Rare event asymptotics for a random walk in the quarter plane

This paper presents a novel technique to derive asymptotic expressions for rare event probabilities for random walks in the quarter plane. For concreteness, we study a tandem queue with Poisson arrivals, exponential service times and coupled processors. The service rate for one queue is only a fraction of the global service rate when the other queue is non empty; when one queue is empty, the other queue has full service rate. The bivariate generating function of the queue lengths gives rise to a functional equation. In order to derive asymptotic expressions for large queue lengths, we combine the kernel method for functional equations with boundary value problems and singularity analysis.