IN this chapter numerical the class of PHlPH however, as special cases of GIGI 1 queues; due to the specific methods efficient solution. known as matrix-geometric is not to present solution. efficient organised Instead, solution insight The aim of this chapter and their matrix-geometric geometric models, queueing methods, together models. is further with their all the known operation, material our aim is to show the usefulness and to show that techniques, are a good alternative we readdress to provide into their This chapter matrix-geometric play an important. | Performance of Computer Communication Systems A Model-Based Approach. Boudewijn R. Haverkort Copyright 1998 John Wiley Sons Ltd ISBNs 0-471-97228-2 Hardback 0-470-84192-3 Electronic Chapter 8 PH PH 1 queueing models IN this chapter we address the class of PH PH 1 queues. These queues can be seen as special cases of G G 1 queues however due to the specific distributions involved efficient numerical algorithms known as matrix-geometric methods can be applied for their solution. The aim of this chapter is not to present all the known material on PH PH 1 queues and their matrix-geometric solution. Instead our aim is to show the usefulness of matrixgeometric methods to provide insight into their operation and to show that PH PH 1 models together with their efficient solution techniques are a good alternative to G G 1 queueing models. This chapter is further organised as follows. In Section we readdress the analysis of the M M 1 queue in a matrix-geometric way . This is used as an introduction to the matrix-geometric analysis of the PH PH 1 queue in Section . Numerical algorithms that play an important role in the matrix-geometric technique are discussed in Section . We then discuss a few special cases in Section . In Section we discuss the caudal curve which plays an interesting role when studying the tail behaviour of queues. We finally comment on additional queueing models that still allow for a matrix-geometric solution in Section . The M M 1 queue Consider an M M 1 queueing model with arrival rate A and service rate p . The Markov chain underlying this queueing model is a simple birth-death process where the state variable denotes the total number of packets in the queueing station. Since the state variable is a scalar such a process is sometimes called a scalar state process. The generator matrix 146 8 PH PH 1 queueing models Figure The state-transition diagram for the M M 1 queue Q of the CTMC underlying the M M 1 queue has the .
