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Reliability and Cost-Effectiveness of a Redundant System with Three Types of Failures and Repairs

 

Dharmvir Singh Vashisth1, A. S. Gousia Banu2

1Department of Mathematics, R.N. Engg. College, Rohtak (Haryana), INDIA.

2Department of Management, Narayana Engineering College (Nec) Nellore (A.P), INDIA

Correspondence addresses

[email protected] , [email protected]

Research Article

 

Abstract:The goal of this paper is to carry out the reliability and cost effectiveness of a two unit cold standby system with allowed downtime and correlated failure and repair times. For this purpose, a stochastic Model is developed in which one unit is operative and other is kept as cold standby. Each unit has three modes of failure viz. normal, degraded and total failure. The time taken in replacement of a failed unit is negligible but it is a random variable. The failure rates are constant and system repair time are arbitrary distributed. The numerical results for reliability and cost-effectiveness are obtained using supplementary variable technique. The repair and failure times are correlated .

 

Introduction:    In most of the reliability models of dissimilar units studied so far  with three modes, viz, normal, partial and total failure have been considered. Gupta et al. [1] have studied two unit stand by system under partial failure and pre-emplive  repair priority. They have all assumed that failure and repair are independent but it looks in practice that they have certain correspondence. Goel et al. [2] have considered that failure and repair time follow a behaviour exponential distribution. Sharma [4] has initiated the study of availability of the system which consist of two identical cold stand by units with constant failure rates. Initially one unit is operative while the other remains as stand by. Each of the units has three modes, i.e., normal, degraded and total failure. The system fails when both the units fail and may also fail due to common cause failure. The time taken in replacement of a failed unit by a stand by unit is not negligible but it is a random variable. Goel et al. [2] also presented a mathematical model for predicting a two identical active units redundant system with three types of failures namely mechanical, catastrophic and human failure. Any units may fail either partially or completely. And the system is only required when all the units fail including the helping unit. The failure rates are constant and system repair times are arbitrarily distributed. He obtained state probabilities by Laplace transforms.

                In this paper, we are interested to analyse the two unit cold stand by system with allowed downtime and correlated failure and repair times. Using this concept of correlation into consideration the joint distribution of failure and repair times is taken to be bivariate exponential distributed. The numerical results for reliability and cost-effectiveness are obtained to improve the importance of the study.

 

Notations:

 (a)  

(b)                       :      Constant failure rate due to catastrophic failure.

(c)                       :       Constant failure rate due to human failure.

(d)   :  Constant failure rate due to mechanical failure.                             (1)

(e)                       : Constant waiting due to mechanical failure of both the units.

(f)                   (at time t the system is in state

(g)          (the system is in  state at time t due to partial mechanical failure of

                                helping unit and elapsed repair time lies in the interval

(h)          (the system is in state  at time t due to the complete mechanical

                                 failure of helping unit and elapsed repair time lies in the interval

                               

(i)         (the system is in the state  at time t due to catastrophic failure and

                                elapsed repair time lies in the interval

(j)          (the system is in the state  at time t and elapsed repair time in the

                               interval

(k)        (the system is in state  at time t due to human error and elapsed

                              repair time lies in the interval

(l)               (at time t, the system is in state  due to partial failure of maintenance

                               helping unit)

(m)             (at time t, the system is in state  due to partial failure of maintenance

                               and complete failure of helping unit).

(n)              (the system is in state  at time t due to complete failure of unit)

(o)  First order probabilities that failed / failed / operable / waited.

(p):system  repaired in time

(q) ,                 unless otherwise stated.

Transition of diagram


 

 

 

 

 

 


 

                                                       

 

 

 

 

 

 

 

 

 

 

 

 

 

 

                                                                  

 

Text Box: Failed state
Text Box: Noited state
Text Box: Degradded  state
Text Box: Operable  state
Rounded Rectangle: Oval:  

 

 

 

 

 

 

 

 

 

 



 

Formulation of Mathematical Models:

 

 Probabilistic consideration and preceding stated procedures yield the following equations:

     

 

  

                          

               

   

   

                                                

                         

    ......(2)

                                                       

                             

                                             

       

 

 

Initial conditions

 

If a final state, it is ..So otherwise 0.

