Stationary-process approximations for the nonstationary Erlang loss model

In this paper we consider the M(t)/G/s/0 model, which has s servers in parallel, no extra waiting space, and i.i.d. service times that are independent of a nonhomogeneous Poisson arrival process. Arrivals finding all servers busy are blocked (lost). We consider approximations for the average blocking probabilities over subintervals (e.g., an hour when the expected service time is five minutes) obtained by replacing the nonstationary arrival process over that subinterval by a stationary arrival process. The stationary-Poisson approximation, using a Poisson (M) process with the average rate, tends to significantly underestimate the blocking probability. We obtain much better approximations by using a non-Poisson stationary (G) arrival process with higher stochastic variability to capture the effect of the time-varying deterministic arrival rate. In particular, we propose a specific approximation based on the heavy-traffic peakedness formula, which is easy to apply with either known arrival-rate functions or data from system measurements. We compare these approximations to exact numerical results for the M(t)/M/s/0 model with linear arrival rate.