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February, 1993 Processing Networks with Parallel and Sequential Tasks: Heavy Traffic Analysis and Brownian Limits
Vien Nguyen
Ann. Appl. Probab. 3(1): 28-55 (February, 1993). DOI: 10.1214/aoap/1177005506


In queueing theory one seeks to predict in quantitative terms the congestion delays that occur when jobs or customers complete for processing resources. At present no satisfactory methods exist for the analysis of systems that allow simultaneous performance of tasks associated with a single job or customer. We present a heavy traffic analysis for the class of homogeneous fork-join networks in which jobs are routed in a feedforward deterministic fashion. We show that under certain regularity conditions the vector of total job count processes converges weakly to a multidimensional reflected Brownian motion (RBM) whose state space is a polyhedral cone in the nonnegative orthant. Furthermore, the weak limits of workload levels and throughput times are shown to be simple transformations of the RBM. As will be explained, the "steady-state throughput time" (a random variable) is expressed in terms of workload levels via the "longest path functional" of classical PERT/CPM analysis.


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Vien Nguyen. "Processing Networks with Parallel and Sequential Tasks: Heavy Traffic Analysis and Brownian Limits." Ann. Appl. Probab. 3 (1) 28 - 55, February, 1993.


Published: February, 1993
First available in Project Euclid: 19 April 2007

zbMATH: 0771.60082
MathSciNet: MR1202514
Digital Object Identifier: 10.1214/aoap/1177005506

Primary: 60K25
Secondary: 60J65 , 60K20

Keywords: Fork-join networks , heavy traffic analysis , performance analysis , processing networks , reflected Brownian motion , sojourn time analysis

Rights: Copyright © 1993 Institute of Mathematical Statistics


Vol.3 • No. 1 • February, 1993
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