Heavily loaded queue coupled to two underloaded queues

We consider a system of three parallel queues with Poisson arrivals and expo- nentially distributed service requirements. The service rate for the heavily loaded queue depends on which of the two underloaded queues are empty. We derive the lowest-order asymptotic approximation to the joint stationary distribution of the queue lengths, in terms of a small parameter measuring the closeness of the heavily loaded queues and the heav- ily loaded queue have geometrically and, after suitable scaling, exponentially distributed lengths, respectively. The expression for the exponential decay rate for the heavily loaded queue involves the solution to an inhomogeneous linear functional equation. Explicit re- results are obtained for this decay rate when the two underloaded queues have vastly different arrival and service rates.