On a Queuing Model with Customers Arriving According to a Diffusion Process

In the case of heavy traffic, the arrival process in a queuing model can be approximated by a diffusion process. In this paper, we consider such a model. The number X(t) of customers in the system at time t evolves according to a degenerate two-dimensional diffusion process. In a special case, the distribution of X(t) is calculated explicitly. Moreover, a stochastic control problem known as a homing problem is formulated, and the equation satisfied by the value function is given. Particular homing problems are solved explicitly.