Abstract
To determine the maximum separation between events for nonrepetitive systems with max and linear constraints, there are the “iterative tightening from above” (ITA) approach and the “iterative tightening from below” (ITB) approach. Since such systems can be formulated as systems constrained by min–max inequalities, this paper gives an algorithm named MMIMaxSep for solving min–max inequalities. The algorithm is a generalization and a mathematically elegant reformulation of Yen et al.’s MaxSeparation algorithm which uses the ITB approach. Our numerical experiments indicate that MMIMaxSep is very efficient. Moreover, MMIMaxSep has a unique advantage of being able to directly handle tree-represented min–max functions, and its complexity is closely related to the complexity of computing cycle time of min–max functions.
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References
Baccelli F, Cohen G, Olsder GJ, Quadrat J-P (1992) Synchronization and Linearity. New York: Wiley
Borriello G (1988) A new interface specification methodology and its application to transducer analysis. Ph.D. Thesis, Univ. California at Berkeley
Brzozowski JA, Gahlinger T, Mavaddat F (1991) Consistency and satisfiablity of waveform timing specifications. Networks 1(1): 91–107
Casavant A (2000) Complexity of the MPL timing analysis algorithm. In: Proceedings ACM/IEEE International Workshop on Timing Issues in the Specification and Synthesis of Digital Systems (Tau). Austin
Chakraborty S, Yun KY, Dill DL (1999) Timing analysis of asynchronous systems using time separation of events. IEEE Transactions on CAD 18(8): 1061–1076
Cheng Y, Zheng D-Z (2004) A cycle time computing algorithm and its application in the structural analysis of min–max systems. Discrete Event Dyn Syst: Theory and Application 14(1): 5–30
Cheng Y, Zheng D-Z (2005) Min–max inequalities and the timing verification problem with max and linear constraints. Discrete Event Dyn Syst: Theory and Applications 15(2): 119–143
Cochet-Terrasson J, Gaubert S, Gunawardena J (1999) A constructive fixed point theorem for min–max functions. Dynam Stability Syst 14(4): 407–433
Cormen TH, Leiserson CE, Rivest RL (1990) Introduction to Algorithms. New York: McGraw-Hill
De Schutter B, De Moor B (1996) A method to find all solutions of a system of multivariate polynomial equalities and inequalities in the max algebra. Discrete Event Dynamic Systems: Theory and Applications 6(2): 115–138
De Schutter B, Heemels WPMH, Bemporad A (2002) On the equivalence of linear complemantarity problems. Oper Res Lett 30(4): 211–222
Gaubert S, Gunawardena J (1998a) The duality theorem for min–max functions. Comptes Rendus de l’Academie des Sciences 326: 43–48
Gaubert S, Gunawardena J (1998b) A non-linear hierarchy for discrete event dynamical systems. In: Proc. 4th Workshop on Discrete Event Systems. Cagliari, Italy
Gunawardena J (1994) Min–max functions. Discrete Event Dyn Syst 4:377–406
Heidergott B, Olsder GJ, van der Woude J (2006) Max Plus at Work. Princeton, NJ: Princeton University Press
Hulgaard H, Burns SM, Amon T, Borriello G (1995) An algorithm for exact bounds on time separation of events in concurrent systems. IEEE Transactions on Computer 44: 1306–1317
Jin F, Hulgaard H, Cerny E (1998) Maximum time separation of events in cyclic systems with linear and latest timing constraints. In: Formal Methods in Computer Aided Design: Second International Conference. Palo Alto, pp 167–184
Kohlberg E (1980) Invariant half-lines of nonexpansive piecewise-linear transformations. Math Oper Res 5(3): 366–372
McMillan KL, Dill DL (1992) Algorithms for interface timing verification. In: Proceedings Int. Conf. on Computer Design: VLSI in Computers and Processors, pp. 48–51
Olsder GJ (1991) Eigenvalues of dynamic max–min systems. Discrete Event Dyn Syst 1: 177–207
Olsder GJ, Perennes S (1997) Iteration of (Min,Max,+) functions. Draft (available online at http://citeseer.nj.nec.com/310205.html)
Walkup EA (1995) Optimization of linear max-plus systems with application to timing analysis. Ph.D. Thesis, Univ. of Washington
Walkup EA, Borriello G (1994) Interface Timing Verification with Combined Max and Linear Constraints. Tech. Rep. 94-03-04, Dept. of Computer Science, Univ. of Washington
Yen T, Ishii A, Casavant A, Wolf W (1998) Efficient algorithms for interface timing verification. Formal Methods in System Design 12(3): 241–265
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This work was supported by the National Natural Science Foundation of China under grant 60404010.
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Cheng, Y., Zheng, DZ. An Algorithm for Timing Verification of Systems Constrained by Min–max Inequalities. Discrete Event Dyn Syst 17, 99–129 (2007). https://doi.org/10.1007/s10626-006-0004-x
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DOI: https://doi.org/10.1007/s10626-006-0004-x