Abstract
We consider in this paper a set of generic tasks constrained by a set of uniform precedence constraints corresponding to a natural generalization of the basic cyclic scheduling problem. The two parameters of any uniform constraint (namely the value and the height) between two tasks may be negative, which allows one to tackle a larger class of practical applications.
We firstly study the structure and the existence of a periodic schedule. A necessary and sufficient condition for the existence of a schedule is then deduced. As there are no resource constraints, tasks following the earliest schedule have minimum average cycle times. The permanent state of the earliest schedule is characterized and we point out that the minimum average cycle times of tasks are equal for the earliest schedule and an optimal periodic schedule. An algorithm to check the existence of a schedule and to compute these minimum average cycle times using linear programming is lastly presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Armstrong, R., Lei, L., & Gu, S. (1994). A bounding scheme for deriving the minimal cycle time of a single-transporter n-stage process with time-window constraints. European Journal of Operational Research, 78, 130–140.
Bacelli, F., Cohen, G., Olsder, G. J., & Quadrat, J.-P. (1992). Synchronisation and linearity: an algebra for discrete event systems. New York: Wiley.
Berge, C. (1970). Graphes et hypergraphes. Paris: Dunod.
Carlier, J., & Chrétienne, P. (1988). Problèmes d’ordonnancement : modèlisation, complexité, algorithmes. Paris: Masson.
Cavory, G., Dupas, R., & Goncalves, G. (2005). A genetic approach to solving the problem of cyclic job shop scheduling with linear constraints. European Journal of Operational Research, 161, 73–85.
Chen, H., Chu, C., & Proth, J.-M. (1998). Cyclic schedule of a hoist with time window constraints. IEEE Transactions on Robotics and Automation, 14(1), 144–152.
Chrétienne, P. (1985). Transient and limiting behavior of timed event graphs. RAIRO Techniques et Sciences Informatiques, 4, 127–192.
Cohen, G., Moller, P., Quadrat, J.-P., & Viot, M. (1989). Algebraic tools for the performance evaluation of discrete event systems. Discrete Event Dynamics Systems, 77(1), 39–58. IEEE proceeding: special issue.
Cormen, T., Leiserson, C., & Rivest, R. (1990). Introduction to algorithms. Cambridge: MIT Press.
Cucu, L., & Sorel, Y. (2003). Schedulability condition for systems with precedence and periodicity constraints without preemption. In Proc. RTS2003 11th conference on real-time and embedded systems.
de Werra, D., Eisenbeis, C., Lelait, S., & Marmol, B. (1999). On a graph-theoretical model for cyclic register allocation. Discrete Applied Mathematics, 93, 191–203.
Gasperoni, F., & Schwiegelshohn, U. (1994). Generating close to optimum loop schedules on parallel processors. Parallel Processing Letters, 4, 391–403.
Govindarajan, R., Altman, E., & Gao, G. R. (1996). A framework for resource-constrained rate-optimal software pipelining. IEEE Transactions on Parallel and Distributed Systems, 7(11), 1133–1149.
Hall, N. G., Lee, T.-E., & Posner, M. (2002). The complexity of cyclic scheduling problems. Journal of Scheduling, 5, 307–327.
Hanen, C., & Munier, A. (1995). Cyclic scheduling on parallel processors: an overview. In P. Chrétienne, E. G. Coffman, J. K. Lenstra, & Z. Liu (Eds.), Scheduling theory and its applications. New York: Wiley.
Hillion, H., & Proth, J.-M. (1989). Performance evaluation of a job-shop system using timed event graph. IEEE Transactions on Automatic Control, 34(1), 3–9.
Lee, T.-E., & Lee, B.-S. (2005). An extension of negative event graphs for discrete event systems with general time constraints (Technical report). Department of Industrial Engineering, KAIST, August 2005.
Lee, T.-E., Lee, H.-Y., & Sreenivas, R. S. (2006). Token delays and generalized workload balancing for timed evant graphs with application to cluster toll operation. In Proc. IEEE international conference on automation science and engineering (pp. 93–99). Shanghai, China, October 2006.
Lee, T.-E., & Park, S.-H. (2005a). An extended event graph with negative places and tokens for time window constraints. IEEE Transactions on Automation Science and Engineering, 2(4), 319–332.
Lee, T.-E., & Park, S.-H. (2005b). Steady state analysis of a timed event graph with time window constraints. In Proc. IEEE international conference on automation science and engineering (pp. 404–409). Edmonton, Canada, August 2005.
Lei, L. (1993). An o(n 2log nlog b) algorithm for determining the optimal integer cyclic transportation schedule with a given route. Computers and Operation Research, 20(8), 807–816.
Levner, E., & Kats, V. (1998). A parametric critical path problem and an application for cyclic scheduling. Discrete Applied Mathematics, 87, 149–158.
McCormick, T., & Rao, U. S. (1994). Some complexity results in cyclic scheduling. Mathematical and Computer Modelling, 20, 107–122.
Middendorf, M., & Timkovsky, V. G. (2002). On scheduling cycle shops: Classification, complexity and approximation. Journal of Scheduling, 5, 135–169.
Sucha, P., Pohl, Z., & Hanzalek, Z. (2004). Scheduling of iterative algorithms on fpga with pipelined arithmetic unit. In Proc. 10th IEEE real-time and embedded technology and applications symposium, RTAS (pp. 404–412).
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was performed during a stay of the author at the INRIA-Rocquencourt center in the AOSTE project and was partially supported by INRIA.
Rights and permissions
About this article
Cite this article
Munier Kordon, A. A graph-based analysis of the cyclic scheduling problem with time constraints: schedulability and periodicity of the earliest schedule. J Sched 14, 103–117 (2011). https://doi.org/10.1007/s10951-009-0159-z
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10951-009-0159-z