On some open problems in reliability theory

TL;DR

This paper investigates a stochastic scheduling model on unreliable machines, deriving explicit solutions in special cases, analyzing asymptotic behavior, and exploring the applicability of semigroup theory, while also identifying open problems.

Contribution

It provides explicit solutions for special cases, analyzes asymptotic behavior, and formulates the model as an abstract Cauchy problem, highlighting limitations of existing mathematical approaches.

Findings

Explicit solution in special case with constant rates

Asymptotic behavior of the solution determined

Laplace transform of the general case obtained

Abstract

We study a stochastic scheduling on an unreliable machine with general up-times and general set-up times which is described by a group of partial differential equations with Dirac-delta functions in the boundary and initial conditions. In special case that the random processing rate of job $i,$ the random up-time rate of job $i$ and the random repair rate of job $i$ are constants, we determine the explicit expression of its time-dependent solution and give the asymptotic behavior of its time-dependent solution. Our result implies that $C_0-$semigroup theory is not suitable for this model. In general case, we determine the Laplace transform of its time-dependent solution. Next, we convert the model into an abstract Cauchy problem whose underlying operator is an evolution family. Finally, we leave some open problems.