COMPARATIVE STUDY OF RELIABILITY MODELS IN RESPECT OF PROFIT AND AVAILABILITY OF SINGLE UNIT SYSTEMS WITH DIFFERENT REPAIR POLICIES.

: In this Research paper two reliability models of a single unit system are analysed in detail using regenerative point technique. The unit fails completely via partial failure. There is single server who appears and disappears randomly from the system. However, he attends the system immediately at complete failure of the unit in Model-I. Server inspects the unit at its partial failure to see the possibility of on-line repair. If online repair is not possible, it is repaired in down state. The server cannot leave the system during inspection and repair. The distributions of failure time, random appearance and disappearance of the server are taken as negative exponential while that of inspection and repair time are arbitrary. All random variables are uncorrelated and independent to each other. The expressions for some reliability and economics measures are derived. The results for a particular case are also obtained to depict the behaviour of MTSF, availability and profit of the system models graphically.


INTRODUCTION
In real life, it is observed that when a system is not repaired at the proper time and partially failed unit is taken out from the system for repair, the system suffers operational as well as economical loss. For example, when online repair of the system of an electric transformer is not done at its partial failure due to the defects in cooling, in temperature indicators and non-functioning of on-load tap changer, then it may be damaged completely. However, on-line repair of the electric transformer is not possible when it fails partially due to the problems in its protective system. In such a situation inspection of the system can be done to reveal the possibility of on-line repair. If on-line repair is not possible, its repair can be done in the down state. Several scholars including R.S. Naidu and M.N. Gopalan [1984] [1], K. Murari and V. Goyal [1984] [2]and Makaddis et al. [1989] [3]. have analysed reliability models ignoring the above observations under the assumptions that repair of the system is possible only at its complete failure and server never attends the system immediately. But sometimes it is not possible for the server to attend the system immediately may because of his pre-occupations. In such a situation, system remains down. Therefore, it becomes necessary to protect the operation of the working unit as long as possible after its partial failure so that reliability and profit of the system may increase. S.K. Singh [1989] [4]evaluated profit of a system with random appearance and disappearance of the server. In view of the above the present study focused on the analysis of two reliability models for a single unit system in which unit fails completely via partial failure. There is a single server who inspects the unit at its partial failure to examine the feasibility of on-line repair. If on-line repair of the unit is not feasible, it is repaired in down state, that is, after stopping the operation of the system. In Model-I, server appears and disappears randomly with some probability while in Model-II, he may be called immediately, whenever unit fails completely. The regenerative point technique is adopted to derive the expressions for some reliability and economic measure such as transition probabilities, mean sojourn times, mean time to system time failure (MTSF), steady state availability, busy period of the server, unexpected number of visits and profit function. The numerical results for a particular case are also obtained for both the models to draw the graphs. System Description and Assumptions 1.
The system consists of two single-unit models. Initially the unit is operative and fails completely via partial failure.

2.
There is a single server who appears and disappears randomly with some probabilities from the system.

3.
The server cannot leave the system while repairing the unit. 4.
The server attends the system immediately when the system fails completely in Model-II. 5.
The repair and inspection are perfect. 6.
The unit under repair at partial failure mode in down-state cannot fail. 7.
Each unit has an exponential distribution of time to failures while distributions of repairs and inspections are arbitrary. 8.
All the random variables are mutually independent.

Busy Period Analysis Let
( )be the probability that the server is busy in repairing the unit at an instant 't' given that the system entered regenerative state at t=0. The recursive relations for ( )are as follows: ( )= ( ) + ∑ qij W6(t)=e-1 For Model-II We can obtain the recursive relations for Bi(t) as given on (11) fori= (0,1,2,3,4,5) and j= {(1.2) ;(,3) ;(3,4) ;(4,5,6) ;(1) ;(1,4) ;(1)}. Also Wi(t) = 0 for i=0,1,2 While remaining are Taking L.T. of the above relation (11) and determine B0 * (s)for each model. Using this, we get in the long run, the time for which server is busy as B0(∞) =  Fig.3 shows the graphical trend of MTSF with respect to failure rate r2 for different sets of values of inspection rete( ), failure rate (r1) and repair rate ( 1). From this , we conclude the MTSF decreases with the increase of failure rates r1 and r2.However, it further increases if repair rate ( 1)increases and decreases by the increase of inspection rate ( ). Fig. 4 and 5 show the behaviour of availability of both models respectively w.r.t. Failure rate r2 and from these figures we see that availability decreases with the increase of failure rate (r2) and increases with the increase of inspection rate ( ). Further, availability of Model-II is greater than that of Model-I. Figures. 6 and 7 represent the variations of profits of both models respectively w.r.t. Failure rate (r2) and cost of busy period of server and under some casesModel-I is profitable as compared to Model-II.