On a Single-Server Finite Queuing Model with State-Dependent Arrival and Service Processes

This paper generalizes the M/G/1 queuing process by considering both the arrival and the service rates as being essentially arbitrary functions of the current number of customers in the system, and by assuming, moreover, that the amount of service demanded by a customer is conditioned by the queue length at the moment service is begun for that customer. Taking the imbedded Markov chain approach, the paper proposes a method for calculating (1) the limiting probability distribution of the congestion and (2) the expected value of the time needed to complete a service. An expression for the distribution of the waiting time is then given and its first moment deduced.