International Journal of Soft Computing and Engineering (IJSCE) ISSN: 2231-2307, Volume-2, Issue-6, January 2013 70 Abstract— In a heterogeneous parallel computer system, the computational power of each of the processors differs from one another. Furthermore, with distributed memory, the capacity of the memory, which is distributed to each of the processors, differs from one another. Using queuing system to describe a distributed memory heterogeneous parallel computer system, each of the heterogeneous processors will have its own heterogeneous queue. The variation of waiting time of heterogeneous parallel computer system with distributed memory needs to be modeled because it will help designers of parallel computer system to determine the extent of variation of the waiting time. It will also help users to know when to realize minimum variation of the waiting time. This paper models the variation of the waiting time of distributed memory heterogeneous parallel computer system using recursive models. It also uses the statistical method of Z-Transform to verify and validate the recursive model. Keywords— distributed memory, heterogeneous parallel computer, parallel computer system, queuing network, recursive models, variation, waiting time, Z-Transform. I. INTRODUCTION A heterogeneous parallel computer system is one in which the computational power of each of the processors differs from one another. With distributed memory, it means that each of the heterogeneous processors has its own memory. Describing the system using queuing network, each of the processors has its own queue. With a round robin scheduling algorithm, processes can be scheduled to the various parallel processors, whenever a process needs to perform an I/O operation, it joins the appropriate I/O queue. Therefore, the queuing network of a heterogeneous parallel computer system consists of parallel processors, parallel processor queues, I/O processors and I/O queues. Suppose there are n different parallel processor queuing systems and k different I/O queuing systems. A queuing system in this context is defined as a processor, together with its own queue. We assume that the various queues are finite [1], [2], [3], [4] i.e. there is a limit to the number of jobs that can be admitted into the queues, and negligible communication overhead. Suppose X 1 , X 2 , X 3 , …, X n , X n+1 , X n+2 , X n+3 , …, X n+K are the maximum number of processes that can be admitted into the respective queues. We assume that processes arrive at the various queues according to Poisson distribution, and they are serviced according to Exponential distribution [5], [6]. Figure 1 illustrates a model of the queuing network of a heterogeneous parallel computer system Manuscript received on January, 2013. Oguike, O. E., Department of Computer Science, University of Nigeria, Nsukka, Enugu State, Nigeria. Agu, M.N., Department of Computer Science, University of Nigeria, Nsukka, Enugu State, Nigeria. Echezona, S.C.., Department of Computer Science, University of Nigeria, Nsukka, Enugu State, Nigeria. with distributed memory. There are different performance metrics of a parallel computer system that can be modeled, however, for distributed memory heterogeneous parallel computer system, variation of waiting time is an important performance metric that needs to be modeled. This is because the various computational resources and processes are heterogeneous, therefore there is need to measure the extent of variation between the heterogeneous computational resources and processes. Figure 1: Queuing network of a heterogeneous parallel computer system with distributed memory II. LITERATURE REVIEW AND LIMITATION OF CURRENT TECHNIQUE Queuing approach has been used extensively in the literature to model the performance of computer systems. However, this has been done in different ways and for different models of computer systems. In [20], the authors used a recursive computation approach to solve the steady state equations, thereby leading to the modeling of the various performance metrics of a multi-terminal system that is subject to breakdown. Furthermore, the author in [24] used a rigorous approach to model the performance of heterogeneous parallel computer system without introducing any constraint on the kind of interconnection between the heterogeneous nodes. Furthermore, in [24], systems with the same interconnection Modeling Variation of Waiting Time of Distributed Memory Heterogeneous Parallel Computer System Using Recursive Models Oguike, O.E., Agu, M.N., Echezona, S.C. cpu queue Parallel Pocessors cpu queue cpu queue cpu queue I/O queue I/O queue I/O queue I/O processors