@MISC{Boudec02basicmin-plus, author = {Jean-yves Le Boudec and Patrick Thiran}, title = {Basic Min-plus and Max-plus Calculus}, year = {2002} }

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Abstract

In this chapter we introduce the basic results from Min-plus that are needed for the next chapters. Max-plus algebra is dual to Min-plus algebra, with similar concepts and results when minimum is replaced by maximum, and infimum by supremum. As basic results of network calculus use more min-plus algebra than max-plus algebra, we present here in detail the fundamentals of min-plus calculus. We briefly discuss the care that should be used when max and min operations are mixed at the end of the chapter. A detailed treatment of Min- and Max-plus algebra is provided in [26], here we focus on the basic results that are needed for the remaining of the book. Many of the results below can also be found in [11] for the discrete-time setting. 3.1 Min-plus Calculus In conventional algebra, the two most common operations on elements of Z or R are their addition and their multiplication. In fact, the set of integers or reals endowed with these two operations verify a number of well known axioms that define alge-braic structures: (Z,+,×) is a commutative ring, whereas (R,+,×) is a field. Here we consider another algebra, where the operations are changed as follows: addition