Predicting departure times in multi-stage queueing systems

We develop an approximation model for the state-dependent sojourn time distribution of customers or orders in a multi-stage, multi-server queueing system, when interarrival and service times can take on general distributions. The model can be used to make probabilistic statements about the departure time of a customer or order, given the number and location of customers currently in process or waiting, and these probabilities can be recomputed while waiting at any point during the sojourn time. The model uses phase-type distributions and a new method to estimate the remaining processing times of customers in service when the sojourn time distribution is computed.