Required work in the M/M/1 queue, with application in IP packet processing

We consider the ordinary M/M/1 queue with the FIFO queueing discipline. It seems that the sum of service times of the customers in the system (or the required work, as we call it briefly) is a random variable that is not cons idered before. In this paper we derive the equilibrium distribution of this variable. The task is not quite trivial because of the dependencies betwee n the elapsed service time and the number of customers in the system. Our m otivation for this problem comes from the performance analysis of a dynamic memory allocation scheme of a packet buffer.