Abstract
Fork/join stations are used to model synchronization constraints between entities in a queuing network. The fork/join station of interest in this chapter consists of a server with zero service times and two input buffers. As soon as there is one entity in each buffer, an entity from each of the buffers is removed and joined together. The joined entity exits the fork/join station instantaneously. Subsequent to its departure, the joined entity forks back into the component entities, which then each get routed to other parts of the network. Fork/join stations find many applications in queuing models of manufacturing and computer systems. In queuing models of assembly systems, the assembly station is typically modeled using a fork/join station (Harrison [7], Latouche [17], Hopp and Simon [8], Rao and Suri [23] Rao and Suri [24]). Fork/join stations are also used model the synchronization constraints in models of material control strategies for multi-stage manufacturing systems (Buzacott and Shanthikumar [5], Di Mascolo et al. [6], Krishnamurthy et al. [11]). In computer systems analysis, queuing networks with fork/join stations have been studied in the context of parallel processing, database concurrency control, and communication protocols (Baccelli et al. [3], Prabhakar et al. [21], Varki [37]).
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References
Albin, S.L., On Poisson Approximations for Superposition of Arrival Processes in Queues. Management Science, 28(2)(1982), 126–137.
Albin, S.L., Approximating a Point Process by a Renewal Process. II. Superposition Arrival Processes to Queues. Operations Research, 32(5)(1984), 1133–1162.
Baccelli, F., Massey, W.A. and Towsley, D., Acyclic Fork/Join Queuing Networks. Journal of ACM, 36(1)(1989), 615–642.
Bhat, U.N., Finite Capacity Assembly Like Queues. Queuing Systems, 1(1986), 85–101.
Buzacott, J.A. and Shanthikumar, J.G., Stochastic Models of Manufacturing Systems, Prentice Hall, New Jersey, 1993.
Di Mascolo, M., Frein, Y. and Dallery, Y., An Analytical Method for Performance Evaluation of Kanban Controlled Production Systems. Operations Research, 44(1)(1996), 50–64.
Harisson, J.M., Assembly-like Queues. Journal of Applied Probability, 10(2)(1973), 354–367.
Hopp, W. and Simon, J.T., Bounds And Heuristics For Assembly-Like Queues. Queuing Systems, 4(1989), 137–156.
Heidelberger, P. and Trivedi, K. S., Queuing Network Models for Parallel Processing with Asynchronous Tasks. IEEE Transactions on Computers, C-31(11)(1983), 1099–1109.
Kamath, M., Suri, R. and Sanders, J. L., Analytical Performance Models for Closed Loop Flexible Assembly Systems. The International Journal of Flexible Manufacturing Systems, 1(1988), 51–84.
Krishnamurthy, A., Suri, R. and Vernon, M., Analytical Performance Analysis of Kanban Systems Using a New Approximation for Fork/Join Stations. Proceedings of the Industrial Engineering Research Conference Dallas, Texas, May 2001.
Krishnamurthy, A., Suri, R. and Vernon, M., Analysis of a Fork/Join Synchronization Station with Coxian Inputs From a Finite Population. Submitted to Annals of Operations Research, (2000).
Krishnamurthy, A., Analytical Performance Models for Material Control Strategies in Manufacturing Systems. PhD. Thesis, Department of Industrial Engineering, University of Wisconsin-Madison, (2002).
Krishnamurthy, A., Suri, R. and Vernon, M., Two-Moment Approximations for the Variability Parameter of the Departure Process from a Fork/Join Station with General Inputs. Technical Report, Department of Industrial Engineering, University of Wisconsin-Madison, (2002).
Kuehn, P.J., Approximate Analysis of General Queuing Networks by Decomposition. IEEE Transactions on Communications, COM-27(1979), 113–126.
Kimura, T., A Two-Moment Approximation for the Mean Waiting Time in the GI/G/s Queue. Management Science, 32(6)(1986), 751–763.
Latouche, G., Queues With Paired Customers. Journal of Applied Probability, 18(1981), 684–696.
Liberopoulous, G. and Dallery, Y., A Unified Framework for Pull Control Mechanisms in Multistage Manufacturing Systems. Annals of Operations Research, 93(2000), 325–355.
Lipper, E.H. and Sengupta, B., Assembly Like Queues With Finite Capacity: Bounds, Asymptotic And Approximations. Queuing Systems, 1(1986), 67–83.
Marshall, K.T., Some Inequalities in Queuing. Operations Research, 16(1968), 661–665.
Prabhakar, B., Bambos, N. and Mountford, T.S., The Synchronization of Poisson Processes and Queuing Networks with Service and Synchronization Nodes. Advances in Applied Probability, 32(2000), 824–843.
Rao, P.C. and Suri, R., Approximate Queuing Network Models of Fabrication/Assembly Systems: Part I-Single Level Systems. Production and Operations Management, 3(4)(1994), 244–275.
Rao, P.C. and Suri, R., Performance Analysis Of An Assembly Station With Input From Multiple Fabrication Lines. Production and Operations Management, 9(3)(2000), 283–302.
Som, P., Wilhelm, P.E. and Disney, R.L., Kitting Process in a Stochastic Assembly System. Queuing Systems, 17(1994), 471–490.
Suri, R., Sanders, J.L. and Kamath, M., Performance Evaluation of Production Networks. In: Handbooks of OR/MS (vol. 4), S. Graves, A. Rinooy Kan and P. Zipkin (eds.), North Holland, 1993.
Suri, R., Diehl, G.W., deTreville, S. and Tomsicek, M., From CAN-Q to MPX: Evolution of Queuing Software for Manufacturing. Interfaces, 25(5)(1995), 128–150.
Takahashi, M., Osawa, H. and Fujisawa, T., A Stochastic Assembly System With Resume Levels. Asia-Pacific Journal of Operations Research, 15(1996), 127–146.
Takahashi, M., Osawa, H. and Fujisawa, T., On A Synchronization Queue With Two Finite Buffers. Queuing Systems, 36(2000), 107–123.
Takahashi, M. and Takahashi, Y., Synchronization Queue With Two MAP Inputs And Finite Buffers. Proceedings of the Third International Conference on Matrix Analytical Methods in Stochastic Models, Leuven, Belgium, 2000.
Whitt, W., Approximating a Point Process by a Renewal Process, I: Two Basic Methods. Operations Research, 30(1)(1982), 125–147.
Whitt, W., The Queuing Network Analyzer. Bell Systems Technical Journal, 62(1983), 2779–2815.
Whitt, W., Approximations to Departure Processes and Queues in Series. Naval Research Logistics Quarterly, 31(1984), 499–521.
Whitt, W., The Best Order of Queues in Series. Management Science, 31(1985), 475–487.
Whitt, W., Approximations for the GI/G/m Queue. Production and Operations Management, 2(2)(1993), 114–161.
Whitt, W., Variability Functions for Parametric Decomposition Approximations of Queuing Networks. Management Science, 41(1995), 1704–1715.
Varki, E., Mean Value Technique For Closed Fork-Join Networks. Proceedings of ACM SIGMETRICS Conference on Measurement and Modeling of Computer Systems, Atlanta, Georgia, 1999.
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Krishnamurthy, A., Suri, R., Vernon, M. (2003). Two-Moment Approximations for Throughput and Mean Queue Length of a Fork/Join Station with General Inputs from Finite Populations. In: Shanthikumar, J.G., Yao, D.D., Zijm, W.H.M. (eds) Stochastic Modeling and Optimization of Manufacturing Systems and Supply Chains. International Series in Operations Research & Management Science, vol 63. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-0373-6_5
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