The principles of Renewal theory are seen in various stochastic processes such as Countable State Markov chains, Continuous Markov processes, Semi-Markov processes, Regenerative processes etc. In one sense, spotting a Renewal process in a complicated stochastic process makes life easy as one can use the concepts from renewal theory to talk about the ensemble average and limiting time averages of the process. This book belongs to the “classics” genre of Renewal theory. Let me attempt to summarize the main chapters of the book.
Chapter 1: Preliminaries
Renewal theory began as the study of some particular probability problems connected with the failure and replacement of components, such as electric light bulbs. Later it became clear that some of the same problems arise also in connection with many other applications of probability theory. The main random variable is X, the failure time. Actually you can think of X as any process that registers a new arrival epoch. The random variable can be discrete or continuous. The book by Feller covers the discrete case and this book is all about continuous case. One of the essential tools to analyze Renewals is “Laplace Transform”. One has to get friendly with Laplace and Inverse Laplace transforms to derived closed form solutions for specific distributions. The three most common distributions used in the book are Exponential, Special Erlangian and Gamma. Why restrict to only these? Well, for one thing, in order to get some insight in to the theory, you work with distributions that have closed form solutions. Secondly, all said and done, the renewal processes with inter-renewal distributions as exponential or gamma are one of the widely studied and applied stochastic processes. To think about it, what’s the point in trying in introducing a theory that has complicated inter-renewal distribution?.
Chapter 2: Fundamental Models
This chapter defines three processes associated with a Renewal process, i.e. inter renewal sequence of IIDS, the renewal epoch sequence and the renewal counting process. All the three describe the same system. It’s like giving three different views to the same process. For example if you take a simple Poisson arrival process where the counting process has a poisson random measure, the interarrival sequence is exponential, the renewal epoch sequence is special erlangian. It is fairly important to switch from one sequence to another at regular intervals so that you can see what’s going on in the process, as well as compute various statistics of interest.
The chapter talks about three types of Renewal processes
Ordinary Renewal process : First failure time has the same distribution as the subsequent failure times
Modified Renewal process / Delayed Renewal process : First failure time has a different distribution than that of subsequence failure times
Equilibrium Renewal process: First failure time has a specific form of distribution that gives nice asymptotic results.
Delayed renewal process is different from Ordinary renewal process in the assumption about the first renewal. In the ordinary renewal process, the first renewal occurs at time 0. In delayed renewal, the first renewal can have any distribution.
Having defined the process, the chapter lists the most common random variables that one needs to get a handle on, to understand any renewal process
Time up to rth renewal.
Number of renewals in time t (Nt).
The renewal function.
Renewal density – Although it is technically not a density.
Higher moments of Nt like expectation and variance.
Backward recurrence time (Age).
Forward recurrence time ( Residual life).
To give a taste of things to come in the book, all the above aspects are derived for a renewal process with inter-renewal times as exponential.
Chapter 3: Distribution of Renewals
This chapter derives the distribution of renewals, i.e. distribution of counting process and distribution of renewal epochs. Since they describe the same system, one can switch between two sequences from time to time to get the distribution of both the processes. Also the approach taken here shows the power of Laplace transforms, i.e. start off with a probability generating function, apply Laplace transform to get an expression that involves Laplace transform of the inter-renewal density function. The resulting expression is an equation between Laplace transform of the generating function of counting process and Laplace transform of the inter-renewal density function. This expression is generic in the sense that you can plug in LT of any inter-renewal distribution and you obtain the LT of renewal counting. All you have to do is to take inverse LT of the expression to get the distribution of renewals. So, the key takeaway is the generic expression of renewal distribution that involves LT of inter-renewal density. These expressions are used throughout the book. In fact the LT of generating functions of all the three types of renewal processes, i.e. Ordinary renewal, Modified renewal and Equilibrium renewal processes are derived. The chapter ends with a Central Limit theorem type statement for the number of renewals in time t.
Chapter 4: Moments of Number of Renewals
The recipe is the same as the previous chapter. Start off with the generating function for the whatever moment you are interested in, E(X^r), take the Laplace transform, get it to a form that involves Laplace transform of the inter-renewal distribution, and then take inverse Laplace to get the moment function. In the context of Renewal theory, the first moment has a special name called “Renewal function”. In fact a ton of problems are solved using a specific form involving renewal function, called the “Renewal equation”. This chapter derives the first moment and the second moment for all the three types of renewal processes, i.e. ordinary+delayed+equilibrium processes.
Although renewal equation is not dealt much in this book, there are a range of problems that can be solved using Renewal equation. The basic funda behind renewal equation is that you get a recurrence relation in discrete or continuous time after conditioning on the first renewal.
Chapter 5: Recurrence Times
In any renewal process, there are some basic questions that are of interest? When is the next renewal going to happen? If I land up at a particular time, what is the time since the last renewal? The former is called the residual life and the latter is the called the age. Both of these are random variables can be formulated using Renewal Reward theory and the distribution of both these variables can be computed. However this book does not use Renewal Reward theory, but builds up on the theory introduced in previous chapters, ie. use Laplace transforms. Somehow I felt this part could have been easily handled using Renewal reward theory instead of the painfully long computations given in the book.
Chapter 6: Superposition of Renewal Processes
This chapter talks about a process that is a result of more than one renewal processes. The resulting process obviously need not be a renewal process. However by assuming independence of constituents, one can come up with some closed form solutions. There is also a mention of the distribution of the interval between successive renewals. Note that since the resulting process is not a renewal process, the distribution is not always easy to derive.
Chapter 7: Alternating Renewal Processes
This is a relatively easier phenomenon to handle than the superposition case. The resulting process is still a renewal process and hence all the random variables that were derived for a single renewal process can be extended easily to this case. This chapter also gives a gentle introduction to Semi-Markov processes. It is in this chapter that renewal equation is used to solve the probability of the system in a particular state. Again solving a renewal equation is made easy by taking a LT of the entire equation.
Chapter 8: Cumulative Processes
At each of the renewal epochs, one can associate a random variable and there are many situations where one is interested in the cumulative sum of this random variable. Bulk arrivals, Poisson Markings, Poisson thinning etc are all examples of cumulative processes. The distribution and the first passage time for these cumulative processes are explored in this section.
Chapter 9 further generalized the renewal process assumptions. Chapter 10 and 11 deal with application of renewal processes to very specific problem areas such as models of failure and replacement strategies, more relevant to manufacturing context.
The core principles of the book are laid out in the first 9 chapters and take up 100 pages. I think it’s better to approach this book after having some decent fundas on Laplace transforms. In fact the cover page of the book illustrates the technique of contour integration that is relevant to obtain inverse Laplace transform. In any case, renewal theory principles are extremely important for someone interested in countable state Markov processes and Continuous time Markov processes.