A.P.A. van Moorsel

Transient solution of Markov models by combining adaptive and standard uniformization

Adaptive uniformization (AU) has been proposed to compute transient measures in continuous-time Markov chains and is especially attractive for solving large and stiff dependability models. The major advantage of AU is that it requires at most as many iterations as standard uniformization (SU), and often far fewer, thus resulting in substantial computation savings. However, this computation gain can be offset by the need to compute more complex jump-probabilities in AU, whose computation is more expensive than computing Poisson probabilities in SU. In particular, AU is computationally superior to SU if and only if the considered time instant is less than some threshold time value. To overcome this drawback, AU and SU are combined (AU/SU) so that AU is used early in the time interval, and SU is used over the rest of the time interval. AU/SU can be implemented so that: (1) the combination introduces only minor computation overhead, (2) the number of iterations required is almost as low as for AU, and (3) the cost of computing the jump probabilities is about as low as for SU. AU/SU yields a strict lower bound of the true result, within any desired pre-specified accuracy. The error bounds include the error introduced when the Fox/Glynn algorithm is used for computing Poisson probabilities; this algorithm is enhanced to optimize its error-bound characteristics. To demonstrate the benefits of AU/SU it is applied to a machine-repair model, using a version of combined AU/SU implemented in UltraSAN, a performance and dependability evaluation software package.

Optimization of reliability allocation and testing schedule for software systems

To ensure an overall reliability of an integrated software system, software components of the system have to meet certain reliability requirements, subject to some testing schedule and resource constraints. The system testing activity can be formulated as a combinatorial optimization problem with known cost, reliability, effort and other attributes of the system components. In this paper, we consider the software component reliability allocation problem for a system with multiple applications. The failure rate of components used to build the applications are related to the testing cost through various types of reliability growth curves. We achieve closed-form solutions to problems where there is one single application in the system. Analytical solutions are not readily available when there are multiple applications; however, numerical solutions can be obtained using a nonlinear programming tool. To ease the specification of the optimization problem, we develop a GUI front-end to existing mathematical software. We present a systematic outline of the problem formulation and solution, and apply this to an example of a telecommunication software system.

Probabilistic evaluation for the analytical solution of large Markov models: Algorithms and tool support

Abstract Over the past decade constantly increasing computer power has made analytic solution of Markovian performance and dependability models more attractive. However, its application for practical systems still needs improvement since detailed and realistic models typically result in Markov (reward) models that are too large to completely generate and store in memory. In this paper we discuss this problem of largeness of Markov reward models and propose solutions if transient measures are considered. We will introduce a novel approach, called probabilistic evaluation, based on the probabilistic verification and validation methods known in the area of communication protocol analysis. In order to apply these ideas to solve for transient reward measures, we develop algorithms that do not rely on a priori generation of the whole state space. Instead, only parts of the state space will be considered at a time, and states that are not needed to get accurate results will not be generated. We will not only deal with acyclic models but allow for general Markov reward models. The algorithms we develop are based on uniformization. For acyclic models we introduce orthogonal uniformization and for the analysis of non-acyclic models we introduce partial uniformization . For the utility of the above new uniformization methods the order in which the states are generated from the high-level description will turn out to be of great importance. We propose the use of a simple heuristic method for state selection, which, after comparison with two other methods, appears to be relatively cheap and close to optimal when considering transient dependability measures. The uniformization and state space exploration methods we develop are implemented in a prototype probabilistic evaluation tool, which is especially useful for experimenting with new algorithms.

Computation of the asymptotic bias and variance for simulation of Markov reward models

The asymptotic bias and variance are important determinants of the quality of a simulation run. In particular, the asymptotic bias can be used to approximate the bias introduced by starting the collection of a measure in a particular state distribution, and the asymptotic variance can be used to compute the simulation time required to obtain, a statistically significant estimate of a measure. While both of these measures can be computed analytically for simple models and measures, e.g., the average buffer occupancy of an M/G/1 queue, practical computational methods have not been developed for general model classes. Such results would be useful since they would provide insight into the simulation time required for particular systems and measures and the bias introduced by a particular initial state distribution. We discuss the numerical computation of the asymptotic bias and variance of measures derived from continuous-time Markov reward models. In particular, we show how both measures together can be efficiently computed by solving two systems of linear equations. As a consequence of this formulation, we are able to numerically compute the asymptotic bias and variance of measures defined on very large and irregular Markov reward models. To illustrate this point, we apply the developed algorithm to queues with complex traffic behavior, different service time distributions, and several alternative scheduling disciplines that may be typically encountered in nodes in high-speed communication networks.

UltraSAN version 3 overview

UltraSAN is a software package for model-based evaluation of systems represented as stochastic activity networks. The software has been implemented in a modular fashion, with clearly delineated interfaces between model specification, construction, and solution. UltraSAN offers an X Windows-based user interface and both analytical and simulation solution modules for transient and steady-state performance, dependability, and performability measures. Furthermore, the tool facilitates graphical representation of the obtained results by its report generator. This paper gives a very brief overview and pointers to more detailed descriptions of the software.