 

Mathematical Analysis:

 

 Taking Laplace of the equation (2)  we obtain

                          

                                                  

           (3)


 

     

                                                                                                 

                                                                   

                                                                                                                                   

                                                                                                                                      

Gupta, Singh and Kishore [2] also obtained the up and down state Probabilitie

where,    (4)

 

  

 

Ergodic Behavior of the System:

 

Goel, Singh and Kishore [2] using base, we have

                                                                                             

has obtained values as follows:

                                                                                                                   

                 

    (5)

                                  

                 

                                                      

                                      

Also,     (6)

                (7)

where,                

 

Particular Cases:

 

(1) Constant Rate Repair: When repair rate is an exponential setting by

 in relations (6) one may get

Goel, Singh and Kishore [2] the following value

                             

           

  (8)

                    

                         

            

      

                         

     

 

 

 

Non Repairable System:

 

The Laplace system of transform of reliability when all repairs of the system are zero, i.e., given by

                                       

where,    ,                

The reliability of the system is

                                                (9)

where, 

                            

Discussion:

 

 We look parameters as follows as Goel, Singh and Kishore [2].

Let

                                                          (10)

The analyses for availability is obtained as follows:

         

                                                                                                                                               (11)

Calculating values of

T

0

1

2

3

4

5

 

0.99

0.94

0.83

0.77

0.67

0.61

The values obtained above show that decreases as time increases. It can also be shown that increase of also causes decrease in

        Let us consider

                                   (12)

On account of (12), (8) and (9), we get

          and also

         

                                            

                                       R(t)

T

0

1

2

3

4

5

R(t)

1.153

0.552

0.233

0.153

0.132

0.128

 


 

We draw a graph in R(t), i.e., reliability time. It is found that reliability decreases with time initially decreases is sharp but gradually it is a uniform decrease.

 

Cost Effectiveness of the system:

 

        Assuming that the service facility is always available and remains busy for time t during interval ] 0, t [. Let  be revariance cost per unit time and service cost per unit times.

The profit function Profit(t)

        Let,  the expected profit are shown in the following table, we get respectively


 

        


 

          Particular values of  we calculate values of  a profit function, we give them in this table


 

.


 

            Table : 1

Sr.No

Time

If

     G(t)

If

     G(t)

If  

       G(t)

1

2

3

4

5

6

7

0

1

2

3

4

5

10

-0.067

-0.0922

-0.148

-0.267

-0.7611

-1.125

-3.933

-0.067

-0.9012

0.733

1.043

1.247

1.378

1.605

-0.067

0.867

1.533

2.237

2.848

3.374

3.606


 

 

                This table shows the expected profit during this interval (0, t) for the fixed value per unit of time. This shows that expected profit v/s time decreases rapidly where service cost and increases if

References:

 

[1]     Gupta P. P., Singh S. B. and Geol C. K., Stand by redundant system involving the concepts of multifailure human failure under head of line repair policy, Bulletin of Pure & Applied Sciences, 20, pp. 345-35, 2001.

[2]     Goel C. K., Singh S. B. and Jai Kishore, Reliability analysis of two units redundant system with three types of failure under waiting time to repair. Acta Ciencia Indica, XXIX,pp. 677-688, 2003.

[3]     Mokaddis G. S. and Tawfak M. L., Some characteristics of two dissimilar unit cold stand by redundant with three modes. Microelectron and Reliability, 38, pp. 487-504, 1996.

[4]     Sharma Viresh: Availability analysis of two unit by system with delayed replacement under perfect switching.  

 

 
 
 
 
 
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