Tailed Probabilities in Queueing Processes

In the study of large scale stochastic networks with resource management, differential equations and mean-field limits are two key techniques. Recent research shows that the expected fraction vector (that is, the tail probability vector) plays a key role in setting up mean-field differential equations. To further apply the technique of tail probability vector to deal with resource management of large scale stochastic networks, this paper discusses tail probabilities in some basic queueing processes including QBD processes, Markov chains of GI/M/1 type and of M/G/1 type, and also provides some effective and efficient algorithms for computing the tail probabilities by means of the matrix-geometric solution, the matrix-iterative solution, the matrix-product solution and the two types of RG-factorizations. Furthermore, we consider four queueing examples: The M/M/1 retrial queue, the M(n)/M(n)/1 queue, the M/M/1 queue with server multiple vacations and the M/M/1 queue with repairable server, where the M/M/1 retrial queue is given a detailed discussion, while the other three examples are analyzed in less detail. Note that the results given in this paper will be very useful in the study of large scale stochastic networks with resource management, including the supermarket models and the work stealing models.