Reliability covering problems

Abstract

Abstract This chapter studies a type of reliability problem in which a given set S needs to be covered by certain of its subsets. The subsets are available for use on a random and independent basis with known probabilities. It is of interest to calculate the probability that the entire set S is ‘covered’ by the operating subsets: i.e., every element of Sis present in some available subset. As an example of such a problem, consider a mass transit system containing a number of bus routes and the stops they serve. Because of maintenance or staffing problems, a particular bus route might not always be in service: specifically the route is only known to operate (or be available) with some fixed probability. A useful performance measure for the transit system is therefore the probability that each stop is served by some operating bus route. There are a number of other possible applications of this model, such as evaluating the reliability of delivery routes (needed to cover customer demands) or determining the reliability of flight schedules for aircraft (serving airports). Another application is discussed more fully in Section 7.1, and a general description of the reliability covering problem is presented in Section 7.2. It is shown that solving the covering problem is, in fact, equivalent to calculating the reliability of a coherent system, which is known to be mathematically difficult. Consequently, the general covering problem is provably hard.