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.

Traditional courses for engineers in filtering and signal processing have been based on elementary linear algebra, Hilbert space theory and calculus. However, the key objective underlying such procedures is the (recursive) estimation of indirectly observed states given observed data. This means that one is discussing conditional expected values, given the observations. The correct setting for conditional expected value is in the context of measurable spaces equipped with a probability measure, and the initial object of this book is to provide an overview of required measure theory. Secondly, conditional expectation, as an inverse operation, is best formulated as a form of Bayes’ Theorem. A mathematically pleasing presentation of Bayes’ theorem is to consider processes as being initially defined under a “reference probability.” This is an idealized probability under which all the observations are independent and identically distributed. The reference probability is a much nicer measure under which to work. A suitably defined change of measure then transforms the distribution of the observations to their real world form. This setting for the derivation of the estimation and filtering results enables more general results to be obtained in a transparent way.

I. J. Good (1950) has shown how we can use probability theory backwards to measure our own strengths of belief about propositions. For example, how strongly do you believe in extrasensory perception?

Extrasensory perception

What probability would you assign to the hypothesis that Mr Smith has perfect extrasensory perception? More specifically, that he can guess right every time which number you have written down. To say zero is too dogmatic. According to our theory, this means that we are never going to allow the robot's mind to be changed by any amount of evidence, and we don't really want that. But where is our strength of belief in a proposition like this?

Our brains work pretty much the way this robot works, but we have an intuitive feeling for plausibility only when it's not too far from 0 db. We get fairly definite feelings that something is more than likely to be so or less than likely to be so. So the trick is to imagine an experiment. How much evidence would it take to bring your state of belief up to the place where you felt very perplexed and unsure about it?

The term ‘paradox’ appears to have several different common meanings. Székely (1986) defines a paradox as anything which is true but surprising. By that definition, every scientific fact and every mathematical theorem qualifies as a paradox for someone. We use the term in almost the opposite sense; something which is absurd or logically contradictory, but which appears at first glance to be the result of sound reasoning. Not only in probability theory, but in all mathematics, it is the careless use of infinite sets, and of infinite and infinitesimal quantities, that generates most paradoxes.

In our usage, there is no sharp distinction between a paradox and an error. A paradox is simply an error out of control; i.e. one that has trapped so many unwary minds that it has gone public, become institutionalized in our literature, and taught as truth. It might seem incredible that such a thing could happen in an ostensibly mathematical field; yet we can understand the psychological mechanism behind it.

How do paradoxes survive and grow?

As we stress repeatedly, from a false proposition – or from a fallacious argument that leads to a false proposition – all propositions, true and false, may be deduced.