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Petri nets applications Part 3

Tham khảo tài liệu 'petri nets applications part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Systolic Petri Nets 71 Fig. 10. Dependency domain for matrix product 3.1.1.2 Step 2 Determining temporal equations The second step consists in determining all possible time functions for a system of uniform recurrent equations. A time function t is from DcZn Zn that gives the processing to perform at every moment. It must verify the following condition If xeD depends on yeD i.e. if a vector dependency i yx exists then t x t y . When D is convex analysis enables to determine all possible quasi-affine time functions. In this aim following definitions are used - D is the subset of points with integer coordinates of a convex poyedral D from Rn. - Sum pi.xi i 1.m is a positive combination of points x1 . xn from Rn if Vi Pl 0 - Sum ai.xi i 1.m is a convex combination of x1 . xn if Sum ai i 1.m 1 - s is a summit of D if s can not be expressed as a convex combination of 2 different points of D - r is a radius of D if VxeD VpieR x pi.r eD - a radius r of D is extremal if it can not be expressed as a positive convex combination of other radii of D. - l is a line of D if VxeD VpieR x pi.leD - if D contains a line D is called a cylinder If we limit to convex polyedral domains that are not cylinders then the set S of summits of D is unique as well as the set R of D extremal radii. D can then be defined as the subset of points x from Rn with x y z y being a convex combination of summits of S and z a positive combination of radii of R. Definition 1. T X a is a quasi-affine time function for D if V9e XT.9 1 VreR XT.r 0 VseS XT.s a Thus for the uniform recurrent equations system defining the matrix product X a time functions meets the following characteristics XT X1 X2 X3 with X1 1 X2 1 X3 1 and X1 X2 X3 1. 72 Petri Nets Applications A possible time function can therefore be defined by XT 1 1 1 with the following 3 radii 1 0 0 0 1 0 and 0 0 1 . 3.1.1.3 Step 3 Creating systolic architecture Last step of the method consists in applying an allocation function s of the network cells. .