An Upper Bound Solution for Homogeneous Fork/Join Queuing Systems

In this paper, we use dynamic-bubblesort technology to analyze general first-in-first-out K-queue homogenous fork/join queuing (HFJ) systems for any K >; 2. Jobs arrive with a mean rate λ and a general arrival distribution. Upon arrival, a job forks into K tasks. Task k, k = 1, 2,...,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 TK. The upper bound solution of general K-queue HFJ systems for any K ≥ 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 a//the simulations is less than 5 percent. The corresponding source codes (reusable) are offered on our website for others to use.

[1]  P. Konstantopoulos,et al.  Stationary and stability of fork-join networks , 1989, Journal of Applied Probability.

[2]  Kishor S. Trivedi,et al.  Analytic Queueing Models for Programs with Internal Concurrency , 1983, IEEE Transactions on Computers.

[3]  Tobias Kiesling Using Approximation with Time-Parallel Simulation , 2005, Simul..

[4]  Donald F. Towsley,et al.  Computing Performance Bounds of Fork-Join Parallel Programs Under a Multiprocessing Environment , 1998, IEEE Trans. Parallel Distributed Syst..

[5]  Lester Lipsky,et al.  Queueing Theory: A Linear Algebraic Approach , 1992 .

[6]  Harry G. Perros,et al.  A Decomposition Procedure for the Analysis of a Closed Fork/Join Queueing System , 1991, IEEE Trans. Computers.

[7]  Asser N. Tantawi,et al.  Approximate Analysis of Fork/Join Synchronization in Parallel Queues , 1988, IEEE Trans. Computers.

[8]  Elizabeth Varki,et al.  Response Time Analysis of Parallel Computer and Storage Systems , 2001, IEEE Trans. Parallel Distributed Syst..

[9]  Alan Weiss,et al.  Allocating Independent Subtasks on Parallel Processors , 1985, IEEE Transactions on Software Engineering.

[10]  Donald F. Towsley,et al.  Acyclic fork-join queuing networks , 1989, JACM.

[11]  Donald F. Towsley,et al.  Analysis of Fork-Join Program Response Times on Multiprocessors , 1990, IEEE Trans. Parallel Distributed Syst..

[12]  Haiying Zhang,et al.  A Fast Simulation for Thousands of General Homogeneous Fork/Join Queues , 2010, 2010 International Conference on Intelligent Systems, Modelling and Simulation.

[13]  Leonard Kleinrock,et al.  Queueing Systems: Volume I-Theory , 1975 .

[14]  Ashok K. Agrawala,et al.  Analysis of the Fork-Join Queue , 1989, IEEE Trans. Computers.

[15]  Ray Jinzhu Chen A Hybrid Solution of Fork/Join Synchronization in Parallel Queues , 2001, IEEE Trans. Parallel Distributed Syst..

[16]  L. Flatto,et al.  Two parallel queues created by arrivals with two demands. II , 1984 .

[17]  L. Flatto,et al.  Erratum: Two Parallel Queues Created by Arrivals with Two Demands I , 1985 .

[18]  Anurag Kumar,et al.  Performance Analysis and Scheduling of Stochastic Fork-Join Jobs in a Multicomputer System , 1993, IEEE Trans. Parallel Distributed Syst..

[19]  Donald F. Towsley,et al.  Bounding the Mean Response Time of the Minimum Expected Delay Routing Policy: An Algorithmic Approach , 1995, IEEE Trans. Computers.

[20]  Simonetta Balsamo,et al.  On Queue Length Moments in Fork and Join Queuing Networks with General Service Times , 1997, Computer Performance Evaluation.

[21]  R. Serfozo,et al.  Response times in M/M/s fork-join networks , 2004, Advances in Applied Probability.

[22]  F. Baccelli,et al.  The fork-join queue and related systems with synchronization constraints: stochastic ordering and computable bounds , 1989, Advances in Applied Probability.

[23]  Kishor S. Trivedi,et al.  Queueing Network Models for Parallel Processing with Asynchronous Tasks , 1982, IEEE Transactions on Computers.

[24]  Asser N. Tantawi,et al.  Performance analysis of parallel processing systems , 1987, SIGMETRICS '87.