Douglas R. Miller

The Randomization Technique as a Modeling Tool and Solution Procedure for Transient Markov Processes

We present a randomization procedure for computing transient solutions to discrete state space, continuous time Markov processes. This procedure computes transient state probabilities. It is based on a construction relating a continuous time Markov process to a discrete time Markov chain. Modifications and extensions of the randomization method allow for computation of distributions of first passage times and sojourn times in Markov processes, and also the computation of expected cumulative occupancy times and expected number of events occurring during a time interval. Several implementations of the randomization procedure are discussed. In particular we present an implementation for a general class of Markov processes that can be described in terms of state space S, event set E, rate vectors R, and target vectors T-abbreviated as SERT. This general approach can handle systems whose state spaces are quite large, if they have sparse generators.

Conceptual modeling of coincident failures in multiversion software

Work by D.E. Eckhardt and L.D. Lee (1985), shows that independently developed program versions fail dependently. The authors show that there is a precise duality between input choice and program choice in this model and consider a generalization in which different versions can be developed using diverse methodologies. The use of diverse methodologies is shown to decrease the probability of the simultaneous failure of several versions. Indeed, it is theoretically possible to obtain versions which exhibit better than independent failure behavior. The authors formalize the notion of methodological diversity by considering the sequence of decision outcomes that constitute a methodology. They show that diversity of decision implies likely diversity of behavior for the different versions developed under such forced diversity. For certain one-out-of-n systems the authors obtain an optimal method for allocating diversity between versions. For two-out-of-three systems there seem to be no simple optimality results which do not depend on constraints which cannot be verified in practice. >

Exponential order statistic models of software reliability growth

Failure times of a software reliability growth process are modeled as order statistics of independent, nonidentically distributed exponential random variables. The Jelinsky-Moranda, Goel-Okumoto, Littlewood, Musa-Okumoto logarithmic, and power law models are all special cases of exponential order statistic models, but there are many additional examples as well. Various characterizations, properties, and examples of this class of models are developed and presented.

A Closed Queueing Network Model for Multi-Echelon Repairable Item Provisioning

Abstract A model of a two-echelon (two levels of repair, one level of supply) repairable-item provisioning system is presented. It is desired to find the capacities of the base and depot repair facilities as well as the spares level which together guarantee a specified system service level at minimum cost. Closed queueing network theory is used to model the stochastic process, and an implicit enumeration algorithm is used to solve the optimization problem.

On the use and the performance of software reliability growth models

Abstract We address the problem of predicting future failures for a piece of software. The number of failures occurring during a finite future time interval is predicted from the number of failures observed during an initial period of usage by using software reliability growth models. Two different methods for using the models are considered: straightforward use of individual models (simple models), and dynamic selection among models based on goodness-of-fit and quality-of-prediction criteria (super models). Performance is judged by the relative error of the predicted number of failures over future finite time intervals relative to the number of failures eventually observed during the intervals. Six simple models and eight super models are evaluated based on their performance on twenty data sets. This study is by no means comprehensive. Some conclusions can be drawn, but many open questions remain regarding the use and the performance of software reliability growth models.

The c — Server Queue with Constant Service Times and a Versatile Markovian Arrival Process

Classical results of C. D. Crommelin on the c-server queue with constant service times and Poisson input are extended to the case of the versatile Markovian arrival process, introduced by the author. The purely probabilistic analysis of a related problem in Markov chains leads to an algorithm for the evaluation of the stationary distributions of the queue length and waiting time at an arbitrary epoch. As an illustration, the algorithmic steps are discussed in detail for the case where the arrivals form a Narkov-modulated Poisson process.

A nonparametric software-reliability growth model

The authors (Proc. Eighth Int. Conf. Software Eng., London, England, p.343-4, 1985) previously introduced a nonparametric model for software-reliability growth which is based on complete monotonicity of the failure rate. The authors extend the completely monotone software model by developing a method for providing long-range predictions of reliability growth, based on the model. They derive upper and lower bounds on extrapolation of the failure rate and the mean function. These are then used to obtain estimates for the future software failure rate and the mean future number of failures. Preliminary evaluation indicates that the method is competitive with parametric approaches, while being more robust. >

Almost Sure Comparisons of Renewal Processes and Poisson Processes, with Application to Reliability Theory

If the interarrival times of a renewal process {Si, i = 0, 1, 2...} have a failure rate function which is bounded away from 0 and ∞, then it is possible to construct nonhomogeneous Poisson processes {Ti0, i = 0, 1, 2,...} and {Ti1, i = 0, 1, 2,...} on the same probability space with {Si, i = 0, 1, 2,...} such that {T00, T10, T20,...} ⊂ {S0, S1, S2,...} ⊂ {T01, T11, T21,...} almost surely. This has applications to the reliability theory of maintained systems. If the components of a maintained coherent system have exponential lifetimes and NWU repair times, then an initially perfect system will have a NBU distribution of time until first system failure. Furthermore, transient availability is greater than steady-state availability.

On Some Common Interests Among Reliability, Inventory, and Queuing

Queuing networks can be used to model maintained systems. Under many conditions, closed-network queuing theory can be applied to ascertain the availability of such systems. Multi-echelon repairable-item inventory systems are one such class. Problems of common interest to the reliability, queuing, and inventory communities are highlighted, and solution techniques for these problems are presented.

On the Quality of Software Reliability Prediction

We suggest that users are interested solely in the quality of predictions which can be obtained from software reliability models. Some ways of analysing the quality of predictions are proposed and several models and inference procedures are compared on real software failure data sets. We conclude that some predictions are extremely poor: notably those arising from ML analysis of the Jelinski-Moranda model. Others seem quite good. We suggest promising areas for future work.