Distribution of the number of customers served during a busy period in a discrete time Geom/Geom/1 queue (Q1860689)

A discrete time single server queue, where input follows a Bernoulli distribution (so interarrival times have geometrical distribution) and service time is geometrical, is considered. Moreover the arrival customer is blocked until the (predefined) slot terminates. A known functional equation for the moment generating function is used to obtain both distribution and first moments of the number of customers served during busy period in an explicit form. The continuous time case is studied as the limit of the above model as length of the slot goes to zero.