Two common properties of the Erlang-B function, Erlang-C function, and Engset blocking function (Q597413)

For the queueing systems with finite capacity a quantity of great interest is the probability \(P_B\), that all the servers are busy. The paper investigates this probability in three cases: (1) for the Erlang loss system M/M/\(m/m\), called Erlang B-function; (2) for the Erlang delay system M/M/\(m\), called Erlang C-function; (3) for the Engset blocking system with \(N\) sources and \(m\) servers and no waiting room (M/M/\(m/m/N\)), called Engset blocking function. In all formulae for \(P_B\) there appear the number of servers \(m\) and the fraction \(a=\lambda /\mu \), where \(\lambda \) and \(\mu \) are the arrival and service rates, respectively. Two intuitively clear properties are proved for these systems: the increasing number of servers reduces, the increasing traffic intensity increases the value of \(P_B\). Based on these properties and recursion for each of these three systems algorithms are developed to find the third one given any two parameters.