Exact Blocking Time Statistics for the Erlang Loss Model

The Erlang loss model is one of the fundamental tools in queueing theory with many applications to communications networks. For example, in a simple cellular voice network, the Erlang-B blocking formula is the traditional approach to model the proportion of time cellular base-stations are fully loaded in the busy hour. Such steady state results do not provide information on such important questions as: how likely is it that the blocking time in the busy hour exceeds some threshold. Hence we look in detail at the blocking time in the busy hour, or any finite period. We derive the exact distribution function and density as well as the moments and moment generating function of the blocking time, denoted X. In addition we derive the probabilities of zero blocking, P(X=0), and complete blocking, P(X=1 hour), in the busy hour.