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Polling systems have been used to model a large variety of applications and much research has been devoted to the derivation of efficient algorithms for computing the delay measures in these systems. Recent research efforts in this area, which have focused on the optimization of these systems, have raised the need for very efficient such algorithms.This work develops the descendant set approach as a general efficient algorithm for deriving all moments of customer delay (in particular, mean delay) in these systems. The method is applied to a very large variety of model variations, including:1) The exhaustive and gated service policies, 2) Fractional service policies, 3) The cyclic visit order, 4) Arbitrary periodic visit orders (polling tables), and 5) Customer routing. For mast of these variations the method significantly outperforms the algorithms commonly used today.

The MX/G/1 queueing system is studied under the following two situations: (1) At the end of a busy period, the server is turned off and inspects the length of the queue every time an arrival occurs. When the queue length reaches, or exceeds, a pre-specified value m for the first time, the server is turned on and serves the system until it is empty. (2) At the end of a busy period, the server takes a sequence of vacations, each for a random amount of time. At the end of each vacation, he inspects the length of the queue. If the queue length is greater than, or equal to, a pre-specified value m at this time, he begins to serve the system until it is empty. For both cases, the mean waiting time of an arbitrary customer for a given value of m is derived, and the procedure to find the stationary optimal policy under a linear cost structure is presented.