Recent evidence suggests that some characteristics of computer and telecommunications systems may be well described using heavy tailed distributions — distributions whose tail declines like a power law, which means that the probability of extremely large observations is non-negligible. For example, such distributions have been found to describe the lengths of bursts in network traffic and the sizes of files in some systems. As a result, system designers are increasingly interested in employing heavy-tailed distributions in simulation workloads. Unfortunately, these distributions have properties considerably different from the kinds of distributions more commonly used in simulations; these properties make simulation stability hard to achieve. In this paper we explore the difficulty of achieving stability in such simulations, using tools from the theory of stable distributions. We show that such simulations exhibit two characteristics related to stability: slow convergence to steady state, and high variability at steady state. As a result, we argue that such simulations must be treated as effectively always in a transient condition. One way to address this problem is to introduce the notion of time scale as a parameter of the simulation, and we discuss methods for simulating such systems while explicitly incorporating time scale as a parameter. 1

We present a family of Markov models for analyzing the performance of parallel /distributed processors that execute a job consisting of N independent tasks in parallel using P processors. The model is a Markov Chain with states representing service and failure rates with k (0 ! k P ) active processors. The task-times and processor failures are both exponentially distributed. We derive a number of expressions to determine the mean execution time, probability of success, work, and other measurable quantities, all conditioned on the job finishing successfully. A prototype, implemented using an extended version of ACMPI, is used for actual experiments that are based on simulated task-times and processor failures. We present our results comparing the analytic model with the prototype for a range of values of processor failure rates. We then discuss extensions of the model and issues related to communication costs, approximations and effect of task-time distributions.

Abstract. Jackson networks have been very successful in so many areas in modeling parallel and distributed systems. However, the ability of Jackson networks to predict performance with system changes remains an open question, since Jackson networks do not apply to systems where there are population size constraints. Also, the product-form solution of Jackson networks assumes steady state systems exponential service centers with FCFS queueing discipline. In this paper, we present a transient model for Jackson networks. The model is applicable under any population size. This model can be used to study the transient behavior of Jackson networks and if the number of customers in the network is large enough, the model accurately approaches the product-form solution (steady state solution). Finally, an approximation to the transient model using the steady state solution is presented. 1.

DRAFT \Gamma WORKING COPY1 ss := steady-state; S-S := Self-Similar; PT := Power Tail; TPT :=Truncated Power Tail; r.v. := random variable; iid := Independent, Identically Distributed; CLT := Central Limit Theorem; SM := Semi-Markov . Key Words: Power-Tail distributions, Telecommunications Networks, World-Wide-Web, Self-Similar Processes, Long-Range Autocorrelation, Buffer Overflow, Transient Behavior 1 Department of Computer Science and Engineering, and Taylor L. Booth Center for Computer Applications and Research, University of Connecticut, Storrs, CT 06269-3155 2 Department of Computer Science, Boston University, Boston, MA 3 Institut fur Informatik, Technische Universitat Munchen, Munchen, Germany List of Figures 1 Truncated power-tail reliabilities RYN for N 2 f1; 2; : : : ; 9; 12; 100g , ff = 1:5 and ` = 0:5 , plotted on a log-log scale. For N = 100, log[R(\Delta)] is a straight line for many orders of magnitude of change in x. Even with only 12 terms, log[R(\Delta)] lo...

In studying or designing parallel and distributed systems one should have available a robust analytical model that includes the major parameters that determines the system performance. Jackson networks have been very successful in modeling parallel and distributed systems. However, the ability of Jackson networks to predict performance with system changes remains an open question, since they do not apply to systems where there are population size constraints. Also, the product-form solution of Jackson networks assumes steady state and exponential service centers or certain specialized queueing disciplines. In this paper, we present a transient model for Jackson networks that is applicable to any population size and any finite workload (no new arrivals). Using several Erlangian and Hyperexponential distributions we show to what extent the exponential distribution can be used to approximate other distributions and transient systems with finite workloads. When the number of tasks to be executed is large enough, the model approaches the product-form solution (steady state solution). We also, study the case where the nonexponential servers have queueing (Jackson networks can’t be applied). Finally, we show how to use the model to analyze the performance of parallel and distributed systems.

In studying or designing parallel and distributed systems one should have available a robust analytical model that includes the major parameters that determines the system performance. Jackson networks have been very successful in modeling parallel and distributed systems. However, Jackson networks have their limitations. In particular, the product-form solution of Jackson networks assumes steady state and exponential service centers with certain specialized queueing discipline. In this paper, we present a performance model that can be used to study the transient behavior of parallel and distributed systems with finite workload. When the number of tasks to be executed is large enough, the model approaches the product-form of Jackson networks (steady state solution). We show how to use the model to analyze the performance of parallel and distributed systems. We also use the model to show to what extent the productform solution of Jackson networks can be used.