Bayesian prediction of waiting times in stochastic models

The authors show how saddlepoint techniques lead to highly accurate approximations for Bayesian predictive densities and cumulative distribution functions in stochastic model settings where the prior is tractable, but not necessarily the likelihood or the predictand distribution. They consider more specifically models involving predictions associated with waiting times for semi-Markov processes whose distributions are indexed by an unknown parameter θ. Bayesian prediction for such processes when they are not stationary is also addressed and the inverse-Gaussian based saddlepoint approximation of Wood et al. (1993) is shown to accurately deal with the nonstationarity whereas the normal-based Lugannani & Rice (1980) approximation cannot. Their methods are illustrated by predicting various waiting times associated with M/M/q and M/G/1 queues. They also discuss modifications to the matrix renewal theory needed for computing the moment generating functions that are used in the saddlepoint methods.