  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 tailed
probability vector) plays a key role in setting up mean-field differential
equations. To further apply the technique of tailed probability vector to deal
with resource management of large scale stochastic networks, this paper
discusses tailed 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 tailed 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 simply. 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.
