- Authors
Fricker, Christine; Guillemin, Fabrice; Robert, Philippe
- Abstract
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 $\mu(1-\varepsilon {\mathbb{E}}(X(t)))$ .
- Subjects
PERTURBATION theory; DIFFUSION; DENSITY functionals; QUEUING theory; DIFFERENTIAL equations; APPROXIMATION theory; FUNCTIONAL analysis
- Publication
Queueing Systems, 2009, Vol 61, Issue 1, p1
- ISSN
0257-0130
- Publication type
Article
- DOI
10.1007/s11134-008-9098-1