One of the most widely used techniques to obtain transient measures is the uniformization method. However, although uniformization has many advantages, the computational cost required to calculate transient probabilities is very large for stiff models. We study efficient solutions that can be applied to an approximate method developed for calculating transient state probabilities of Markov models and cumulative expected reward measures over a finite interval. Our work is based on a method that approximates the state probabilities at time t by the state probabilities calculated at a random time with Erlangian distribution. The original method requires an inversion of a matrix obtained from the state transition rate matrix that destroys special structures such as sparseness and banded matrices. This precludes the use of the technique for large models. In our work we propose efficient solutions that can take advantage of special structures. Finally, we present examples that show that the proposed technique is computationally very efficient for stiff models when compared with uniformization.

Software validation is an important activity in order to test whether or not the correct software has been developed. Several testing techniques have been developed, and one of these is statistical usage testing (SUT). The main purpose of SUT is to test a software product from a user's point of view. Hence, usage models are designed and then test cases are developed from the models. Another advantage of SUT is that the reliability of the software can be estimated. In this paper, Markov chains are used to represent the usage models. Several approaches using Markov chains have been applied. This paper extends these approaches and presents a new approach to estimate the reliability from Markov chains. The reliability estimation is implemented in a new tool for statistical usage testing called MaTeLo. The tool is developed in a joint European project involving six industrial partners and two university partners. The purpose of the tool is to provide an estimate of the reliability and to automatically produce test cases based on usage models described as to Markov models.

M.D. Beaudry (1978) proposed a simple method of computing the distribution of performability in a Markov reward process. Two extensions of Beaudry's approach are presented. The authors generalize the method to a semi-Markov reward process by removing the restriction requiring the association of zero reward to absorbing states only. The algorithm proceeds by replacing zero reward nonabsorbing states by a probabilistic switch; it is therefore related to the elimination of vanishing states from the reachability graph of a generalized stochastic Petri net and to the elimination of fast transient states in a decomposition approach to stiff Markov chains. The use of the approach is illustrated with three applications.>

We consider a queueing system with a general service distribution having a possibility of feedback. The customers belong to the same class, and the queueing discipline is first-in, first-out. There is only one server in the station. The facility is rendered Markovian by means of fictitious stages. The input flow depends on the state of the station. It is shown that the equilibrium probabilities can be simply expressed by means of a matrix product. Two particular cases are studied.

In this paper, we present an approximate solution for the asymptotic behavior of relatively general queueing networks. In the particular case of networks with general service time distributions (i.e., fixed routing matrix, one or many servers per station, FIFO discipline), the application of the method gives relatively accurate results in a very short time. The approximate stationary state probabilities are identified with the solution of a nonlinear system. The proposed method is applicable to a larger class of queueing networks (dependent routing matrix, stations with fimite capacity, etc.). In this case, the structure of the network studied must satisfy certain decomposability conditions.