A representative of a major publishing house is on her way home from a conference in Singapore, excited about the possibility of a new book series. On the flight home to New York she opens her blackberry organizer, adding names of new contacts, and is disappointed to realize she may have caught the bug that was bothering her friend Alex at the cafè near the conference hotel. When she returns home she will send Alex an email to see how she's doing and to make sure this isn't a case of some new dangerous flu.

Of course, the publisher is aware that she is part of an interconnected network of other business men and women and their clients: Her value as an employee depends on these connections. She depends on the transportation network of taxis and airplanes to get her job done and is grateful for the most famous network today that allows her to contact her friend effortlessly even when separated by thousands of miles. Other networks of even greater importance escape her consciousness, even though consciousness itself depends on a highly interconnected fabric of neurons and vascular tissue. Communication networks are critical to support the air traffic controllers who manage the airspace around her. A supply chain of manufacturers makes her book business possible, as well as the existence of the airplane on which she is flying.

In the preceding chapters we learned that resources in a network must be shared among different classes of customers. This sharing is performed through scheduling at each station in the network.

Scheduling is just one of many decision processes encountered in typical applications. In the Internet there are many paths between nodes, and hence protocols must be constructed to determine appropriate routes. In a power distribution system there may be many generators that can meet current demands distributed across the power grid. A manufacturing system may have redundant processing equipment, or multiple vendors, and this then leads to a network somewhat more complex than those considered in the previous two chapters.

Figure 6.1 shows a network with eight nodes, four arrival streams, and ten links. The high congestion between nodes 1 and 3 can be modeled through an additional linear constraint on the rate vector ζ in a fluid model. There are many routes from node 1 to node 8, even though node 4 is temporarily unavailable. This example demonstrates that there may be many equilibria in a routing model; the best route for a given user will depend upon the current environment.

Most of the concepts introduced in previous chapters, such as stabilizability and workload relaxations, will be extended to this more general setting. Consideration is largely restricted to a fluid model since the definitions are most transparent in this setting.

Network models are used to describe power grids, cellular telecommunications systems, large-scale manufacturing processes, computer systems, and even systems of elevators in large office buildings. Although the applications are diverse, there are many common goals:

1. (i) In any of these applications one is interested in controlling delay, inventory, and loss. The crudest issue is stability: do delays remain bounded, perhaps in the mean, for all time?

2. (ii) Estimating performance, or comparing the performance of one policy over another. Performance is of course context-dependent, but common metrics are average delay, loss probabilities, or backlog.

3. (iii) Prescriptive approaches to policy synthesis are required. A policy should have reasonable complexity; it should be flexible and robust. Robustness means that the policy will be effective even under significant modeling error. Flexibility requires that the system respond appropriately to changes in network topology, or other gross structural changes.

In this chapter we begin in Section 1.1 with a survey of a few network applications, and the issues to be explored within each application. This is far from comprehensive. In addition to the network examples described in the Preface, we could fill several books with applications to computer networks, road traffic, air traffic, or occupancy evolution in a large building.

Although complexity of the physical system is both intimidating and unavoidable in typical networks, for the purposes of control design it is frequently possible to construct models of reduced complexity that lead to effective control solutions for the physical system of interest.

In this chapter we introduce many of the modeling and control concepts to be developed in this book through several examples. The examples in this chapter are extremely simple, but are intended to convey key concepts that can be generalized to more complex networks. We will return to each of these examples over the course of the book to illustrate various techniques.

A natural starting point is the single server queue.

Modeling the single server queue

The single server queue illustrated in Fig. 2.1 is a useful model for a range of very different physical systems. The most familiar example is the single-line queue at a bank: Customers arrive to the bank in a random manner, wait until they reach the head of the line, are served according to their needs and the abilities of the teller, and then exit the system. In the single server queue we assume that there is a single line, and only one bank teller. To understand how delays develop in this system we must look at average time requirements of customers and the rate of arrivals to the bank. Also, variability of service times or interarrivals times of customers has a detrimental effect on average delays.

Even in this very simple system there are control and design issues to consider. Is it in the best interest of the bank to reserve a teller to take care of customers with short time requirements?