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
Free choice Petri nets
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
Closed form approximations for steady state probabilities of a controlled fork-join network
ICFEM'10 Proceedings of the 12th international conference on Formal engineering methods and software engineering
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The performance analysis of parallel systems which are synchronised is an important but challenging task. This is because product form solutions are not available when synchronisation is required. The system of interest is a controlled fork-join network, comprising two parallel processes. Each process is governed by an exponentially distributed delay. The system has a capacity, N, which cannot be exceeded. Thus only N requests for service can be handled concurrently, so that further requests are blocked. The arrival of requests is also governed by an exponential distribution. We model this class of system with a Generalized Stochastic Petri Net (GSPN) which includes a two branch fork-join structure controlled by an environment that enforces the capacity constraint. This GSPN is thus parameterised by the capacity, N. We derive the parametric reduced reachability graph for arbitrary N, and show that it has (N + 1)2 markings and one strongly connected component. We also prove that the GSPN is bounded by N, and thus show that one process cannot lead the other by more than N. We obtain the associated family of continuous time Markov chains, and derive the family of global balance equations for arbitrary N. We solve these equations for the steady state probabilities for N = 1 and 2, and then present a theorem for the general form of the solution for arbitrary N 2 in terms of ratios of polynomials in the transition rates. A scheme of 21 relationships between these polynomials is also obtained. Finally we explore some asymptotic behaviour of the steady state probabilities and relate them to previously obtained closed form approximate solutions.