Asymptotically Optimal Repair Policies in Two-tier Reliability Systems and Optimal Control to Multiserver Queuing Systems.

 Abstract

 A continuously operating system consisting of N K_i-out-of-N_i, i=1,…, N subsystems is considered. The system functions when at least K out of its N component-subsystems are operational. The components of all subsystems are assumed identical with life times independent exponentially distributed random variables. A single repairman maintains the system and repair times are assumed independent exponentials.

In the present thesis, we examine the optimal repair allocation policy for systems consisting of highly reliable components. (A) For parallel systems, an index type repair allocation policy is constructed which maximizes the expected discounted system operation time for component failure rate sufficiently small and discount rate sufficiently small or sufficiently large. Also, we give a partial characterization of the policy, which maximizes the system reliability at any time instant. (B) For serial systems, we show that there exists (and we describe it explicitly) a repair allocation policy, which maximizes the reliability of the system at any time instant. (C) These results are extended to the problem of controlling the corresponding network of parallel queues in a scheduling problem with long mean arrival times and in its dual routing problem for long mean processing times. (D) We give the optimal feasible assembly plan of systems connected in series or in parallel. (E) Finally, we examine the general case for any K-out-of-N system.