Journal of Scientific & Industrial Research Vol. 73, October 2014, pp. 653-655 Cost-Benefit Analysis of a Computer System with Priority to S/W Replacement over H/W Repair Activities Subject to Maximum Operation and Repair Times A Kumar* and S C Malik Department of Statistics, M.D. University, Rohtak-124001, Haryana (India) Received 24 August 2012; revised 28 December 2013; accepted 12 June 2014 The emphasis of the present study is on the evaluation of reliability measures of a computer system with independent hardware (h/w) and software (s/w) failures. A single repair facility is provided immediately for conducting repair activities of h/w and s/w. Preventive maintenance of the system is done after a maximum operation time. Replacement of the h/w by new one is made in case server fails to complete its repair in a given maximum time. However, only replacement of the s/w is done by new one giving some replacement time. Priority is given to s/w replacement over h/w repair activities. The failure time distribution of the h/w and s/w follows negative exponential while the distributions of preventive maintenance, repair and replacement time are taken as arbitrary with different probability density functions. Illustration for a particular case is given to show the graphical behaviour of some important reliability measures. Keywords: Computer System, Maximum Operation and Repair Times, Preventive Maintenance and Cost- Benefit Analysis. Introduction The use of computer systems has been increasing day by day in most of the critical and sensitive areas like aircraft, automobiles and home appliances. And, impact of their failures may be costly and dangerous to the society. It is, therefore, of great importance to handle such systems with very care and high reliability. The reliability of these systems can be improved by adopting some suitable techniques including redundancy and other cost-effective repair policies. The technique of redundancy has not been used much more in case of computer systems. However, Friedman and Tran (1992) tried to develop a combined reliability model for the whole system in which hardware and software components work together without considering the concept of redundancy. Recently, Malik and Kumar [2011, 2012] investigated reliability models for a computer system with preventive maintenance and repair subject to maximum operation and repair times. In view of the above observations and facts in mind, the concentration of the present paper is on the evaluation of some important reliability measures of a computer system with cold standby redundancy. A reliability model is developed considering independent h/w and s/w failures. A single repair facility is provided who conducts preventive maintenance after maximum —————— *Author for correspondence Email: [email protected] operation time and also makes replacement of the h/w by new one in case its repair is not possible by him in a given maximum time. Priority is given to s/w replacement over h/w repair activities. All random variables are statistically independent. The distributions of failure time and rate by which system undergoes for preventive maintenance are taken as negatively exponential. The distributions of h/w repair, s/w replacement and h/w replacement are assumed as arbitrary with different probability density functions. The system passes through the following transition states: S0(N0, Cs), S1(N0, Pm), S2(N0,HFur),S3(N0,SFurp),S4(N0,HFurp),S5(HFUR,W Pm),S6(HFwr,PM),S5(SFURP,HFwr),S8(PM,SFwrp), S9(SFURP,WPm),S10(SFURP,SFwrp),S11(HFwr,SFur p),S12(HFUR,HFwr),S13(PM,Wpm),S14(HFurp,HFWR ),S15(HFurp,Wpm),S16(HFURP,Wpm),S17(HFwrp,SFu rp) and S18(HFURP,HFwr). On the basis of these transition states following reliability measures are obtained using semi-Markov process and regenerative point technique: Mean Time to System Failure (MTSF) Let i(t) be the cdf of first passage time from the regenerative state i to a failed state. Regarding the failed state as absorbing state, we have the following recursive relations for i (t) as i t    Qi, j t   j t    Qi,k t  j k ...(1) 654 J SCI IND RES VOL 73 OCTOBER 2014 Where j is an un-failed regenerative state to which the given regenerative state i can transit and k is a failed state to which the state i can transit directly. The MTSF can be obtained by taking LST of above relation (1) which is given by ~ 1  0 ( s) s o s MTSF = lim Steady State Availability Let Ai(t) be the probability that the system is in upstate at instant 't' given that the system entered regenerative state i at t = 0. The recursive relations for Ai (t) are given as Ai  t   M i  t    qi(,nj)  t  A j  t  ... (2) j Where j is any successive regenerative state to which the regenerative state i can transit through n transitions. Mi(t) is the probability that the system is up initially in state Si  E is up at time t without visiting to any other regenerative state, we have by taking LT of above relations (2) and solving for A0* ( s) , the steady state availability is given by A0 ()  lim sA0* ( s). s 0 Busy Period Analysis for Server Let BiP (t ) BiR (t ) BiS (t ) and BiHRp (t ) be the probabilities that the server is busy in Preventive maintenance of the system, repairing the unit due to hardware failure, replacement of the software and hardware components at an instant ‘t’ given that the system entered state i at t = 0. The recursive relations for BiP (t ) BiR (t ) BiS (t ) and BiHRp (t ) are formulated in same manner as in steady state availability. By using Laplace transformation on recurrence relations and solving for Bi*P ( s ) Bi*R ( s ) Bi*S ( s) and Bi*HRp ( s) , the time for which server is busy due to repair and replacements respectively is given by B0H  lim sB0*H ( s) = s 0 N 3H , B0S  lim sB0*S ( s) s 0 D2 N 3S NR , B0R  lim sB0*R ( s)  3 = D2 D2 s 0 N3HRp HRp * HRp And B  lim sB0 (s)  0 D2 s 0 Expected Number of Replacements and Visits by the Server Let RiH (t ) and RiS (t ) be the expected number of replacements of the hardware and software components by the server. Let Ni (t ) be the expected number of visits by the server in (0, t] given that the system entered the regenerative state i at t = 0. The recursive relations for RiH (t ) RiS (t ) and Ni (t ) , are formulated in same manner as in steady state availability. By using Laplace transformation on recurrence relations and solving for Fig.1—Profit Vs. Preventive Maintenance Rate (α) KUMAR & MALIK : COST-BENEFIT ANALYSIS OF A COMPUTER SYSTEM RiH (t ) , RiS (t ) and Ni (t ) the time for which server is busy due to repair and replacements respectively is given by R0H ()  lim sR0H ( s) = s 0 R0S ()  lim sR0S ( s) = s 0 N 4H and D2 N 4S D2 Cost- Benefit Analysis The profit incurred to the system model in steady state can be obtained as Conclusion Considering exponential distribution for all random variables, the study reveals that a computer system in which chances of h/w failure are high can be made more profitable by using cold standby redundancy, by conducting preventive maintenance and by giving priority to s/w replacement over h/w repair activities. References 1 2 P  K0 A0  K B  K 2 B  K3 B P 1 0 HRp 4 0 K B R 0 S 0  K5 R0H  K6 R0S  K7 N0 … (3) K0 = Revenue per unit up-time of the system and Ki = Cost per unit time for which server is busy due to various repair activities. 655 3 Friedman M A & Tran P, Reliability Techniques for Combined Hardware/Software Systems, Proc Annual Reliability and Maintability Symp(1992) pp.290-293. Malik S C & Kumar A, Profit Analysis of a Computer System with Priority to Software Replacement over Hardware Repair Subject to Maximum Operation and Repair Times, Int J Engine Sci Technol, 3, No. 10(2011) pp. 7452- 7468. Malik S C & Kumar A, Stochastic Modeling of a Computer System with Priority to PM over S/W Replacement Subject to Maximum Operation and Repair Times, Int J Comput Applicat, 43 (3)(2012), pp. 27-34.