Abstract
Fork/join stations are commonly used to model the synchronization constraints in queuing models of computer networks, fabrication/assembly systems and material control strategies for manufacturing systems. This paper presents an exact analysis of a fork/join station in a closed queuing network with inputs from servers with two-phase Coxian service distributions, which models a wide range of variability in the input processes. The underlying queue length and departure processes are analyzed to determine performance measures such as throughput, distributions of the queue length and inter-departure times from the fork/join station. The results show that, for certain parameter settings, variability in the arrival processes has a significant impact on system performance. The model is also used to study the sensitivity of performance measures such as throughput, mean queue lengths, and variability of inter-departure times for a wide range of input parameters and network populations.
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References
Altiok, T. (1996). Performance Analysis of Manufacturing Systems. Springer Series in Operations Research, New York: Springer.
Baccelli, F., W.A. Massey, and D. Towsley. (1989). “Acyclic Fork/Join Queuing Networks.”Journal of the ACM 36, 615–642.
Bhat, U.N. (1986). “Finite Capacity Assembly like Queues.” Queuing Systems 1, 185–101.
Buzacott, J.A. and J.G. Shanthikumar. (1993). Stochastic Models of Manufacturing Systems. Englewood Cliffs, NJ: Prentice-Hall.
Di Mascolo, M., Y. Frein, and Y. Dallery. (1996). “An Analytical Method for Performance Evaluation of Kanban Controlled Production Systems.” Operations Research 44(1), 50–64.
Disney, R.L. and R.L. Kiessler. (1987). Traffic Processes in Queuing Networks: A Markov Renewal Approach. Johns Hopkins University Press.
Harrison, J.M. (1973). “Assembly-like Queues.” Journal of Applied Probability 10(2), 354–367.
Heidelberger, P. and K.S. Trivedi. (1983). “Queuing Network Models for Parallel Processing with Asynchronous Tasks.” IEEE Transactions on Computers C-31(11), 1099–1109.
Hopp, W.J. and J.T. Simon. (1989). “Bounds and Heuristics for Assembly-Like Queues.” Queuing Systems 4, 137–156.
Kamath, M., R. Suri, and J.L. Sanders. (1988). “Analytical Performance Models for Closed-Loop Flexible Assembly Systems.” International Journal of Flexible Manufacturing Systems 1(1), 51–84.
Kashyap, B.R.K. (1965). “A Double Ended Queuing System with Limited Waiting Space.” Proceedings of the National Institute of Science (India) 31, 559–570.
Krishnamurthy, A., R. Suri, and M. Vernon. (2001). “Analytical Performance Analysis of Kanban Systems Using a New Approximation for Fork/Join Stations.” In Proceedings of the Industrial Engineering Research Conference, Dallas, TX.
Krishnamurthy, A. (2002). “Analytical Performance Models for Material Control Strategies in Manufacturing Systems.” Ph.D. Thesis, Department of Industrial Engineering, University of Wisconsin-Madison.
Krishnamurthy, A., R. Suri, and M. Vernon. (2003). “Two-Moment Approximations for Throughput and Mean Queue Length of a Fork/Join Station with General Inputs from Finite Populations.” In J.G. Shanthikumar, D.D. Yao, and W.H.M. Zijm (eds.), Stochastic Modeling and Optimization of Manufacturing Systems and Supply Chains. Kluwer International Series in Operations Research and Management Science, pp. 87-126.
Latouche, G. (1981). “Queues With Paired Customers.”Journal of Applied Probability 18,684–696.
Liberopoulous, G. and Y. Dallery. (2000). “A Unified Framework for Pull Control Mechanisms in Multistage Manufacturing Systems.” Annals of Operations Research 93, 325–355.
Lipper, E.H. and B. Sengupta. (1986). “Assembly Like Queues with Finite Capacity: Bounds, Asymptotics and Approximations.” Queuing Systems 1, 67–83.
Prabhakar, B., N. Bambos, and T.S. Mountford. (2000). “The Synchronization of Poisson Processes and Queuing Networks with Service and Synchronization Nodes.”Advances in Applied Probability 32, 824–843.
Rao, P.C. and R. Suri. (1994). “Approximate Queuing Network Models of Fabrication/Assembly Systems: Part I-Single Level Systems.” Production and Operations Management 3, 244–275.
Rao, P.C. and R. Suri. (2000). “Performance Analysis of an Assembly Station with Input from Multiple Fabrication Lines.” Production and Operations Management 9(3), 283–302.
Som, P., W.E. Wilhelm, and R.L. Disney. (1994). “Kitting Process in a Stochastic Assembly System.” Queuing Systems 17, 471–490.
Takahashi, M., H. Osawa, and T. Fujisawa (1996). “A Stochastic Assembly System with Resume Levels.” Asia-Pacific Journal of Operations Research 15, 127–146.
Takahashi, M., H. Osawa, and T. Fujisawa. (2000). “On a Synchronization Queue with Two Finite Buffers.” Queuing Systems 36, 107–123. auTakahashi, M. and Y. Takahashi. (2000). “Synchronization Queue with Two MAP Inputs and Finite Buffers.” In roceedings of the Third International Conference on Matrix Analytical Methods in Stochastic Models, Leuven (Belgium), pp. 375-390.
Varki, E. (1999). “Mean Value Technique for Closed Fork-Join Networks.” In Proceedings of ACM SIGMETRICS Conference on Measurement and Modeling of Computer Systems, Atlanta, GA, pp. 103-112.
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Krishnamurthy, A., Suri, R. & Vernon, M. Analysis of a Fork/Join Synchronization Station with Inputs from Coxian Servers in a Closed Queuing Network. Annals of Operations Research 125, 69–94 (2004). https://doi.org/10.1023/B:ANOR.0000011186.14865.19
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DOI: https://doi.org/10.1023/B:ANOR.0000011186.14865.19