A large family of semi-classical polynomials: the perturbed Chebyshev
Proceedings of the fourth international symposium on Orthogonal polynomials and their applications
Sojourn times in a processor sharing queue with service interruptions
Queueing Systems: Theory and Applications
Integration of streaming services and TCP data transmission in the Internet
Performance Evaluation - Performance 2005
Integration of streaming services and TCP data transmission in the Internet
Performance Evaluation - Performance 2005
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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 驴(X(t)), where 驴(x) is some bounded function and μ0. 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 驴(x) is defined by 驴(x)=1驴驴((x 驴 a/驴)驴(驴b/驴)) for some positive real numbers a, b and 驴, 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 驴 is replaced with 驴(x)=1驴驴 x for sufficiently small 驴. We finally perform a perturbation analysis of this latter solution for small 驴. 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 $\mu(1-\varepsilon {\mathbb{E}}(X(t)))$ .