STOCHASTIC ANALYSIS OF A TWO UNIT SYSTEM WITH VACATION FOR THE REPAIR FACILITY AFTER m REPAIRS

Stochastic analysis of a two unit system with vacation period for the repair facility after the completion of m repairs is studied. All the underlying distributions are assumed to be non-Markovian. The reliability and availability analysis for such a system is studied. A numerical illustration is given.


INTRODUCTION
Two unit standby redundant repairable systems have attracted the attention of many applied probabilists and system analysts. In the literature available so far on such systems, it is clear that the repair facility is continuously available to attend to the repair of the failed units. However, it is reasonable to expect that a vacation might be needed for the repair facility after 6 repairs, before the next repair could be taken up. Such a vacation period certainly arises in many mechanical and electrical systems. This principle of a vacation period was first introduced by Subramanian and Sarma [2].
Recent developments in the modeling and analysis of highly reliable systems require the use of some more sophisticated models. According to Platis et al. [1] Electricité De France (EDF) used this for the homogeneous Markov approach to model an electrical substation in order to evaluate some measures of system performance. However, in this model, the underlying distributions are all non-Markovian, in the sense that, all the underlying distributions in this model are arbitrarily distributed.
In this paper, the concept of a vacation period for the repair facility after the completion of 6 repairs, and the arbitrary distributions described above have been introduced. The reliability and availability measures have been obtained using the regeneration point technique and product densities. A numerical example illustrates the results in section 7.

SYSTEM DESCRIPTION
(a) The system consists of two identical units: one unit is operating on-line and the other is kept in cold standby. Either unit performs the system functions satisfactorily.
(b) Switch is perfect and switchover is instantaneous.
(c) After the completion of 6 repairs, the repair facility is not available for a random time, denoted by 'vacation time' , the duration of which is governed by an arbitrarily distributed r.v. with p.d.f. E.

AUXILIARY FUNCTIONS
Let us define the following auxiliary functions, which will be used in the reliability and availability analysis.
We know that the events . , 2 are regenerative. Since . can occur in two ways, it may or may not be a regenerative event.
Functions s o E| and s @ E| For ' 2c c c 6 c let Considering all the 6 cases D to I in figure 2, we get Using probabilistic arguments, and observing that the events . c . e c c . 6 successively occur following an . 2 event, we have

RELIABILITY ANALYSIS
We observe that the . 2 events constitute a renewal process. For the sake of simplicity we assume that an . 2 event has occurred at | ' f Let To derive an expression for the reliability -E| of the system, we consider the following mutually exclusive and exhaustive cases: -no . 2 event occurs up to |c or -at least one . 2 event occurs in Efc |o Accordingly we have

AVAILABILITY ANALYSIS
This is defined as the 'probability that the system is able to operate within the tolerances at a given instant of time'. In symbols the pointwise availability is: Using renewal theoretic arguments we get : 8 The steady state availability " can be obtained using the relation   unit is swited on-line.

NUMERICAL ILLUSTRATION
A numerical example of some results obtained for the model are given in this section. For this purpose we assume that The special case of 6 ' is considered.