Abstract
We introduce the First Fit Matching Periods algorithm for rate-monotonic multiprocessor scheduling of periodic tasks with implicit deadlines and show that it yields asymptotically optimal processor assignments if utilization values are chosen uniformly at random. More precisely we prove that the expected waste is upper bounded by \(\mathcal{O}(n^{3/4} (\log n)^{3/8})\). Here the waste denotes the ratio of idle times, cumulated over all processors and n gives the number of tasks.
The algorithm can be implemented to run in time \(\mathcal{O}(n \log n)\) and even in the worst case, an asymptotic approximation ratio of 2 is guaranteed. Experiments yield an average waste proportional to n 0.70, indicating that the above upper bound on the expected waste is almost tight.
While such average-case analyses are a classical topic of Bin Packing, to the best of our knowledge, this is the first result dealing with a theoretical average-case analysis for this scheduling problem, which was described by Liu and Layland more than 35 years ago and has received a lot of attention, especially in the real-time and embedded-systems community.
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References
Liu, C.L., Layland, J.W.: Scheduling algorithms for multiprogramming in a hard-real-time environment. J. ACM 20(1), 46–61 (1973)
Lehoczky, J.P., Sha, L., Ding, Y.: The rate monotonic scheduling algorithm: Exact characterization and average case behavior. In: IEEE Real-Time Systems Symposium (1989)
Lehoczky, J.P.: Fixed priority scheduling of periodic task sets with arbitrary deadlines. In: IEEE Real-Time Systems Symposium, pp. 201–213 (1990)
Korst, J., Aarts, E.H.L., Lenstra, J.K., Wessels, J.: Periodic multiprocessor scheduling. In: Aarts, E.H.L., van Leeuwen, J., Rem, M. (eds.) PARLE 1991. LNCS, vol. 505, pp. 166–178. Springer, Heidelberg (1991)
Audsley, A.N., Burns, A., Richardson, M., Tindell, K.: Applying new scheduling theory to static priority pre-emptive scheduling. Software Engineering Journal, 284–292 (1993)
Oh, Y., Son, S.H.: Allocating fixed-priority periodic tasks on multiprocessor systems. Real-Time Syst. 9(3), 207–239 (1995)
Davari, S., Dhall, S.K.: On-line algorithms for allocating periodic-time-critical tasks on multiprocessor systems. Informatica (Slovenia) 19(1) (1995)
Liebeherr, J., Burchard, A., Oh, Y., Son, S.H.: New strategies for assigning real-time tasks to multiprocessor systems. IEEE Trans. Comput. 44(12), 1429–1442 (1995)
Korst, J., Aarts, E., Lenstra, J.K.: Scheduling periodic tasks with slack. INFORMS J. Comput. 9(4), 351–362 (1997)
Oh, D.I., Baker, T.P.: Utilization bounds for N-processor rate monotone scheduling with static processor assignment. Real-Time Systems (1998)
Baruah, S., Goossens, J.: Scheduling real-time tasks: Algorithms and complexity. In: Leung, J.Y.T. (ed.) Handbook of Scheduling — Algorithms, Models, and Performance Analysis. Computer and Information Science Series, vol. 28. Chapman & Hall/CRC, Boca Raton (2004)
Fisher, N., Baruah, S.: A fully polynomial-time approximation scheme for feasibility analysis in static-priority systems with arbitrary relative deadlines. In: ECRTS 2005. IEEE Computer Society, Los Alamitos (2005)
Leung, J., Kelly, L., Anderson, J.H.: Handbook of Scheduling: Algorithms, Models, and Performance Analysis. CRC Press, Inc., Boca Raton (2004)
Garey, M.R., Graham, R.L., Johnson, D.S., Yao, A.C.C.: Resource constrained scheduling as generalized bin packing. J. Combin. Theory Ser. A 21, 257–298 (1976)
Johnson, D.S.: Near-optimal bin packing algorithms. PhD thesis, MIT, Cambridge, MA (1973)
Shor, P.W.: The average-case analysis of some on-line algorithms for bin packing. In: FOCS 1984, Singer Island, FL. IEEE, Los Alamitos (1984)
Frederickson, G.N.: Probabilistic analysis for simple one- and two-dimensional bin packing algorithms. Information Processing Letters 11(4/5), 156–161 (1980)
Knödel, W.: A bin packing algorithm with complexity O(n logn) and performance 1 in the stochastic limit. In: Gruska, J., Chytil, M.P. (eds.) MFCS 1981. LNCS, vol. 118, pp. 369–378. Springer, Heidelberg (1981)
Lueker, G.S.: An average-case analysis of bin packing with uniformly distributed item sizes. Technical Report 181, Dept. Inf. and CS, University of California at Irvine (1982)
Fernandez de la Vega, W., Lueker, G.S.: Bin packing can be solved within 1 + ε in linear time. Combinatorica 1(4), 349–355 (1981)
Karmarkar, N., Karp, R.M.: An efficient approximation scheme for the one-dimensional bin-packing problem. In: FOCS 1982, pp. 312–320. IEEE, Los Alamitos (1982)
Coffman Jr., E.G., Garey, M.R., Johnson, D.S.: Approximation algorithms for bin-packing—an updated survey. In: Algorithm design for computer system design. CISM Courses and Lectures, vol. 284, pp. 49–106. Springer, Vienna (1984)
Eisenbrand, F., Rothvoß, T.: Static-priority Real-time Scheduling: Response Time Computation is NP-hard. In: IEEE Real-Time Systems Symposium, RTSS (2008)
Eisenbrand, F., Rothvoß, T.: A PTAS for static priority real-time scheduling with resource augmentation. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 246–257. Springer, Heidelberg (2008)
Coffman, E.G.J., So, K., Hofri, M., Yao, A.C.: A stochastic model of bin-packing. Inf. Control 44, 105–115 (1980)
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Karrenbauer, A., Rothvoß, T. (2009). An Average-Case Analysis for Rate-Monotonic Multiprocessor Real-Time Scheduling. In: Fiat, A., Sanders, P. (eds) Algorithms - ESA 2009. ESA 2009. Lecture Notes in Computer Science, vol 5757. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04128-0_39
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