Max Plus at Work: Modeling and Analysis of Synchronized by Bernd Heidergott

By Bernd Heidergott

Trains pull right into a railroad station and needs to stay up for one another ahead of leaving back so as to enable passengers switch trains. How do mathematicians then calculate a railroad timetable that effectively displays their comings and goings? One procedure is to take advantage of max-plus algebra, a framework used to version Discrete occasion structures, that are compatible to explain the ordering and timing of occasions. this can be the 1st textbook on max-plus algebra, offering a concise and self-contained creation to the topic.

Applications of max-plus algebra abound on the earth round us. site visitors structures, computing device conversation platforms, creation traces, and flows in networks are all in response to discrete even platforms, and therefore could be with ease defined and analyzed via max-plus algebra.

The e-book involves an advent and 13 chapters in 3 elements. half One explores the creation of max-plus algebra and of method descriptions established upon it. half bargains with a true software, particularly the layout of timetables for railway networks. half 3 examines quite a few extensions, comparable to stochastic structures and min-max-plus platforms. The textual content is appropriate for last-year undergraduates in arithmetic, and every bankruptcy offers workouts, notes, and a reference part.

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Extra info for Max Plus at Work: Modeling and Analysis of Synchronized Systems: A Course on Max-Plus Algebra and Its Applications

Sample text

Corresponding to, say [5], has cyclicity 3. 's have cyclicity 1 by definition. 's. Hence, O"Q(A) = 6. 3. 3. Based on the reduced graph, let [i1] = [8] = {8}, [i2] = [9] = {9}, [i 3] = [5] = {5, 6, 7}, [i 4 ] = [10] = {10}, and [i 5 ] = [2] = {1, 2, 3, 4}. org/terms 36 CHAPTER2 another; then the following normal form of A is the result 'I c- I cc lc c c c c c c c c c c c c c c c c 9 c c c c c c \ c c c c c c c c c c c 0 c c c c c c c c c c c c c c c c c c c 0 6 -5 ~ c c c c c 9 c c c c 16 c c c 0 c 4 c c c c c c c c c c c c c c c -3 c c c 0 c In particular; the diagonal blocks of this normal form of A, starting in the upper left corner and going down to the lower right corner; are given by An, A22, A 33 , A44, and A55, respectively.

What is the maximum weight of a path from 1 to 3 of length 4? 8. 1. 9. Let A=(::). Show that A has eigenvalues c and e, and give one corresponding eigenvector for each eigenvalue. 10. For the next matrices A, investigate the existence and uniqueness of a solution of the equation x = A 181 x EB b with b = u. If a solution exists, give the complete solution set of the equation. A=( -2 -2 -1 11. 11, and verify that this vector is the greatest solution of the inequality A 181 x :::; b. Show that for the inequality A 181 x ~ b no (unique) smallest solution exists.

This is just like in conventional algebra with + replaced by max and x by + . , has n rows and m columns. 8); then the elements of A® Bare given by [A® B] 11 = e ® ( -1) EB e ® 1 = max(O- 1, -oo + 1) = -1, [A® Bh2 = e ® 11 EB e ® e = max(O + 11, -oo- oo) = 11, [A®B]2 1 =3®(-1)EB2®1 =max(3-1,2+1) = 3, and [A® B]22 =-3 ® 11 EB 2 ® e = max(3 + 11,2- oo) = 14, yielding, in matrix notation, A® B = ( -;1 ~~ ) . Notice that the matrix product in general fails to be commutative. Indeed, for the above A andB B ® A = ( 1; 1e3 ) =/= A ® B.

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