Instability of Subcritical Queueing Networks

This chapter is broken into three sections, according to the order of appearance and content of the examples. In Section 3.1, the first basic examples of unstable subcritical queueing networks are given. These consist of examples in [LuK91] and [RyS92] for static priority disciplines and [KuS90] for a clearing policy. In Section 3.2, examples of unstable subcritical FIFO networks are given, which consist of examples in [Br94a,b] and [Se94]. Section 3.3 discusses examples for other disciplines that illustrate features of interest. They consist of an unstable network of Kelly type from [DaWe96], and examples from [Du97] and [Ba98], where the region of stability of certain networks is examined in greater detail.

The intent of this chapter is to provide an elementary introduction to the subject and at the same time impart some feeling for the development of this subject in the 1990's. An omission with regard to the latter is the interaction with contemporary developments for heavy traffic limits. This includes the example in [DaWa93], which highlighted the general lack of understanding of the asymptotics for even simple multiclass queueing networks. This material requires additional terminology and new concepts. Since it lies outside the scope of these lectures, we omit it with some regret.