On the numerical solution of functional equations with application to response time distributions

A unified approach is developed to solve functional equations defining generating functions. Such equations are often constructed as a means to solve recurrence relations, such as those arising from queue length probabilities and response time probability distributions in Markov models. Many such equations have been obtained over several decades. Some have been solved analytically, some numerically and for some no tractable solution has been found. Our unified approach is able to provide accurate numerical solutions to such equations, even when they include derivatives of the generating function. It solves the JSQ model with two queues for the first time, utilizing a novel "partial" generating function related to the one the functional equation defines. Numerical results, displayed in tables of moments and graphs of probability density functions, show good accuracy against simulations with 500 000 regenerative cycles.