Studies On Batch Arrival M~X/M/1Queue In A Diffusion Random Environment

We study in this paper an Mx/M/l queue whose server rate depends on the state of an independent OU diffusion process{X(t), t∈[O,T]} so that its value at time t is f is μΦ(X(t)), where Φ(x) is some bounded function and μ> O. We establish the differential system for the conditional probability density functions of the couple (X(t),L(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((xΛ(a/ε))V(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 in this paper, which provides a foundation for the study of the system performance.