Calculating transient characteristics of the Erlang loss model by numerical transform inversion

We show how to compute the time-dependent blocking probability given an arbitrary initial state, the distribution of the time that all servers first become busy given an arbitrary initial state, the time-dependent mean number of busy servers given an arbitrary initial state, and the stationary covariance function for the number of busy servers over time in the Erlang loss model by numerically inverting the Laplace transforms of these quantities with respect to time. Algorithms for computing the transforms are available in the literature, but they do not seem to be widely known. We derive a new revealing expression for the transform of the covariance function. We show that the inversion algorithm is effective for large systems by doing examples with up to 10,000 servers. We also show that computations for very large systems (e.g., 10⁶ servers) can be done with computations for moderately sized systems (e.g., 10²–10³ servers) and scaling associated with the heavy-traffic limit involving convergence of a normalized process to the reflected Ornstein–Uhlenbeck diffusion process. By the same reasoning, the Erlang model computations also can be used to calculate corresponding transient characteristics of the limiting reflected Ornstein–Uhlenbeck diffusion process.