Approximate Analysis of Fork/Join Synchronization in Parallel Queues
IEEE Transactions on Computers
Performance Analysis of Parallel Processing Systems
IEEE Transactions on Software Engineering
Analysis of the Fork-Join Queue
IEEE Transactions on Computers
Acyclic fork-join queuing networks
Journal of the ACM (JACM)
A Decomposition Procedure for the Analysis of a Closed Fork/Join Queueing System
IEEE Transactions on Computers
Interpolation approximations for symmetric Fork-Join queues
Performance '93 Proceedings of the 16th IFIP Working Group 7.3 international symposium on Computer performance modeling measurement and evaluation
Mean value technique for closed fork-join networks
SIGMETRICS '99 Proceedings of the 1999 ACM SIGMETRICS international conference on Measurement and modeling of computer systems
Computing Performance Bounds of Fork-Join Parallel Programs Under a Multiprocessing Environment
IEEE Transactions on Parallel and Distributed Systems
Response Time Analysis of Parallel Computer and Storage Systems
IEEE Transactions on Parallel and Distributed Systems
Approximate closed-form aggregation of a fork-join structure in generalised stochastic petri nets
valuetools '06 Proceedings of the 1st international conference on Performance evaluation methodolgies and tools
Queueing models of RAID systems with maxima of waiting times
Performance Evaluation
A study of the convergence of steady state probabilities in a closed fork-join network
ATVA'10 Proceedings of the 8th international conference on Automated technology for verification and analysis
On parametric steady state analysis of a generalized stochastic petri net with a fork-join subnet
PETRI NETS'11 Proceedings of the 32nd international conference on Applications and theory of Petri Nets
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Our work is motivated by just-in-time manufacturing systems, where goods are produced on demand. We consider a class of products made from two components each manufactured by its own production line. The components are then assembled, requiring synchronisation of the two lines. The production lines are coordinated to ensure that one line does not get ahead of the other by more than a certain number of components, N, a parameter of the system. We assume that the statistics of the processes follow exponential distributions, with requests to manufacture the product arriving at a rate λ0 and the two production lines having rates λ1 and λ2. Generalised Stochastic Petri Nets (GSPN) are used to model this system where N is the initial marking of a control place. TimeNET is used to calculate the stationary token distribution of the GSPN as N increases, revealing convergence of the steady state probabilities. We characterise the range of rates for which useful convergence occurs using a large number of TimeNET runs and show how these results can be used to approximate the steady state probabilities for arbitrarily large N, to a desired level of accuracy. Further, for λ0 min(λ1, λ2) we discover geometric progressions in the steady state probabilities once they have converged. We use these progressions to derive closed form approximations, the accuracies of which increase as N increases.