Calculating time-dependent performance measures for the M/M/1 queue

Methods are discussed for computing transient performance measures for the M/M/1 queue. These performance measures are often expressed in terms of modified Bessel functions without any discussion about computation. In fact, a common expression for the probability transition function of the M/M/1 queue length process has an infinite sum of modified Bessel functions. For actually generating numbers, however, it is convenient to use numerical integration with associated integral representations, as was first pointed out by P.M. Morse (Oper. Res., vol.3, p.255-61, 1955)