Response time distributions and network perturbation into product-form

Two new methodological results are obtained: first, a way to perturb a network into one with a product-form solution for its equilibrium state probabilities, and secondly, a new compositional approach to deriving corresponding response time distributions. The Reversed Compound Agent Theorem (RCAT) is used to construct suitable perturbations in near-product-form networks that render them separable by satisfying the conditions of this theorem. Response time calculations in stochastic networks are usually developed in terms of sample path analyses beginning in an equilibrium state. We consider the joint probability distribution of the sojourn times of a tagged task at each node of a path in a network and observe that this is the same in both the forward and reversed processes. Therefore if the reversed process is known, each node-sojourn time can be taken from either process. In particular, the reversed process can be used for the first node in a path and the forward process for the other nodes in a recursive analysis. This approach derives, quickly and systematically, existing results for response time probability densities in tandem, open and closed tree-like, and overtake-free Markovian networks of queues. An example shows that the technique is far more widely applicable, constructing a perturbed network with productform, a new result in its own right, and then finding a very simple expression for its response time probability density function.