Review of

**Max Plus at Work: Modeling and Analysis of Synchronized Systems: A Course on Max-Plus Algebra and Its Applications**, by Bernd Heidergott, Geert Jan Olsder and Jacob van der Woude, Princeton University Press, Princeton, New Jersey, 2006. 213 pp., $80.00 (hardbound). ISBN 0691117632.

The definition of a max-plus algebra has several
components. Define two elements ɛ ≡ -∞ and

e ≡ 0, form the extended set R

a, b ɛ R

e ≡ 0, form the extended set R

_{max}as the real numbers unioned with { ɛ } and then for anya, b ɛ R

_{max}define the operations
a + b ≡ max(a,b) and a ×
b ≡ a + b, where the use of the ‘+’ here means standard addition.

Clearly, a + ɛ = ɛ + a =
a and a × ɛ = ɛ × a = ɛ. It is easy to verify that a max-plus algebra satisfies
many of the standard algebraic properties of numbers such as associativity,
existence of a zero and a unit element. The algebraic structure that it forms
is called a semiring.

**Note**: Altering the definitions of ɛ ≡ +∞ and a + b ≡ min(a,b) we have a min-plus algebra. The min-plus algebra is not considered in this book.

With this definition, the operations are
extended to the formation of vectors and matrices, with the standard operations
on matrices and the determination of eigenvalues and eigenvectors. Other
operations on matrices based on these operations are performed.

The matrices are then used to model systems of
sequential events on connected locations such as railway systems. Essentially they
can be used on everything that can be modeled as a flow through a network,
where there is some form of timetable or temporal distinction.

The value of the max-plus algebra is clear,
the example of the Dutch national railway system is used throughout the book to
illustrate how it is used. The level of difficulty is stated as at the last
year of undergraduate mathematics and this is accurate. Provided the student
has had sufficient exposure to graph theory and abstract and linear algebras,
the material will be understandable. Exercises are given at the end of the
chapters but no solutions are included.

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