By Bernd Heidergott
Trains pull right into a railroad station and needs to look forward to one another earlier than leaving back in an effort to enable passengers swap trains. How do mathematicians then calculate a railroad timetable that thoroughly displays their comings and goings? One process is to exploit max-plus algebra, a framework used to version Discrete occasion platforms, that are well matched to explain the ordering and timing of occasions. this can be the 1st textbook on max-plus algebra, offering a concise and self-contained advent to the topic.
Applications of max-plus algebra abound on the earth round us. site visitors platforms, machine communique structures, construction strains, and flows in networks are all in response to discrete even structures, and therefore could be comfortably defined and analyzed through max-plus algebra.
The booklet comprises an creation and 13 chapters in 3 components. half One explores the creation of max-plus algebra and of approach descriptions established upon it. half offers with a true software, particularly the layout of timetables for railway networks. half 3 examines numerous extensions, corresponding to stochastic structures and min-max-plus platforms. The textual content is acceptable 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
Example text
More precisely, in studying the shape of the upper contour the actual height of the heap is disregarded. To that end, the vector of relative differences in Xrt (w ), called the shape vector, is denoted by s( w ). 4 is obtained by letting the boldfaced line (the upper contour) sink to the ground level, yielding the vector s (w) = (0, 1, 1, 0, 0) T. More formally, the shape vector is defined as rER. Suppose that the sequence in which the pieces appear cannot be controlled (their arrivals may be triggered by an external source).
Hence, the shape is equal to v for any k. The projective space turns out to be a convenient mathematical space for speaking about limits of sequences { x(k) : k E N} stemming from max-plus recurrence relations. Notice that it actually has not been explained what it means when the above limit is said to exist. 4. 5 EXERCISES 1. 1 are indeed semirings. 2. Compute the following: (a) -8 ® E (b) c-1)®! ( ~fH} (c) the product of the next two matrices ( 4 2 -1 E E E 8 7 ~) 3. , Fermat's theorem is not true over Rmax· 4.
If a graph g = (N, 'D) is not strongly connected, then not all nodes of N communicate with each other. In this case, given a node, say, node i, it is possible to distinguish the subset of nodes that communicate with i from the subset of nodes that do not communicate with i. In the first subset all nodes communicate with each other, whereas in the second subset not all nodes necessarily communicate with each other. In the latter case a further subdivision of the nodes is possible. Repeated application of the previous idea therefore yields that the node set N can be partitioned as N1 UN2 U · · · UN'q, whereNr, r E q, denotes a subset of nodes that communicate with each other but not with other nodes of N.