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Top1. Introduction
The fuzzy set theory introduced by Zadeh in 1965 has been applied successfully in various fields and distributed computing system is one of them. DCS is used to describe a system whenever there are several computers interacted in some fashion so that a program runs on a system with multiple processors.DCS provides faster computation by facilitating parallel execution of tasks of a program. Program partitioning into the tasks and their allocations are two major steps in designing of DCS. If these steps are not done properly, an increase in the number of processor in the system may actually result in the decreases of the total throughput due to the saturation effect that arises from excessive inter-processor communication. The excessive inter-processor communication is always costliest and least reliable factor in DCS. Therefore, an efficient task allocation strategy is required for the proper utilization of computational resources and minimization of inter-processor communication that arises when the interacting tasks reside on different processors.
Task allocation policy can be static or dynamic, depending upon the time at which the allocation decisions are made. Dynamic algorithms are employed more frequently because they offer a better efficiency, having the flexibility of tasks migration. In addition, usually dynamic algorithms are neither very complicated to construct nor costly to run. Hence dynamic algorithms have the potential to outperform static algorithms in general. Stone (1977) has suggested an efficient optimal algorithm for the problem of assigning tasks to two processors by making the best use of the well-known network flow algorithm in the related two-terminal network graph. A shortest tree algorithm for optimal assignment of the tasks has been developed by Bokhari (1981). Towsley (1986) generalized the result of Bokhari to the case of series parallel structures. Bokhari (1979) analyzed the problem of dynamic assignment in a two processors system, which permits relocation of tasks from one processor to the other at some points during the execution of the program. Yadav et al. (2008) generalized the result of Bokhari (1979) for NP- hard 
system.
In a linear array network when any two non-adjacent processors communicate with each other, the inter-mediate processor must participate in the communication. Thus the communication cost per unit information transferred between any two processors increases linearly as the distance between them increases. Cho et al. (1994) generalized the result of C. H. Lee et al. (1992) for the general structure problem in a linear array network with any number of machines. In Sinclayer (1988), Rotithor (1994), Vidyarthi and Tripathi (2001), and Yadav, Singh, and Pradhan (2012), several approaches are used to find the tasks assignment for minimizing the total cost of running program. V. Sriramdas et al. (2014) developed a tasks allocation model treating allocation factors as fuzzy numbers. Recently Kumar (2015) has developed an effective tool for handling the fuzzy tasks allocation problem. The system reliability of a DCS can be measured by probability of the successful execution of its tasks. A model for allocating the tasks to processors in heterogeneous DCS with the goal of maximizing the system reliability has been developed in Kang, He, and Wei (2013).