Mean Sojourn Time in Multi Stage Fork-Join Network: The Effect of Synchronization and Structure

Mean Sojourn Time in Multi Stage Fork-Join Network: The Effect of Synchronization and Structure

Yonit Barron (Department of Statistics, University of Haifa, Haifa, Israel)
DOI: 10.4018/IJORIS.2015070104

Abstract

Fork-Join queue networks (F-J) have received increasing attention during the last Decade, due to their ability to model parallel and distributed computer processing, supply chains and assembly systems. However, most research is focused on a single stage processing, and only scant work exists on F-J with two or more stages. In this paper, the author investigates (through simulation) the performance behavior of a multi-stage system; in particular, the performance of a synchronized system is compared to an unsynchronized system regarding three major factors: (1) the number of parallel tasks; (2) the number of serial stages and (3) the utilization.
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1. Introduction

Fork-Join (F-J) queuing networks are prevalent models in a variety of communication and manufacturing systems. F-J network in the realm of data communication typically represents the processing of computer programs, data packets, telephone calls, etc., that involve parallel multitasking, and the splitting and joining of information. In manufacturing, a F-J network, called an assembly network, represents the assembly of several parts and sub-assemblies into a product (or a system) that requires several parts to be processed simultaneously at separate work stations or plant locations. (e.g., Baccelli et al., 1989b; Matta et al., 2005; Krishnamurthy & Suri, 2005; Ramakrishnan & Krishnamurthy, 2008, 2012). In logistics, a supply chain F-J network typically represents the arrival of an order composed of several different items or products from a vendor, or synchronization between arriving and departing vehicles at the transshipment location (e.g., Srinivasa et al., 2001).

The basic F-J queue, illustrated in Figure 1, is a one-stage network. Jobs (or entities) arrive at the fork node, where each incoming job is split into m tasks that are sent to m parallel servers.

Figure 1.

The basic one-stage Fork-Join queuing network with m servers

A task may have to wait in a queue until its server finishes all previous tasks. After the task completes its service, the task must wait (in a second queue) until all the other m-1 tasks (of the same job) complete their process. When all m tasks of the same job are completed, they rejoin (synchronize) at the join node. The time elapsing from entering the fork node until departing the join node is denoted as the sojourn time.

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