PG and PCG Performance Evaluation

To give a comparison for PG and PCG, we implement EDF according to the following rules presented by Funk [9]:

1

• No processor is idled while there is an active job awaiting execution.

• When fewer than m jobs are active, they are required to execution upon the fastest processors while the slowest are idled.

• Higher priority jobs are executed on faster processors.

To analyze the schedulability of EDF on uniform multiprocessors, as shown in Figure 5.2, we generate 1000000 set of tasks and processors for each pair of tasks and processors. In Figure 5.2(a), the number of tasks and processors is equal, in Figure 5.2(b), the number of tasks is 10. It is easy to show while the number of tasks and processors increase, EDF will miss deadline because the complexity of assignment raising. EDF only considers about the deadline of each task, it will miss deadline while the urgent tasks have not be executed.

90

number of tasks and processors

0

1 2 3 4 5 6 7 8 9 10

b f

numberoftasksandprocessors numberofprocessors (a)thenumberoftasksandprocessorsisequal (b)thenumberoftasksisfixed

Figure 5.2: Schedulability of EDF on uniform multiprocessors

90

number of tasks and processors

0

1 2 3 4 5 6 7 8 9 10

numberofprocessors

numberoftasksandprocessors p

(a)thenumberoftasksandprocessorsisequal (b)thenumberoftasksisfixed

Figure 5.3: Schedulability of PG

To analyze the schedulability of PG and PCG on uniform multiprocessors, as shown in Figure 5.3 and Figure 5.4, we generate 1000000 set of tasks and processors for each pair of tasks and processors. In Figure 5.3(a), the number

90

number of tasks and processors

0

1 2 3 4 5 6 7 8 9 10

numberofprocessors

numberoftasksandprocessors p

(a)thenumberoftasksandprocessorsisequal (b)thenumberoftasksisfixed

Figure 5.4: Schedulability of PCG

of tasks and processors is equal, in Figure 5.3(b), the number of tasks is 10.

In Figure 5.4(a), the number of tasks and processors is equal, in Figure 5.4(b), the number of tasks is 10. PG and PCG could schedule all the set of tasks and processors.

Although PG and PCG is schedulable for all cases on uniform multiproces-sors, the times of context switches is increasing based on the number of T-L_er planes. As shown in Figure 5.5 and Figure 5.6, we generate 100000000 set for each pair of tasks and processors. In Figure 5.5(a), the number of tasks and processors is equal, in Figure 5.5(b), the number of tasks is 10. In Figure 5.6(a), the number of tasks and processors is equal, in Figure 5.6(b), the number of tasks is 10. We could figure out the times of context switching by PG and PCG is larger than EDF, While the number of tasks and processors increase, the

80

Figure 5.5: Performance Analysis of PG while Comparing to EDF

70

Figure 5.6: Performance Analysis of PCG while Comparing to EDF

times of context switching will increase, too. This is because We generate all the tasks randomly, the period of them is different, PG and PCG will generate lots of T-L_er planes and have lots of scheduling within each plane. It shows

although the schedulability of PG and PCG is optimal, the time complexity of

number of tasks and processors

0

Figure 5.7: The number of increasing cases between PCG and PG

25

number of tasks and processors

0

Figure 5.8: Performance Analysis between PCG and PG

Now we want to discuss about the performance between PG and PCG as shown in Figure 5.7 and Figure 5.8. In Figure 5.7(a), the number of tasks and processors is equal, in Figure 5.7(b), the number of tasks is 10. In Figure 5.8(a), the number of tasks and processors is equal, in Figure 5.8(b), the number of tasks is 30. It shows that PCG give better performance than PG, and when the number of tasks and processors increase, PCG will not give better performance.

This is because the number of T-L_er planes is significant, although PCG give an upper bound n in a T-L^er plane, when the number of T-L_er planes even larger, it could not be sure the performance of it is still good.

While we are based on the concept of T-L planes, the bottleneck of our scheduling algorithm is the number of T-L_er planes. When the period of each task is harmonic, the number of T-L_er planes will decrease. On the other hand, when the period of each task is diverse, the number of T-L_er planes will increase dramatically.

