Product-form solutions for a class of structured multi-dimensional Markov processes

Motivated by queueing systems with heterogeneous parallel servers, we consider a class of structured multi-dimensional Markov processes whose state space can be partitioned into two parts: a finite set $V$ containing boundary states and a set $W$, which has one infinite dimension and a fixed number of finite dimensions. Using a separation of variables technique we show that the equilibrium distribution can be represented as a linear combination of product forms. For an important subclass of queueing systems, we characterize explicitly the waiting time distribution in terms of a mixture of exponentials.