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 μϕ(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−εx 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-\eps\E(X(t)))