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

Sven Erlander
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Sven Erlander: Stockholm University, Stockholm 23, Sweden

Operations Research, 1965, vol. 13, issue 5, 734-746

Abstract: 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 χ 1 ( 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 χ 1 ( 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 .

Date: 1965
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