V. D. Dinopoulou^*, C. Melolidakis^**

 ^*Dept. of Production Engineering and Management,
Technical University of Crete, GR-73100 Chania, Greece
^**Dept. of Mathematics, University of Athens, 15785 Athens, Greece

A continously operating system consisting of N K_i-out-of-N_i subsystems connected in series is considered. The components of all subsystems are assumed identical with life times independent exponentially distributed random variables. The system is maintained by a single repaiman and the repairtimes are assumed independent exponentials. In the present paper, we examine the optimal repair allocation policy for highly reliable systems. In particular, we show that there exists (and we describe it explicitly) an easy to compute repair allocation policy, which maximizes the reliability of the system at any time instant t. This policy is unique and depends only on the state and the structure of the system (and not on the failure and repair rates). We also compute the leading term of a power series expansion of the reliability of the system at an arbitrary time instant t under the optimal policy. Finally, these results are extented 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.

Keywords: Controlled Markov Processes, Reliability, Queueing System.