Symbolic moment calculation for the sojourn time in M/G/1 queues with Bernoulli feedback

In queueing systems with feedback of output customers, the time spent by a customer in the system from arrival to final departure (the sojourn time) is of a primary interest. The first and second moments of the sojourn time in a first-come first-served M/G/1 queue with Bernoulli feedback were given by Takacs [1962]. By using the symbolic formula manipulation functions of Mathematica, we can calculate higher-order moments of the sojourn time for Takacs's model. We show a Mathematica program for this calculation as well as the explicit result for the third moment. The moments of the sojourn time can also be obtained for the exhaustive and gated service systems with multiple server vacations. An attempt shown in this paper suggests a new technique for symbolic computational queueing theory.