The Remaining Busy Period for a Single Server Queue with Poisson Input

Consider a single-server queuing process with Poisson input and general service time distribution. Define the remaining busy period χ(t) as the time from a given instant t until the server becomes idle for the first time. Let ξ(t) be the number of customers in the system at time t. Let χ₁(t) be the remaining service time for the customer being served at time t. In this paper we shall study the stochastic law of the remaining service time χ₁(t) given that ξ(0) = i and ξ(t) = r. The result will then be applied to the problem of finding the joint distribution of the remaining busy period χ(t) and the number of customers served during χ(t) given that ξ(0) = i and ξ(t) = r.