In this chapter we first study how variability propagates through tandem networks of \(G/G/c\) queues in Section 7.1. We then explore deterministic queueing networks in Section 7.2, which, despite their simplicity, already pose interesting challenges. Next, we analyze open networks of \(M/M/c\) stations in Section 7.3. The analysis of such networks involves an equation of the type \(\lambda = \gamma + \lambda P\), where \(\lambda\) and \(\gamma\) are row vectors and \(P\) is a (stochastic) matrix. In Section 7.4, we concentrate on the solution of this equation.

We point out that the techniques developed in this chapter extend (way) beyond just queueing theory; they are worth memorizing. Thus, while this chapter closes our journey through the study of queueing systems, it is a first step toward a much longer journey into the diverse applications of probability theory. For further reading, we recommend the following books.

• You can find very nice discussions of networks of \(M/M/\infty\), chemical reactions, population dynamics, and Petri nets in J. C. Baez and J. Biamonte, Quantum Techniques for Stochastic Mechanics, 2019, which is freely available on arXiv.
• Simple queueing networks (networks that satisfy the so-called local balance) can be modeled as electrical networks. For this, see P. Doyle and J. Laurie Snell, Random Walks and Electrical Networks, 1984, which you can download for free from the homepage of Doyle.
• In more general terms, queueing systems or networks are examples of Markov processes. A particularly nice book on these topics is J. R. Norris, Markov Chains, 1997. The material of this chapter can be couched in the theory of martingales and optimal stopping, which find many applications in areas such as inventory theory, decision theory, and quantitative finance.