Analysis of a Computer System with Software Redundancy Subject to Hardware Preventive Maintenance

:- The present work is devoted to the analysis of a computer system with software redundancy subject to hardware preventive maintenance. For this purpose, a system model has been developed by considering software redundancy. There is an independent hardware and software failure in the system model. The preventive maintenance of the hardware component has been conducted after a specific operation time ‘t’ (called maximum operation time). A single server is provided immediately to the system for carrying out hardware repair, software up-gradation and hardware preventive maintenance as and when needed. The repair activities conducted by the server are perfect. The failure time of the hardware and software components follow negative exponential distribution whereas the distributions for hardware repair, software up-gradation and preventive maintenance times are taken as arbitrary with different probability density functions. Some important reliability characteristics of the system models have been examined stochastically in steady state for particular values of various parameters and costs by using semi-Markov process and regenerative point technique. Graphs are drawn to depict the behaviour of mean time to system failure (MTSF), availability and profit function under different sets of assumptions on the parameters. The profit of the present model has also been compared with the model as discussed in the research paper of [5].


I. INTRODUCTION
In past few decades, the importance of computer widely increases in most of areas such as companies, medicals, banks, etc. In the previous research based on computer system, some stochastic models for a computer system have been developed and analyzed in steady state by considering the aspects of component wise redundancy in cold standby, priority in repair disciplines and maximum repair time to hardware. And, a computer system would be more profitable if redundancy is provided to the hardware along with maximum repair time rather than redundancy to the software. [1] examined a two-unit standby redundant system with preventive maintenance. [2] Analyzed a two dissimilar cold standby system with preventive maintenance and replacement of standby. On the other hand, the deterioration rate of a system can be reduced by conducting its preventive maintenance after a maximum operation time. [3] obtained reliability measures of a system under preventive maintenance. Hence, it becomes necessary to examine the effect of preventive maintenance on reliability measures of a computer system with the concepts of preventive maintenance and software redundancy. However, [4] tried to develop stochastic models of operating systems with preventive maintenance and priority subject to maximum operation and repair times. [6] established reliability measures of a computer system with hardware redundancy subject to preventive maintenance. The present work is devoted to the analysis of a computer system with software redundancy subject to hardware preventive maintenance. For this purpose, a system model has been developed by considering software redundancy in cold standby with the concept of hardware preventive maintenance. Hardware and software failures are independent in the system model. The preventive maintenance of the hardware component has been conducted after a specific operation time 't' (called maximum operation time). There is a single server who visits the system immediately for carrying out hardware repair, software up-gradation and hardware preventive maintenance as and when needed. The repair activities conducted by the server are perfect. The failure time of the hardware and software components follow negative exponential distribution whereas the distributions for hardware repair, software up-gradation and preventive maintenance times are taken as arbitrary with different probability density functions. Some important reliability characteristics of the system models have been examined stochastically in steady state for particular values of various parameters and costs by using semi-Markov process and regenerative point technique. Graphs are drawn to depict the behaviour of mean time to system failure (MTSF), availability and profit function under different sets of assumptions on the parameters. The profit of the present model has also been compared with the model as discussed in the research paper of [5].
: Probability that the server is busy in repairing the unit due to hardware failure at an instant 't' given that the system entered state Si at t = 0.
S i B (t) : Probability that the server is busy in upgradation of the software at an instant 't' given that the system entered the regenerative state Si at t = 0. ( ) : Probability that the server is busy in preventive maintenance of the hardware given that the system entered state at t=0. (1) It can be easily verified that p01+p02+p03 = p10= p20+p24+p25+p26 = p20+p23.4+p22.5+p21.6 = p30 = p43=p52 =1 The mean sojourn times (μi) in the state Si are

C. Steady State Availability
The recursive relations for ( ) are given as: Taking LT of equation (7) and solving for * 0 () As, the steady state availability is given by (19)

F. Expected Number of Software Up-gradations
The recursive relations for NSU i (t) are given as: It is analyzed that the mean time to system failure (MTSF), availability and profit go on decreasing with the increase of failure rates (λ1 and λ2) and the rate ( 0 ) by which hardware undergoes for preventive maintenance after a pre specific operation time't' while their values keep on moving up with the increase of hardware repair rate (α), software up-gradation rate (θ) and preventive maintenance rate (γ) provided system has more chances of hardware failure. Further, it is interesting to note that a computer system has more values of these reliability measures when it has more chances of software failure than that of hardware failure. The graphical presentation of the results related to these reliability measures obtained for the system model is shown in the figures 2 to 4.

VIII. COMPARATIVE STUDY OF PROFIT OF SYSTEM MODELS
The profit of the present model has been compared with that of the model discussed in the research paper Munday and Malik (2015). It is revealed that the present model is less profitable. And, hence we can say that the concept of hardware preventive maintenance in a computer system with software redundancy in cold standby is not much helpful in making the system more profitable. The behavior of the profit difference of the system model with respect to hardware failure rate (λ1) has been shown graphically in figure 5.