On a generic class of two-node queueing systems

This paper analyzes a generic class of two-node queueing systems. A first queue is fed by an on-off Markov fluid source; the input of a second queue is a function of the state of the Markov fluid source as well, but now also of the first queue being empty or not. This model covers the classical two-node tandem queue and the two-class priority queue as special cases. Relying predominantly on probabilistic argumentation, the steady-state buffer content of both queues is determined (in terms of its Laplace transform). Interpreting the buffer content of the second queue in terms of busy periods of the first queue, the (exact) tail asymptotics of the distribution of the second queue are found. Two regimes can be distinguished: a first in which the state of the first queue (that is, being empty or not) hardly plays a role, and a second in which it explicitly does. This dichotomy can be understood by using large-deviations heuristics

Extremes and fluid queues

In queueing theory, the main entity is a station where service is provided; examples include counters, call centers, elevators, and traffic lights. Customers seeking this service arrive at the system, where they wait if the service facility cannot immediately allocate the required amount of service. They depart after being served. In most of these applications, it is intrinsically uncertain at what time customers arrive and how long they need to be served. This explains why probability theory plays an important role in the analysis of queueing systems.\ud In order to design these systems optimally, it is desirable to know how this randomness influences the performance of the system. Therefore, it is crucial to study system characteristics that reflect this performance. For instance, one may wish to analyze the probability distributions of the queue length or the waiting time of a customer, as a function of the random arrival process and service requirements.\ud The investigation of such a performance measure starts with a reformulation of the problem into mathematical terms. On an appropriate level of abstraction, the analysis no longer involves queues; rather a problem of (applied) probability theory needs to be solved.\ud The system characteristics that we encounter in this thesis are all related to so-called extremes as a result of this translation. An inherent advantage of investigating queues through examining extremes is the wide applicability of the resulting theory. For instance, many of our results are also relevant for risk theory and financial mathematics; even though it may be unclear upfront how these fields relate to customers waiting in a line, extremes also play a pivotal role in these theories. Since our results are often illustrated with queueing examples, we first briefly discuss how queues are related to extremes