Stability of Queueing Networks

The approach we employ in this chapter to study the stability of queueing networks employs fluid limits and fluid models. Fluid models were discussed briefly in Section 1.3; we will examine both fluid limits and fluid models in detail here. Employing these tools, one can reduce the study of queueing networks to their simpler deterministic analogs. The basic theory is given here; applications will be given in Chapter 5.

Independently, [St95] and [Da95] developed criteria for the stability of queueing networks, in terms of the stability of limits of rescaled solutions of the network. (In the terminology of this chapter, [St95] used fluid limits and [Da95] used both fluid limits and fluid models.) [St95] assumed exponential distributions for the interarrival and service times; [Da95] considered more general distributions, but at the price of requiring the use of Markov processes with general state space. Applications illustrating this approach are given in [Da95].

The material in this chapter is based on the approach taken in [Da95]. The main theorem of the chapter, Theorem 4.16, corresponds to Theorem 4.2 in [Da95]. Our approach here is a modification of that in [Br98a]. Care has been taken to give a detailed presentation of the material, including that cited in [Da95]. As a consequence, the chapter is quite long; in the remainder of the introduction, we summarize its contents.