A Two Dissimilar Unit Parallel System with Administrative Delay in Repair

Various authors including (Gopalan and D’Souza, 1975), (Goel and Gupta, 1983), (Gupta and Singh, 1985), have analysed the two-unit cold standby system models assuming that the repair facility is instantaneously available so that the repair is started immediately upon the failure of a unit provided that he/she is not busy in repairing the other unit. In this paper when an operative unit fails, the repair is no available immediately due to some administrative action. In this paper investigate a two non-identica unit parallel system model assuming the concept of administrative delay for getting the repairman available with the system. Each unit has two modes Normal (N) and total failure (F). Using the supplementary variable technique the following reliability characteristics of interest have been obtained. Reliability of the system and mean time to system failure (MTSF). (i) Point wise availability of the system in terms of its Laplace Transform. (ii) Expected up time of the system in the interval t,0. (iii) Actual steady-state probabilities for the system being in various states. (iv) Expected busy period of the repairman in the interval t,0. (v) Expected profit incurred by the system in t,0 and in steady state. The behaviour of MTSF and profit function have also been studies through graphs in respect of some parameters in a particular case.

1 : Constant failure rate of unit-1. 2 : Constant failure rate of unit-2. 1 : Constant rate of arrival of the repairman at the system when one unit is failed. 2 : Constant rate of arrival of the repairman at the system when both units are failed. xgxii, : General rate of repair and p.d.f. of repair time if unit-i. i = 1, 2 xgx, : General rate of simultaneous repair and corresponding p.d.f. of repair time of both the units. tPj : Probability that the system is in state js at epoch t. j = 0, 1, 2,…7 dxtxQk, : Probability that the system is in state ks at epoch t and has sojourned in this state for during between x and dxx. * : Laplace transformation. O : Operative W : Waiting for repair r : Under repair

Possible states of the system and transition into the states for this model are shown in fig. 1. There are 53210,,,,sssss are up states. States 764,,sss are down states.

Simple probabilistic considerations give the following equations, the integral are all from 0 to . tdxxtxQttPttP3321001 tOtdxxtxQtdxxtxQ,,725 tOttPttPttP1012111 tOttPttPttP2011221 tOtxtxQtttxQ12331,, tOttPttPttP212441 tOtxtxQtttxQ21551,, tOttPttPttP222661 tOxtxQtttxQ1,,77 Thus, we have

sPsAsPssP*01*012

1*1



sPsAsPssP*02*011

2*2





ssP*03*02

2*1

12

11*31  

 



  

sPsAsPsssP*04*0122

21*4







ssP*05*01

1*2

11

21*51  

 



  

sPsAsPsssP*06*0112

21*6



sgsAsAsAsAsP*07*0 *

3251642*71  



 

 

11

1*221

12

2*11121*0 



1*3251*642sgsAsAsgsAsA The probability that the system will be in state 0s in the long run is given by

*0000limlim

000044332211AAAAAAAA 000776655AAAAAA

*

TRLTsR* 02x 02*5*3*2*1*0*xsPsPsPsPsPsR

*

1

11

1*222

12

2*1112153211









 



Busy period of repairman in repairing a unit-1 during t,0 is

b30

1

ssPsAsb/*03*1 Busy period of repairman in repairing a unit-2 during t,0 is

b50

2

ssPsAsb/*05* Busy period of repairman in repairing two units during t,0 is

b70

3

ssPsAssPsnb//*0*7*

tP = Total revenue in t,0 – Expected cost of repair in t,0 tKtKtKtKbbbup3322110 where

up

0



0K= revenue per-unit up-time 1K= per-unit of time repair cost of failed unit-1 2K= per-unit of time repair cost of failed unit-2 3K= per-unit of time repair cost of both the failed unit-1 and unit-2. In steady state



tPPtlim

0736241053221PAKAKAKKAAAA

All the repair time distribution are exponential. xxxexgexgexg212211

212 1221

221

1

   



21

21*212

12*1 

2112

113



2111

215





212212

1121132

 





221211

1212152

 



Relation between failure and profit from fig. 2 MTSF decreases when failure rate increases. In fig. 3 if failure rate increases then profit decreases.

Chung, Who Kee (1990). ‘A reliability analysis of a K-out of – N:G redundant system with common – cause failure and critical human error.’ Microeletorn – Reliab, 30, p. 237. Goel, L. R. and R. Gupta (1983). ‘A multi-component two unit cold standby system with three modes’ Microelectron Reliab. 23(5), pp. 799-803. Goel, L. R. R. Gupta and S. K. Singh (1985). ‘Profit analysis of a cold standby system with two repair distribution’, Microelectron. Reliab, 25, pp. 467-472. Gopalan, M.N. and C.A. D’Souza (1975). ‘Probability analysis of a system with two dissimilar units subject to preventive maintenance and a single service facility’, Operations Research, 23, pp.

544-548.

Mittal, S.K. & Surbhi Gupta et. al. (2006). Analytical Behaviour of a parallel redundant complex system involving Environmental Failure under had of line Repair Discipline. Acta Cinecia India. Vol XXX, II M, No I. Wang, K. H. (2004). A survey of Maintenance Polices of Deteriorating System, Euro. J. Res. Vol 139

(3).

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