ISBIS4 Abstract

Contact Author's Name: Lidia Filus
Title of Abstract: Construction of a New Class of Non-Markovian Stochastic Processes with Applications to Reliability of Systems with Repair
Author(s): Jerzy Filus*, Lidia Filus**
Affiliation: *Oakton Community College, Des Plaines, IL, USA; **Northeastern Illinois University, Chicago, IL, USA

A new class of discrete time non-markovian stochastic processes is constructed in order to model the maintenance procedures of deteriorating, repairable systems with fully observable states. Suppose there is a predetermined fixed finite set (\'list\') of possible causes or symptoms of any failure that may occur during the system\'s operation. Any non-empty subset of this set will be called \"failure type\". With any such failure type there is to be associated a (finite) set of available repairs. One repair from this set is to be chosen to put the system on. The non-negative r.vars. (the components X(1), X(2),... of the constructed stochastic process) are to be interpreted as lengths of time between consecutive failures (each followed by a repair). We assume that X(1) has a weibullian pdf, while for k = 2,3,... , X(k) has a weibullian conditional pdf, given the times X(1),...,X(k-1) whose values at \'the moment\' (k-1) are considered to be known. The dependences structure among the r.vars. X(1),...X(k), k = 3,4,... is characterized by the following two requirements:
1) the parameters of the conditional weibullian pdf of X(k) are continuous \"parameter functions\" whose arguments are values of X(1),..., X(k-1),
2) the parameter functions themselves contain parameters that depend on a given set of (k-1) ordered pairs:
[type of j-th failure; a corresponding choice of repair], j = 1,...,(k-1). Notice that the conditional pdf of
X(k) (k = 3,4,...) depends on the entire past of the system operation history, i.e., not only on the system\'s state at time (k-1). This relaxation of the usual markovian type requirements is expected to enhance the accuracy of predictions and thereby improve control of the system\'s performance after, say, the (k-1)th repair (so that a \"more rational\" choice of the underlying repairs may be expected). We assume furthermore that with each type of repair a certain cost is associated. This may give rise to various optimization problems that may, for example, be aimed at balancing total repair costs with gains in efficiency of system performances.

The underlying stochastic processes class may also serve as a model in an actuarial framework. One only needs to reinterpret some basic notions, preserving most of the formal structure.

We would like to mention that the obtained new stochastic processes are also \"nice\" as mathematical entities, and easy in analytical handling. Many of the stochastic quantities such as moments, correlations, and regressions can readily be obtained by simple calculations. Similarly, the models obtained by replacing the underlying weibullian pdfs for the ones belonging to some other pdf classes such as those of gammas, generalized gammas or the gaussians turn out to be nice and easy in analytical calculations.

Application of the models constructed probably goes far beyond the reliability and the actuarial frameworks. The construction of an extension of the class of (discrete time) stochastic processes towards analog processes with continuous time can readily be demonstrated.