Subscribe

Issue No.05 - May (2011 vol.22)

pp: 874-878

Ray Jinzhu Chen , Xiamen University, Xiamen

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/TPDS.2010.168

ABSTRACT

In this paper, we use dynamic-bubblesort technology [4] to analyze general first-in-first-out K-queue homogenous fork/join queuing (HFJ) systems for any K \ge 2. Jobs arrive with a mean rate \lambda and a general arrival distribution. Upon arrival, a job forks into K tasks. Task k,k= 1,2, \ldots,K, is assigned to the kth queuing system, which is a first-in-first-out server with a general service distribution and an infinite capacity queue. A job leaves the HFJ system as soon as all its tasks complete their service. We mathematically prove an upper bound solution for the mean response time that we denote by T_K. The upper bound solution of general K-queue HFJ systems for any K \ge 2 is very simple and practical—one only needs to simulate a small number of queues (e.g., 16 queues). The tightness is evaluated by comparing with the simulation of thousands of queues for three different HFJ cases. The maximum offset for our upper bounds over all the simulations is less than 5 percent. The corresponding source codes (reusable) are offered on our website for others to use.

INDEX TERMS

Modeling and prediction, queuing theory, performance evaluation, parallel simulation.

CITATION

Ray Jinzhu Chen, "An Upper Bound Solution for Homogeneous Fork/Join Queuing Systems",

*IEEE Transactions on Parallel & Distributed Systems*, vol.22, no. 5, pp. 874-878, May 2011, doi:10.1109/TPDS.2010.168REFERENCES

- [1] F. Baccelli, A. Makowski, and A. Shwartz, "The Fork-Join Queue and Related Systems with Synchronization Constraints: Stochastic Ordering, and Computable Bounds,"
Advances in Applied Probability, vol. 21, pp. 629-660, Sept. 1989.- [2] F. Baccelli, W.A. Massey, and D. Towsley, "Acyclic Fork-Join Queueing Networks,"
J. ACM, vol. 36, no. 3, pp. 615-642, July 1989.- [3] S. Balsamo and I. Mura, "On Queue Length Moments in Fork and Join Queuing Networks with General Service Times,"
Computer Performance Evaluation Modeling Techniques and Tools, Springer, pp. 218-231, 1997.- [4] R.J. Chen, "A Hybrid Solution of Fork/Join Synchronization in Parallel Queues,"
IEEE Trans. Parallel and Distributed Systems, vol. 12, no. 8, pp. 829-845, Aug. 2001.- [5] R.J. Chen, H. Zhang, and H. Hu, "A Fast Simulation for Thousands of General Homogeneous Fork/Join Queues,"
Proc. Int'l Conf. Intelligent Systems, Modelling and Simulation (ISMS '10), pp. 300-305, 2010.- [6] R.J. Chen, "Simulation Code for Thousands of Fork/Join Queues," http://software.xmu.edu.cn/ViewArticleShow.aspx?aid=3078 , May 2009.
- [7] R.J. Chen and C. Jiang, "Simulation Code for an Upper Bound Solution," http://software.xmu.edu.cn/ViewArticleShow. aspx?aid=3259 , Mar. 2010.
- [8] L. Flatto and S. Hahn, "Two Parallel Queues Created by Arrivals with Two Demands I,"
SIAM J. Applied Math., vol. 44, pp. 1041-1053, Oct. 1984.- [9] L. Flatto, "Two Parallel Queues Created by Arrivals with Two Demands II,"
SIAM J. Applied Math., vol. 45, pp. 861-878, Oct. 1985.- [10] P. Heidelberger and K.S. Trivedi, "Queueing Network Models for Parallel Processing with Asynchronous Tasks,"
IEEE Trans. Computers, vol. C-31, no. 11, pp. 1099-1109, Nov. 1982.- [11] P. Heidelberger and K.S. Trivedi, "Analytic Queueing Models for Programs with Internal Concurrency,"
IEEE Trans. Computers, vol. C-32, no. 1, pp. 73-82, Jan. 1983.- [12] T. Kiesling, "Using Approximation with Time-Parallel Simulation,"
Simulation, vol. 81, no. 4, pp. 255-266, 2005.- [13] C. Kim and A.K. Agrawala, "Analysis of the Fork-Join Queue,"
IEEE Trans. Computers, vol. 38, no. 2, pp. 250-255, Dec. 1989.- [14] L. Kleinrock,
Queueing Systems Volume I: Theory. John Wiley & Sons, 1975.- [15] S.S. Ko and R.F. Serfozo, "Response Times in M/M/s/Fork/Join Networks,"
Advances in Applied Probability, vol. 36, pp. 854-871, 2004.- [16] P. Konstantopoulos and J. Walrand, "Stationary and Stability of Fork-Join Networks,"
J. Applied Probability, vol. 25, pp. 604-614, 1989.- [17] C.P. Kruskal and A. Weiss, "Allocating Independent Subtasks on Parallel Processors,"
IEEE Trans. Software Eng., vol. SE-11, no. 10, pp. 1001-1016, Oct. 1985.- [18] A. Kumar and R. Shorey, "Performance Analysis and Scheduling of Stochastic Fork-Join in a Multicomputer System,"
IEEE Trans. Parallel and Distributed Systems, vol. 4, no. 10, pp. 1147-1164, Oct. 1993.- [19] L. Lipsky,
Queueing Theorem, a Linear Algebraic Approach. Macmillan, 1992.- [20] Y.C. Liu and H.G. Perros, "A Decomposition Procedure for the Analysis of a Closed Fork/Join Queueing System,"
IEEE Trans. Computers, vol. 40, no. 3, pp. 365-370, Mar. 1991.- [21] J.C.S. Lui, R.R. Muntz, and D. Towsley, "Bounding the Mean Response Time of a Minimum Expected Delay Routing System: An Algorithmic Approach,"
IEEE Trans. Computers, vol. 44, no. 12, pp. 1371-1382, Dec. 1995.- [22] J.C.S. Lui, R.R. Muntz, and D. Towsley, "Computing Performance Bounds of Fork-Join Parallel Programs under a Multiprocessing Environment,"
IEEE Trans. Parallel and Distributed Systems, vol. 9, no. 3, pp. 295-311, Mar. 1998.- [23] R. Nelson and A.N. Tantawi, "Approximate Analysis of Fork/Join Synchronization in Parallel Queues,"
IEEE Trans. Computers, vol. 37, no. 6, pp. 739-745, June 1988.- [24] R. Nelson, D. Towsley, and A.N. Tantawi, "Performance Analysis of Parallel Processing Systems,"
IEEE Trans. Software Eng., vol. 14, no. 4, pp. 532-540, Apr. 1988.- [25] D. Towsley, C.G. Rommel, and J.A. Stankovic, "Analysis of Fork-Join Program Response Times on Multiprocessors,"
IEEE Trans. Parallel and Distributed Systems, vol. 1, no. 3, pp. 286-303, July 1990.- [26] E. Varki, "Response Time Analysis of Parallel Computer and Storage Systems,"
IEEE Trans. Parallel and Distributed Systems, vol. 12, no. 11, pp. 1146-1161, Nov. 2001. |