Source: Adv. in Appl. Probab.
Volume 36, Number 3
We study a fork-join processing network in which jobs arrive
according to a Poisson process and each job splits into m
tasks, which are simultaneously assigned to m nodes that
operate like M/M/s queueing systems. When all of its tasks
are finished, the job is completed. The main result is a
closed-form formula for approximating the distribution of the
network's response time (the time to complete a job) in
equilibrium. We also present an analogous approximation for the
distribution of the equilibrium queue length (the number of jobs
in the system), when each node has one server. Kolmogorov-Smirnov
statistical tests show that these formulae are good fits for the
distributions obtained from simulations.
Ayhan, H. and Seo, D. W. (2001). Laplace transform and moments of waiting times in (max,+) linear systems with Poisson input. Queueing Systems 37, 405--438.
Baccelli, F. and Brémaud, P. (1994). Elements of Queueing Theory. Springer, New York.
Baccelli, F. and Makowski, A. M. (1985). Simple computable bounds for the fork--join queue. In Proc. Johns Hopkins Conf. Inf. Sci., Johns Hopkins University Press, Baltimore, MD.
Baccelli, F., Makowski, A. M. and Shwartz, A. (1989). The fork--join queue and related systems with synchronization constraints: stochastic ordering and computable bounds. Adv. Appl. Prob. 21, 629--660.
Balsamo, S., Donatietllo, L. and Van Dijk, N. M. (1998). Bound performance models of heterogeneous parallel processing systems. IEEE Trans. Parallel Distributed Systems 9, 1041--1056.
Chen, R. J. (2001). A hybrid solution of fork/join synchronization in parallel queues. IEEE Trans. Parallel Distributed Systems 12, 829--845.
Flatto, L. (1985). Two parallel queues created by arrivals with two demands II. SIAM J. Appl. Math. 45, 861--878.
Mathematical Reviews (MathSciNet): MR804012
Flatto, L. and Hahn, S. (1984). Two parallel queues created by arrivals with two demands I. SIAM J. Appl. Math. 44, 1041--1053. Erratum: 45, 168 (1985).
Mathematical Reviews (MathSciNet): MR759714
Haji, R. and Newell, G. F. (1971). A relation between stationary queue and waiting time distributions. J. Appl. Prob. 8, 617--620.
Mathematical Reviews (MathSciNet): MR293745
Knessl, C. (1991). On the diffusion approximation to a fork and join queueing model. SIAM J. Appl. Math. 51, 160--171.
Konstantopoulos, P. and Walrand, J. (1989). Stationary and stability of fork--join networks. J. Appl. Prob. 26, 604--614.
Krishnamurthy, A., Suri, R. and Vernon, M. (2004). Analysis of a fork/join synchronization station with inputs from Coxian servers in a closed queueing network. Ann. Operat. Res. 125, 69--94.
Kulkarni, V. G. (1995). Modeling and Analysis of Stochastic Systems. Chapman and Hall, London.
Kumar, A. and Shorey, R. (1993). Performance analysis and scheduling of stochastic fork--join jobs in a multicomputer system. IEEE Trans. Parallel Distributed Systems 4, 1147--1164.
Kushner, H. J. (2001). Heavy Traffic Analysis of Controlled Queueing and Communication Networks (Appl. Math. (New York) 47). Springer, New York.
Law, A. M. and Kelton, W. D. (1991). Simulation Modeling and Analysis. McGraw-Hill, New York.
Mathematical Reviews (MathSciNet): MR630193
Nelson, R. and Tantawi, A. N. (1988). Approximation analysis of fork/join synchronization in parallel queues. IEEE Trans. Comput. 37, 739--743.
Nguyen, V. (1993). Processing networks with parallel and sequential tasks: heavy traffic analysis and Brownian limits. Ann. Appl. Prob. 3, 28--55.
Raghavan, N. R. S. and Viswanadham, N. (2001). Generalized queueing network analysis of integrated supply chains. Internat. J. Production Res. 39, 205--224.
Ross, S. M. (1996). Stochastic Processes, 2nd edn. John Wiley, New York.
Serfozo, R. F. (1999). Introduction to Stochastic Networks (Appl. Math. (New York) 44). Spinger, New York.
Tan, X. and Knessl, C. (1996). A fork--join queueing model: diffusion approximation, integral representations and asymptotics. Queueing Systems 22, 287--322.
Varma, S. and Makowski, A. (1994). Interpolation approximations for symmetric fork--join queues. Performance Evaluation 20, 245--265.
Zhang, Z. (1990). Analytical results for waiting time and system size distributions in two parallel queueing systems. SIAM J. Appl. Math. 50, 1176--1193.