The notes at the end of the course are slightly improved from those posted at the time of Lecture 1. I thank those of you who have spotted some typos.

You can now also look at the overhead projector slides that I sometimes used in lectures, such as those today that I used to summarise four Part II courses that you may like to study next year: Probability and Measure, Applied Probability, Optimization and Control, and Stochastic Financial Models.

In my discussion of random walk and electrical networks in Section 12.4 I appealed to

**Rayleigh's Monotonicity Law**: " if some resistances of a circuit are increased (decreased) the resistance between any two points of the circuit can only increase (decrease)." A proof of this "obvious" fact can be constructed by (i) proving Thomson's Principle: "Flows determined by Kirchhoff's Laws minimize energy dissipation", and then (ii) showing that Thomson's Principle implies Rayleigh's Monotonicity Law. You can read the details of this in Doyle and Snell Random walks and electric networks, pages 51–52.

In Section 12.2 I mentioned Burke's output theorem (1956) which says that the output process of a $M/M/1$ queue in equilibrium is a Poisson process with the same rate as the input. In writing "$M/M/1$" the "$M$"s mean Markovian (i.e. a Poisson input process of rate $\lambda$ and i.i.d. exponentially distributed service times with parameter $\mu$ (where $\mu > \lambda$), and the "$1$" means a single server. In queueing theory this very useful notation is known as Kendall's notation. For example, a $D/G/m$ queue is one in which arrivals are separated by a deterministic time, there are general service times and $m$ servers working in parallel.)

I remarked (just for fun) that

**queueing**is the only common word in the OED with five vowels in a row. Obscure words are ones like "miaoued" (what the cat did).

I once proved a generalization of Burke's output theorem that holds even when the queue has not reached equilibrium (see: The interchangeability of $\cdot/M/1$ queues in series, Weber, 1979). Suppose we have two single-server first-in-first-out queues in series, which we might write as $\cdot/M/1 \to /M/1$. The customers' service times in the first queue are i.i.d. exponentially distributed with parameter $\lambda$ and in the second queue they are i.i.d. exponentially distributed with parameter $\mu$. On finishing service in the first queue a customer immediately joins the second queue. Suppose the system starts with $N$ customers in the first (upstream) queue and no customers in the second (downstream) queue. My theorem says that all statistics that we might measure about the departure process from the second queue are the same if $\lambda$ and $\mu$ are interchanged. Thus by observing the process of departures from the second queue we cannot figure out which way around the two $/M/1$ servers are ordered. For example, the time at which we see the first departure leave the second queue has expected value $1/\lambda + 1/\mu$ (which is symmetric in $\lambda$ and $\mu$). Its moment generating function is $\lambda\mu(\theta-\lambda)^{-1}(\theta-\mu)^{-1}$. The expected time at which the second departure takes place is

$$

\frac{2 \lambda^2+3 \lambda\mu+2 \mu^2}{\lambda^2 \mu+\lambda \mu^2}.

$$All other statistics are also symmetric in $\lambda$ and $\mu$. It was by computing such quantities that I first formulated the conjecture that this might be true. Burke's theorem is a corollary of my result that can be obtained by thinking about $N$ tending to infinity (can you see how?)

Markov chains have been studied for about 100 years, but there are still interesting things to discover. An important on-going research area is Markov chain Monte Carlo.