The time-dependent Erlang loss model with retrials

We consider a generalization of the classical Erlang loss model with both retrials of blocked calls and a time-dependent arrival rate. We make exponential-distribution assumptions so that the number of calls in progress and the number of calls in retry mode form a nonstationary, two-dimensional, continuous-time Markov chain. We then approximate the behavior of this Markov chain by two coupled nonstationary, one-dimensional Markov chains, which we solve numerically. We also develop an efficient method for simulating the two-dimensional Markov chain based on performing many replications within a single run. Finally, we evaluate the approximation by comparing it to the simulation. Numerical experience indicates that the approximation does very well in predicting the time-dependent mean number of calls in progress and the times of peak blocking. The approximation of the time-dependent blocking probability also is sufficiently accurate to predict the number of lines needed to satisfy blocking probability requirements.