~ 1 Pergamon Microelectron. Reliab., Vol. 37, No. 5, pp. 739-741, 1997 Copyright © 1997 ElsevierScienceLtd Printed in Great Britain.All rights reserved 0026 2714/97$17.00+.00 PII: S0026-2714(96)00098-4 A DELAY-DIFFERENTIAL EQUATION RELATED TO A RENEWABLE PARALLEL SYSTEM E. J. VANDERPERRE Faculty of Engineering, King Mongkut's Institute of Technology. Ladkrabang, 10520 Bangkok, Thailand S. VANNAKRAIROJN Faculty of Information Technology, King Mongkut's Institute of Technology, Ladkrabang, 10520 Bangkok, Thailand and S. S. MAKHANOV Department of Civil Engineering, Faculty of Engineering, King Mongkut's Institute of Technology, Ladkrabang, 10520 Bangkok, Thailand (Received fi~r publication 2 May 1996) Abstract--We consider Gaver's parallel system sustained by a cold stand by unit and attended by two identical repairmen. The system satisfies the usual conditions (i.i.d. random variables, perfect repair, instantaneous and perfect switch, queueing). Each operative unit has a constant failure rate and a deterministic repair time. We analyze the total joint idle time of both repairmen during the survival time of the system. The analysis requires the solution of a so-called delay-differential equation. A numerical example illustrates the structure of the solution for some particular values of the underlying parameters. Copyright (" 1997 Elsevier Science Ltd. INTRODUCTION Parallel systems (for instance two power generators, in active redundancy [1], connected with the light-plant of a tunnel) are widely used to increase the reliability of industrial plants. Gaver's two-unit parallel system [2], sustained by a cold or warm standby unit [1] and attended by a single repair facility, henceforth called an S-system, has received considerable attention [3-9]. As a variant, we recently proposed to analyze the reliability of Gaver's parallel system sustained by a cold standby unit but attended by two identical repairmen, called a T-system [10]. The T-system satisfies the usual conditions (i.i.d. random variables, perfect repair [11], in- stantaneous and perfect switch [1], queueing). Both repair facilities are jointly busy, if and only if, at least two units are in failed state. In any other case, at least one repairman is idle. It is evident that the proposed T-system reduces the waiting time for repair with respect to a similar S-system. Therefore, a T-system improves the reliability of the corresponding S-system. At present, we analyze the total joint idle time of both repairmen during the survival time of the T-system. Apart from the usual conditions, we suppose that each operative unit has a constant failure rate [12] and a deterministic repair time. The analysis requires the solution of a so-called delay-differential equation. A numerical example illustrates the structure of the solution for some particular values of the underlying parameters. FORMULATION Consider a T-system satisfying the usual conditions. Each operative unit has a constant failure rate 2 > 0 and a deterministic repair time distribution R(t - to), where R(.) is the heaviside unit-step function with jump at t = to > 0. In order to describe the different states of the T-system, we introduce a stochastic process {Nr, t~>0 } with arbitrary state space '~A, B, C, D} c [0, ~) characterized by the following events. {Nr = A}: 'Both repairmen are idle at time t, i.e. two units are operating in parallel sustained by a cold standby unit.' Figure 1 shows a functional block diagram of the T-system operating in the renewal state A. {N, = B~: 'One repairman is busy at time t, i.e. two units are operating in parallel and one unit is in repair.' Note that by assumption, both repairmen are statistically equal. Hence, it is by no means necessary to specify which repairman is busy or idle at time t. Figure 2 shows a functional block diagram of the T-system operating in state B. {N t = C}: 'Both repairmen are jointly busy and only one unit is operative at time t.' [N~ = D}: 'Both repairmen are jointly busy and a 739