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Numerical Methods for Solving Discrete Event Systems

Winfried Grassmann and Javad Tavakoli
Publication Date: 
Number of Pages: 
CMS/CAIMS Books in Mathematics
[Reviewed by
Bill Satzer
, on
Discrete event systems describe the evolution of states in time where the states are discrete and event-driven and depend on the asynchronous occurrence of discrete events in time. Transitions from state to state in systems like this are essentially determined by mechanisms that are driven entirely by the events themselves. Natural examples of such systems are a queue of customers arriving and forming lines at a bank, management of inventory through a manufacturing facility, or movement of data through a packet-switched network like the Internet. Some of the systems the authors consider are repair depots (where responses may depend on several queues of available replacement parts).
Discrete event simulation has been used for some time to model such systems. In the early days of the Internet this was used extensively to estimate end-to-end-delay of data packets moving across the net and encountering many queues in tandem. At that time (several years ago when I was actively working with network designers to improve user response time), it was a important tool used to model interactions between hundreds of internet sites where today there are millions of such interactions. At the time – and even now – simulation was the most appropriate tool when the number of events is very large.
The current book considers how one might formulate and solve discrete event systems using numerical methods but without simulation of those discrete events. The authors focus on methods rather than models; their intended audience is those who have a model to solve and want to find methods to do that. Markov chains are the primary tools they use to formulate their approaches. They argue that their approach has special advantages, especially for smaller systems and where higher precision is needed 
This book presents algorithms that are deterministic with no randomization. (By contrast, discrete event simulation typically relies on randomness and Monte Carlo methods.) The authors formulate the systems they consider as Markov chains and solve them as such. Queueing systems are the primary examples of discrete event systems considered here. But the authors go beyond standard models of queueing theory. In particular they consider situations where arrival times at queues are not Poisson-distributed and service times are not exponentially distributed.
The book avoids advanced mathematical methods and as such is accessible to students with a good background in probability. By its nature, it is rather specialized and likely to be most useful to those interested in modeling small event-driven systems. The authors describe many variations on their methods and provide a number of numerical examples. It would have been helpful if they had also included some more specific examples showing where the additional precision of their methods has value.
Bill Satzer (, now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications. He did his PhD work in dynamical systems and celestial mechanics.