## Dissertations from ProQuest

#### Title

The joint imputed Poisson distribution and its application to minimum cost maintenance scheduling

1993

Article

#### Degree Name

Doctor of Philosophy (Ph.D.)

#### Department

Interdepartmental Studies

#### First Committee Member

Moshe F. Friedman - Committee Chair

#### Abstract

Sophisticated electrical systems generally have circuit breakers designed to avert extensive damage by shutting the system down in the event of power surges. One such example is the network employed by electric utility companies to provide power to their customers. Circuit breakers, installed at various locations along the network, protect sensitive electrical units from random overload due to lightning strikes.When a circuit breaker has unknowingly worn out, a subsequent lightning strike produces a very costly "blackout," causing considerable damage to and a lengthy repair time for the system. There are only two ways to discover a non-functioning circuit breaker; either by suffering a blackout, or by detecting it during periodic testing. Maintenance costs for checking, and when necessary, replacing worn out circuit breakers must be balanced against the potentially more expensive costs of blackouts.The problem is modeled as two interrelated Poisson processes. We define the random variable M as the number of lightning strikes (imputing process with observable occurrences) and N as the number of circuit breaker failures (imputed process with occurrences that cannot be observed) which occur during the time period (0,$\tau$).General formulas for the Joint Imputed Poisson Distribution P(m,n;$\tau$) and Conditional Imputed Poisson Distribution P(n$\vert$m;$\tau$) are motivated and proven. The thesis also develops expressions for both marginal distributions, P(m;$\tau$) which is simple Poisson, and P(n;$\tau$) the Imputed Poisson Distribution. Computational efforts, complete with computer graphics, reveal the general behavior of these distributions for selected values of their parameters. In addition, exact expressions for the expectations, variances, and covariance of the number of occurrences in the interrelated processes are also derived.The thesis concludes with a determination of the optimal period $\tau\sp*$ between circuit breaker maintenance checks that minimizes the total expected cost per unit time on (0,$\tau\sp*$). While it is in general impossible to derive a closed form expression of $\tau\sp*$, we demonstrate that the expected total cost per unit time is a unimodal function of $\tau.$ Then an iterative procedure to obtain $\tau\sp*$, based on Newton's Method, is detailed.

#### Keywords

Mathematics; Operations Research