Pergamon Microelectron. Reliab., Vol. 36, No. 2, pp. 255-259, 1996 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0026-2714/96 $9.50+.00 0026-2714(95)00006-2 ON THE CHARACTERISTICS OF A SYSTEM HAVING MASTER AND HELPING UNIT S. K. Singh and Sheeba G. Nair School of Studies in Statistics, PT. Ravishankar Shukla University, Raipur (M.P.) 492010, India (Received for publication 1 December 1994) Abstract--A system having one master and one helping unit with two failure modes--partial and total--is analysed. The helping unit is used to support the master unit in operation. Whenever the helping unit fails it is either repaired or replaced with probability p(q). Failure time distributions are taken to be negative exponential whereas repair time distributions are taken to be arbitrary. Using the regeneration point technique, several system characteristics such as mean time to system failure, availability, busy period of the repairman, etc. are obtained. Finally, some graphs are drawn in order to highlight the important results in particular cases. 1. INTRODUCTION Two unit cold standby systems with two modes, operative and failed, have been widely studied by several authors, in the field of reliability. Procter and Singh [1] have first incorporated the concept of three modes--normal, partial failure and total failure-- and obtained several characteristics of the system under study. In all the standby systems [1-8], standby units are used only to increase the reliability of the system. However, there are many electrical and electronic systems in which reliability may also be increased by making some arrangements with the system. For example in a T.V. set, a stabilizer is used to control the power supply. Besides this, in many systems operation of the main unit depends upon the helping unit. For example, in a steel plant essential materials for processing needed for the main steel processing units are supplied by different attached helping units. A very few attempts have been made to study the models related to helping units. The purpose of the present paper is to study a system having one master and one helping unit. Initially both units are in operation, Whenever the master unit works without the helping unit, its failure rate increases. There is a single repair facility which repairs all the failed units. Priority in repair is given to the helping unit. After repair each unit acts as new. The replacement time of the helping unit is taken to be negative exponential. Using the regeneration point technique the following measures of system effective- ness are obtained: mean time to system failure (MTSF); pointwise availability in (0, t] and in steady state; expected busy period of the repairman in (0, t] and in steady state; expected busy period of the repairman under replacement in (0, t] and in steady state; and expected profit earned by the system in (0, t] and in steady state. 2. NOTATION fl, (i = 1, 2) h,d P q yi(t), Gi(t) (i = 1, 2) ga(t), G3(t) partial failure rate of the master unit complete failure rate of the master unit from normal/partially failure mode constant failure and replacement rate of the helping unit probability [failed helping unit under repair] probability [failed helping unit under replace- ment] p.d.f, and c.d.f, of repair rate of the partially/ completely failed master unit p.d.f, and c.d.f, of repair rate of the completely failed helping unit Symbols used for the states M0, M, M.r, Mpw. Mr, Mwr I4o, nr, t4~ master unit is under operation/good and non- operative master unit fails partially and is under repair/ waiting for repair master unit fails completely and is under repair/waiting for repair helping unit is under operation/fails and is under repair/fails and is under replacement Other notations are similar to Ref. [8]. Possible transitions between states along with the transition rates are shown in Fig. 1. 3. TRANSITION PROBABILITIES AND MEAN SOJOURN TIMES Simple probabilistic considerations yield the following expressions for non-zero transition prob- abilities Pi/ fo Pol = ~l e-('~+Pl+h)f dt = ~lAl-ll; similarly Po~ = fllAll 1, Po3 = phAll 1, Po4 = qhAxt x 255