A time-dependent busy period queue length formula for the M/Ek/1M/Ek/1 queue

In this paper, a closed-form time-dependent busy period queue length probability is obtained for the M/Ek/1M/Ek/1 queue. This probability is frequently needed when we compare the length of the busy period and the maximum amount of service that can be rendered to the existing customers. The transient probability is given in terms of the generalized modified Bessel function of the second type of Griffiths et al. (2006a). The queue length probability for the M/M/1M/M/1 queue is also presented as a special case.