The Erlang B formula represents the steady-state blocking probability in the Erlang loss model or M/M/s/s queue. We derive asymptotic expansions for the offered load that matches, for a given number of servers, a certain blocking probability. In addressing this inversion problem we make use of various asymptotic expansions for the incomplete gamma function. A similar inversion problem is investigated for the Erlang C formula.

We consider the random variable Z n,α = Y 1 + 2 α Y 2 + . . . + n α Y n , with α ∈ R and Y 1 , Y 2 , . . . independent and exponentially distributed random variables with mean one. The distribution function of Z n,α is in terms of a series with alternating signs, causing great numerical difficulties. Using an extended version of the saddle point method, we derive a uniform asymptotic expansion for P(Z n,α < x) that remains valid inside (α ≥ −1/2) and outside (α < −1/2) the domain of attraction of the central limit theorem. We discuss several special cases, including α = 1, for which we sharpen some of the results in [7].