On the remaining service time upon reaching a given level in M/G/1 queues

The ability of effectively finding the distribution of the remaining service time upon reaching a target level in M/G/1 queueing systems is of great practical importance. Among other things, it is necessary for the estimation of the Quality-of-Service (QoS) provided by Asynchronous Transfer Mode (ATM) networks. The previous papers on this subject did not give a comprehensive solution to the problem. In this paper an explicit formula for this distribution is given. This formula is general as it includes any initial level of the length of the queue, any type of service distribution (heavy tails) and any traffic intensity ρ. Moreover, it is easy to use and fast in computation. To show this several numerical examples are presented. In addition, a solution of the similar problem in G/M/1 queues (which is the distribution of the remaining interarrival time) is given.