A simple technique for computing mean performance measures of closed single-class fork-join networks with exponential service time distribution is given here. This technique is similar to the mean value analysis technique for closed product-form networks and iterates on the number of customers in the network. Mean performance measures like the mean response times, queue lengths, and throughput of closed fork-join networks can be computed recursively without calculating the steady-state distribution of the network. The technique is based on the mean value equation for fork-join networks which relates the response time of a network to the mean service times at the service centers and the mean queue length of the system with one customer less. Unlike product-form networks, the mean value equation for fork-join networks is an approximation and the technique computes lower performance bound values for the fork-join network. However, it is a good approximation since the mean value equation is derived from an equation that exactly relates the response time of parallel systems to the degree of parallelism and the mean arrival queue length. Using simulation, it is shown that the relative error in the approximation is less than 5% in most cases. The error does not increase with each iteration.

Abstract—This paper introduces queuing network models for the performance analysis of SPMD applications executed on generalpurpose parallel architectures such as MIMD and clusters of workstations. The models are based on the pattern of computation, communication, and I/O operations of typical parallel applications. Analysis of the models leads to the definition of speedup surfaces which capture the relative influence of processors and I/O parallelism and show the effects of different hardware and software components on the performance. Since the parameters of the models correspond to measurable program and hardware characteristics, the models can be used to anticipate the performance behavior of a parallel application as a function of the target architecture (i.e., number of processors, number of disks, I/O topology, etc). Index Terms—Single program multiple data (SPMD), multiple instruction multiple data (MIMD), performance model, queuing network model, fork-join queues, mean value analysis (MVA), parallel I/O, synchronization overhead, speedup surface. 1

In this paper, we first give a characterization of a class of probability transition matrices having closed-form solutions for transient distributions and the steady-state distribution. We propose to apply stochastic comparison approach to construct bounding chains belonging to this class. Therefore bounding chains can be analyzed efficiently through closed-form solutions in order to provide bounds on the distributions of the considered Markov chain. We present algorithms to construct upper bounding matrices in the sense of the ≤st and ≤icx order.

The fork-join queue models parallel resources where arriving jobs divide into various number of sub-tasks that are assigned to unique devices within the parallel resource. Each device in the parallel resource is modeled  ¢¡ £  ¢¡¥ ¤ by queueing servers. A job completes execution and departs the parallel resource after all its sub-tasks complete execution. This paper analyzes ¦-server fork-join queues where arriving jobs divide into ¤¨§�©�§ are assigned to unique servers of the fork-join queue. There is no known closed-form solution for ¦��� � fork-join queues. The paper presents an O(log K) algorithm for computing the mean response time pessimistic and optimistic bounds and for computing the mean response time approximation of the fork-join queue. The error bounds for the response time bounds and approximation are presented. Index Terms: fork-join synchronization, performance evaluation, parallel computer and storage systems. 1

family of fast and accurate non-iterative bounds on closed queueing network performance metrics that can be used in the on-line optimization of distributed applications. Compared to state-of-the-art techniques such as the Balanced Job Bounds (BJB), the GB achieve higher accuracy at similar computational costs, limiting the worst-case bounding error typically within 5%−13 % when for the BJB it is usually in the range 15%−35%. Optimization problems that are solved with the GB bounds return solutions that are much closer to the global optimum than with existing bounds. We also show that the GB technique generalizes as an accurate approximation to closed fork-join networks commonly used in disk, parallel and database models, thus extending the applicability of the method beyond the optimization of basic product-form networks. Index Terms — Non-iterative bounds, performance optimization, closed queueing networks, fork-join systems.

Stochastic bounds are a promising method to analyze QoS requirements. Indeed it is sufficient to prove that a bound of the real performance satisfies the guarantee. However, the time and space complexity issues are not well understood so far. We propose a new algorithm to derive a strong stochastic bound of a Markov chain, using a matrix pattern specifing the structural properties a bounding matrix should comply with. Thus we can obtain a simpler Markov chain bounding for which the numerical computation of the steady-state solution is easier. 1.