The Ph-t/Ph-t/infinity queueing system: Part I - the single node

We develop a numerically exact method for evaluating the time-dependent mean, variance, and higher-order moments of the number of entities in a Ph-t/Ph-t/infinity queueing system. We also develop a numerically exact method for evaluating the distribution function and moments of the virtual sojourn time for any time t; in our setting, the virtual sojourn time is equivalent to the service time for virtual entities arriving to the system at that time t. We include several examples using software that we have developed and have put in downloadable form in the Online Supplement to this paper on the journal's website.