Chapter 6

Conclusions

Although feasible on-line scheduling algorithm for uniform multiprocessors is difficult, we provide a novel T-L_er plane model for uniform multiprocessors to observe the behavior of task and processor easily. We present the Precaution Greedy algorithm, which is the first optimal dynamic-priority scheduling algo-rithm for uniform multiprocessors and the Precaution Cut Greedy scheduling algorithm, which is also optimal and with the times of rescheduling decreased dramatically. We also prove the optimality of the above algorithms and an upper bound n of the times of rescheduling in a T-L^er plane. Finally we give an experimental evaluation for PG, PCG, and EDF scheduling, prove PCG will give better performance than PG. We believe the results might be applicable to current asymmetric multicore platforms of similar uniform multiprocessors, where the processing units are capable of executing the same instruction with different rates, rising the performance in parallel and decreasing the times of context switching. Because the simplicity of our results, it might be also

appli-cable to the most complicated unrelated parallel machines while each task Tⁱ completes (r^i,j × t) units of execution by executing on processor P^j for t time units, the execution work for each task will be an constant value, therefore, we might migrate the same model on unrelated multiprocessors.

Bibliography

[1] Advanced Micro Devices (AMD). Multi-core processors - the next evolution in computing.

[2] S.K. Baruah, N.K. Cohen, C.G. Plaxton, and D.A. Varvel. Proportion-ate progress: A notion of fairness in resource allocation. Algorithmica, 15(6):600–625, June 1996.

[3] S.K. Baruah and J. Goossens. Preemptive scheduling of uniform processor systems. Journal of the ACM, 25(1):92–101, Jan. 1978.

[4] S.K. Baruah and J. Goossens. Rate-monotonic scheduling on uniform mul-tiprocessors. IEEE Transactions on Computers, 52(7):966–970, Jul 2003.

[5] J.M. Calandrino, D. Baumberger, S. Tong Li, Hahn, and J.H. Anderson.

Soft real-time scheduling on performance asymmetric multicore platforms.

Real Time and Embedded Technology and Applications Symposium, April 2007.

[6] A Chandra, M Adler, and P Shenoy. Deadline fair scheduling: bridging the theory and practice of proportionate pair scheduling in multiprocessor

systems. Real-Time Technology and Applications Symposium, pages 3–14, June 2001.

[7] Hyeonjoong Cho, Binoy Ravindran, and E. Douglas Jensen. An optimal real-time scheduling algorithm for multiprocessors. RTSS, pages 101–110, Oct. 2006.

[8] M.L. Dertouzos and A.K Mok. Multiprocessor online scheduling of hard-real-time tasks. IEEE Transcation on Software Engineering, 15(12):1497–

1506, Dec. 1989.

[9] S. Funk, J. Goossens, and S. Baruah. On-line scheduling on uniform mul-tiprocessors. RTSS, pages 183–192, Dec. 2001.

[10] Dorit S. Hochbaum and David B. Shmoys. A polynomial approximation scheme for scheduling on uniform processors: Using the dual approximation approach. SIAM Journal on Computing, 17(3):539–551, 1988.

[11] Philip Holman and James H. Anderson. Adapting pfair scheduling for symmetric multiprocessors. Journal of Embedded Computing, 1(4):543–564, 2005.

[12] Edward C. Horvath, Shui Lam, and Ravi Sethi. A level algorithm for preemptive scheduling. Journal of the ACM, 24(1):32–43, January 1977.

[13] Klaus Jansen and Lorant Porkolab. Improved approximation schemes for scheduling unrelated parallel machines. ACM symposium on Theory of computing, 1999.

[14] Jan Karel Lenstra, David B. Shmoys, and ´Eva Tardos. Approximation algorithms for scheduling unrelated parallel machines. Mathematical Pro-gramming, 46(1), Jan. 1990.

[15] C. L. Liu and J. Layland. Scheduling algorithms for multiprogramming in a hard real-time environment. Journal of the ACM, 10(1):46–61, 1973.

[16] Jane W.S. Liu. Real-Time Systems. Prentice Hall, 2000.

[17] R.R. Muntz and E.G. Coffman. Optimal preemptive scheduling on two-processor systems. Computers, IEEE Transactions on, (11):1014–1020, Nov. 1969.

[18] B. Srivastava. An effective heuristic for minimising makespan on unre-lated parallel machines. The Journal of the Operational Research Society, 49(8):886–894, Aug. 1998.