The Classical Networks

In this chapter, we discuss two families of queueing networks whose Markov processes are positive recurrent when ρ < e, and whose stationary distributions have explicit product-like formulas. The first family includes networks with the FIFO discipline, and the second family includes networks with the PS and LIFO disciplines, as well as infinite server (IS) networks. We introduced the first three networks in Section 1.2; we will define infinite server networks shortly. These four networks are sometimes known as the "classical networks". Together with their generalizations, they have had a major influence on the development of queueing theory because of the explicit nature of their stationary distributions. For this reason, we present the basic results for the accompanying theory here, although only the FIFO discipline is HL. These results are primarily from [BaCMP75], and papers by F. P. Kelly (such as [Ke75] and [Ke76]) that led to the book [Ke79].

In the remainder of the chapter, we present the background for these results and the accompanying theory. In Section 2.2, we give certain basic properties of stationary and reversible Markov processes on countable state spaces that we will use later. Sections 2.3 and 2.4 apply this material to obtain generalizations of the node-level results in Section 2.1 to the two families of interest to us. The first family, homogeneous nodes, includes FIFO nodes under certain restrictions, and the second family, symmetric nodes, includes PS, LIFO, and IS nodes.

The main source of the material in this chapter is [Ke79]. Section 2.2 is essentially an abridged version of the material in Chapter 1 of [Ke79]. Most of the material in Sections 2.3-2.5 is from Sections 3.1-3.3 of [Ke79], with [Wa88], [ChY01], [As03], and lecture notes by J.M. Harrison having also been consulted. The order of presentation here, starting with nodes in Sections 2.3 and 2.4 and ending with quasi-stationarity in Section 2.5, is different.