On the diffusion approximation to two parallel queues with processor sharing

A two-dimensional queuing model is considered in which the server in the second queue helps the server in the first queue during periods when the second queue is empty. The system is analyzed in the heavy traffic limit and explicit approximate solutions are obtained to the resulting diffusion equations using singular perturbation methods. Approximate asymptotic formulas are obtained for the stationary distribution of the number of customers as well as for some first-passage-time problems associated with the busy period. It is shown that these formulas reduce to the asymptotic expansions of the exact solutions, when the latter are available