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?

This chapter develops extensions of the fluid and stochastic network models to capture a wider range of activities. As in the previous chapters we allow scheduling and routing. In the demand-driven models considered in this chapter we also permit “admission control” of raw material arriving to the network so that the total amount of material in the system can be regulated. Although manufacturing systems will motivate most of the discussion in this chapter, power distribution systems as described in Section 2.7 and some communication systems can be modeled as demand-driven networks.

Figure 7.1 illustrates a typical example of the class of models to be investigated. In this 16-buffer network there are two sources of exogenous demand, and the release of two different types of raw material into the system is controlled. At two of the five stations there are multiple buffers so that scheduling is required, and routing is controlled at the exit of Station 3.

This is the most complex example considered in any detail in this book, although it is far simpler than a typical semiconductor wafer fab as described in Section 1.1.1. The International Semiconductor Roadmap for Semiconductors (ITRS) provides an annual assessment of the challenges facing the semiconductor industry [279]. In recent years their reports have contained some recurring themes:

Contention for resources There may be dozens of different product flows in a single factory.

Chapter 4 touches on many of the techniques to be developed in this book for controlling large interconnected networks. The fluid model was highlighted precisely because control is most easily conceptualized when variability is disregarded. The infinitehorizon optimal control problem with objective function defined in (4.37) can be recast as an infinite-dimensional linear program when c is linear. In many examples, such as the simple re-entrant line introduced in Section 2.8, a solution is explicitly computable. The MaxWeight policy and its generalizations are universally stabilizing, in the sense that a single policy is stabilizing for any CRW scheduling model satisfying the load condition ρ• < 1 along with the second moment constraint E[∥A(1)∥2] < ∞.

What is missing at this stage is any intuition regarding the structure of “good policies” for a network with many stations and buffers. In this chapter we introduce one of the most important concepts in this book, the workload relaxation. Our main goal is to construct a model of reduced dimension to simplify computation of policies, and to better visualize network behavior.

In the theory of optimization, a relaxation of a given model is simply a new model obtained by removing constraints. In the case of networks there are several classes of constraints that complicate analysis:

1. (i) The integer constraint on buffer levels.

2. (ii) Constraints on the allocation sequence determined by the constituency matrix.

3. (iii) State space constraints, including positivity of buffer levels, as well as strict upper limits on available storage.

The chief motivation for performance evaluation is to compare candidate policies. For example, many of the policies described in Chapters 4 and 10 depend upon static or dynamic safety-stock parameters, and we would like to know how to choose the best parameter values in order to optimize performance.

We have seen in Chapter 8 that linear programming techniques can provide bounds on performance for the CRW model. This approach can be successfully applied in network models with many buffers and stations. However, linear programming techniques are not flexible with respect to the operating policy. For example, in order to distinguish similar policies with different safety-stock levels, constraints must be introduced in the LP specific to each safety-stock parameter. It is not clear how to introduce such constraints in the approaches that have been developed to date.

While not a topic of this book, there are classes of networks for which the invariant measure π is known. The crucial property required is reversibility, from which it follows that π has a product form, π(x) = π1(x1) … πℓ(xℓ) for x ∈ X [502, 290]. Outside of this very special class of models the computation of π is essentially impossible in large networks. We are thus led to simulation techniques to evaluate performance.

The simulation techniques surveyed in this chapter all involve a Markov chain X on a state space X.

In social contexts, the diffusion of information and/or behavior often exhibits features that do not match well those of the epidemic models discussed in Chapter 3. This concerns, specifically, the transmission mechanism contemplated by those models, which was assumed to be independent of the local (neighborhood) conditions faced by the agents concerned.

In the epidemic formulation of diffusion, the transmission of infection (or information) to a healthy (or uniformed) agent is tailored to her total exposure, i.e. the absolute number of infected neighbors. But in the spread of many social phenomena – mainly if there is a factor of persuasion or coordination involved – relative considerations tend to be important in understanding whether some new behavior or belief is adopted. Generically, we shall speak of these relative considerations operating on the diffusion process as neighborhood effects. They are the object of the present chapter, where we study their implications in a number of different setups.

NEIGHBORHOOD-DEPENDENT DIFFUSION IN RANDOM NETWORKS

In this starting section, we revisit the epidemic models considered in Chapter 3, now under neighborhood effects. First, in Subsection 4.1.1, the focus is on a SIR-like context where diffusion is irreversible and thus one may naturally conceive of diffusion as occurring through a wave of a certain reach. Next, in Subsection 4.1.2, we turn to a setup where diffusion spreads in a way akin to SIS-epidemics, with adoption/infection being only temporary. In this latter case, the concern is the extent of long-run prevalence.

This chapter initiates our study of diffusion and models it as a process whose spreading mechanism is independent of any neighborhood considerations. This means that the procedure by which the process propagates from a certain node to any one of its neighbors is unaffected by the conditions prevailing in the neighborhoods of those two nodes – thus, in particular, it is unrelated to the states displayed by their other neighboring nodes. In this sense, one can conceive the phenomenon as akin to biological infection, a process that is often mediated through local contact at a rate that depends on the aggregate exposure to infected neighbors. With this analogy in mind, we shall label such a process as epidemic diffusion. But, of course, diffusion in socioeconomic environments is often different, subject to neighborhood (as well as payoff-related) considerations. To study it under these conditions, therefore, we need a different framework of analysis, which is introduced in Chapter 4.

ALTERNATIVE THEORETICAL SCENARIOS

Epidemiology is an old and well-established field of research, both empirical and theoretical. Its canonical models fall into two categories:

• SIR (susceptible-infected-recovered), where the life history of each node (or agent) passes from being susceptible (S), to becoming infected (I), to finally being recovered (R), always moving in a unidirectional fashion.

• SIS (susceptible-infected-susceptible), where each node passes from being susceptible (S) to turning infected (I), to becoming again susceptible (S), thus allowing for a bidirectional transition between the two possible states.

The previous chapter has modeled diffusion and play as “reactive” phenomena, i.e. processes that unfold while agents respond to their current neighborhood conditions. In a polar fashion, another important (but “proactive”) phenomenon that is often mediated through the social network is search. A paradigmatic instance of it arises when a certain agent/node faces a problem (or query) whose solution (answer) is to be found somewhere else in the social network. This is reminiscent of the famous experiment set up by Milgram (1967) [200] – recall Section 1.1 – where the task was to direct a letter to a “distant” individual through a chain of social acquaintances. In present times, the internet represents a search scenario where analogous issues appear. In this case, a typical problem consists of finding a desired piece of information by searching through the hyperlinks that connect the various webpages in the huge WWW network.

The effectiveness of search in these setups is inherently affected by network considerations. In general, of course, the way agents access disperse information must be crucially shaped by the architecture (topological characteristics) of the social network. Another consideration, equally important, is the knowledge that agents have on the social network itself. In line with our emphasis on complexity, the natural assumption to make in this respect is that such information is purely local. Sometimes, however, agents may be in the position to rely on some underlying “reference structure” (e.g. the arrangement of individuals along spatial or professional dimensions) to guide and thus improve their search.