László Lakatos

Introduction to Queueing Systems with Telecommunication Applications

The book is composed of two main parts: mathematical background and queueing systems with applications. The mathematical background is a self containing introduction to the stochastic processes of the later studies queueing systems. It starts with a quick introduction to probability theory and stochastic processes and continues with chapters on Markov chains and regenerative processes. More recent advances of queueing systems are based on phase type distributions, Markov arrival processes and quasy birth death processes, which are introduced in the last chapter of the first part. The second part is devoted to queueing models and their applications. After the introduction of the basic Markovian (from M/M/1 to M/M/1//N) and non-Markovian (M/G/1, G/M/1) queueing systems, a chapter presents the analysis of queues with phase type distributions, Markov arrival processes (from PH/M/1 to MAP/PH/1/K). The next chapter presents the classical queueing network results and the rest of this part is devoted to the application examples. There are queueing models for bandwidth charing with different traffic classes, slotted multiplexers, ATM switches, media access protocols like Aloha and IEEE 802.11b, priority systems and retrial systems. An appendix supplements the technical content with Laplace and z transformation rules, Bessel functions and a list of notations. The book contains examples and exercises throughout and could be used for graduate students in engineering, mathematics and sciences.

Some Aspects of Waiting Time in Cyclic-Waiting Systems

We consider a queueing system with Poisson arrivals and exponentially distributed service time and FCFS service discipline. The service of a customer is started at the moment of arrival (in case of free system) or at moments differing from it by the multiples of a given cycle time T (in case of occupied server or waiting queue). The waiting time is always the multiple of cycle time T, one finds its generating function and mean value. The characteristics of service are illustrated by numerical examples. If we measure the waiting time by means of number of cycles, we can optimize the cycle time T.

On the Waiting Time in the Discrete Cyclic–Waiting System of Geo/G/1 Type

We continue to examine a discrete time queueing system where the service of a customer may start at the moment of arrival or at moments differing from it by the multiples of a given cycle time. We find the distribution and the mean value of waiting time in the case of general service time distribution.

Introduction to Queueing Systems

The theory of queueing systems dates back to the seminal work of A. K. Erlang (1878–1929), who worked for the telecom company in Copenhagen and studied telephone traffic in the early twentieth century. To this day the terminology of queueing theory is closely related to telecommunications (e.g., channel, call, idle/busy, queue length, utilization).

On the Queue Length in the Discrete Cyclic-Waiting System of Geo/G/1 Type

We consider a discrete time queueing system with geometrically distributed interarrival and general service times, with FCFS service discipline. The service of a customer is started at the moment of arrival (in case of free system) or at moments differing from it by the multiples of a given cycle time T (in case of occupied server or waiting queue). Earlier we investigated such system from the viewpoint of waiting time, actually we deal with the number of present customers. The functioning is described by means of an embedded Markov chain considering the system at moments just before starting the services of customers. We find the transition probabilities, the generating function of ergodic distribution and the stability condition. The model may be used to describe the transmission of optical signals.

A Discrete Waiting Time Model for Optical Signals

We consider a discrete time queueing system where the service of a customer may start at the moment of arrival or at moments differing from it by the multiples of a given cycle time. One finds the distribution of waiting time and its mean value. These results give possibility for the numerical optimization of cycle length. The original model was raised in connection with the landing process of airplanes, but it appears to be an exact model to describe the functioning of a node at the transmission of optical signals.

Markovian Queueing Systems

Queueing systems whose underlying stochastic process is a continuous-time Markov chain (CTMCs) are the simplest and most often used class of queueing systems. The analysis of these systems is based on the essential results available for the analysis of CTMCs. As a consequence, several interesting properties of these queueing systems can be described by simple closed-form analytical expressions both in transient (as a function of time and initial state) and in steady state.

Introduction to Stochastic Processes

When considering technical, economic, ecological, or other problems, in several cases the quantities \(\left \{{X}_{t},\;t \in \mathcal{T}\right \}\) being examined can be regarded as a collection of random variables. This collection describes the changes (usually in time and in space) of considered quantities. If the set \(\mathcal{T}\) is a subset of the set of real numbers, then the set \(\left \{t \in \mathcal{T}\right \}\) can be interpreted as time and we can say that the random quantities X t vary in time. In this case the collection of random variables \(\left \{{X}_{t},\;t \in \mathcal{T}\right \}\) is called a stochastic process. In mathematical modeling of randomly varying quantities in time, one might rely on the highly developed theory of stochastic processes.

Applied Queueing Systems

Traditional telephone networks were designed to implement a single type of communication service, i.e., the telephone service. Today’s telecommunication networks implement a wide range of communication services. In this section we introduce Markov models of communication services that compete for the bandwidth of a finite-capacity communication link.

Markov Chains with Special Structures

The previous chapter presented methods for analyzing stochastic models where some of the distributions were other than exponential. In these cases the analysis of the models is more complex than the analysis of Markov models. In this chapter we introduce a methodology to extend the set of models that can be analyzed by Markov models while the distributions can be other than exponential.