Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 12, Issue 2 (December 2017), pp. 803 - 812 Applications and Applied Mathematics: An International Journal (AAM) Evaluation of Some Reliability Characteristics of a Single Unit System Requiring Two Types of supporting Device for Operations Ibrahim Yusuf1 and Nura Jibrin Fagge2 1 Department of Mathematical Sciences Bayero University Kano Nigeria iyusuf.mth@buk.edu.ng; 2 Department of Mathematics Kano State College of Arts, Science and Remedial Studies Kano Nigeria njfagge1969@gmail.com; Received: September 13, 2016; Accepted: May 23, 2017 Abstract This study presents the reliability assessment of a single unit connected to two types of external supporting devices for its operation. Each type of external supporting device has two copies I and II on standby. First order differential difference equations method is used to obtain the explicit expression for the steady-state availability, busy period due to failure of type I and II supporting devices of repairmen, steady-state availability and profit function. Based on assumed numerical values given to system parameters, graphical illustrations are given to highlight important results. Comparisons are performed to highlight the impact of unit failure and repair rates on availability and profit. Keywords: Reliability; availability; profit; supporting device; single unit MSC 2000 No.: 90B25, 60K10, 62N05 1. Introduction Reliability is vital for proper utilization and maintenance of any system. It involves technique for increasing system effectiveness through reducing failure frequency and maintenance cost. Reliability assessment of a system provides insight into the probability that the system will be available to be committed to a specified requirement. Systems are usually studied with the intention to evaluate their reliability characteristics in terms of busy period of repairman, steady-state availability and generated revenue. There exist systems that cannot work without the help of external supporting devices connected to such systems. 803 804 Ibrahim Yusuf and Nura Jibrin Fagge These external supporting devices are systems themselves that are liable to failure and so they require preventive maintenance to improve their reliability. Examples of such systems can be seen in aircrafts, nuclear plants, satellites, electric generators, computer systems, power plants, manufacturing systems, and industrial systems. Improving the reliability of such systems with their supporting device is vital in ensuring quality of products. Availability and profit of an industrial system are becoming an increasingly important issue. Where the availability of a system increases, the related profit will also increase. On improving the reliability and availability of a system/subsystem, the production and associated profit will also increase. Increase in production leads to the increase of profit. This can be achieved by maintaining reliability and availability at the highest level. To achieve high production and profit, the system should remain operative for a maximum possible duration. It is important to consider profit as well as the quality requirement. Damcese and Helmy (2010) studied the reliability of systems with mixed standby components. Fathabadi and Khodaei (2012) evaluated the reliability of network flows with stochastic capacity and cost constraint. Gurov and Utkin (2012) studied reliability of loadshare system with piecewise constant load. Hajeeh (2012) dealt with availability of a system with different repair options. Hu et al. (2012) presented availability analysis and design optimisation for a repairable series-parallel system with failure dependencies. Jain and Rani (2013) studied the availability analysis for repairable system with warm standby, switching failure and reboot delay. Kadiyan et al. (2012) presented the reliability and availability of uncaser system of brewery plant. Khalili-Damghani and Amiri (2012) investigated multi-objective reliability redundancy allocation series-parallel problem using efficient epsilon-constraint. Kimura et al. (2011) investigated the reliability of a server system with asynchronous and synchronous. Kiran et al. (2013) performed reliability modelling of mechatronic system based on theoretic approach. Krishnan and Somasundaram (2012) studied reliability and profit of k-out-of-N system with sensor. Pandey et al. (2011) discussed the reliability analysis of a series and parallel network using triangular intuitionistic fuzzy sets. Ram (2010) discussed the reliability measures of three-state complex system. Upadhyay et al. (2013) performed reliability modelling of component based software system. Many research results have been reported on system reliability in the presence of supporting device. These include Yusuf et al (2015) who performed comparative analysis of MTSF between systems connected to supporting device for operation. Yusuf et al (2016) performed reliability computation of a linear consecutive 2-out-of-3 system in the presence of supporting device. Yusuf (2016) presented reliability evaluation of a parallel system with a supporting device and two types of preventive maintenance. The problem considered in this paper is different from the works of the authors discussed above. In this paper, a single unit system connected to two types of dissimilar supporting devices is considered and its corresponding mathematical models are derived. The focus of our analysis is primarily to capture the effect of both type I and II failure and repair rates on availability for different values of main unit failure and repair rates. The organization of the paper is as follows: Section 2 contains a description of the system under study. Section 3 presents formulations of the models. The results of our numerical simulations are presented in Section 4. Finally, we make some concluding remarks in Section 5. AAM: Intern. J., Vol. 12, Issue 2 (December 2017) 805 2. Description of the Model In this paper, a single unit system is considered. It is assumed that the system must work with one copy of both type I and II supporting devices. It is also assumed that each type of supporting device has a copy on standby and the switching is perfect. Both the units and supporting devices are assumed to be repairable. Each of the primary supporting devices fails independently of the state of the other and has an exponential failure distribution with parameter 1 and 2 for type I and II supporting devices, respectively. Whenever a primary supporting device fails, it is immediately sent to repair with parameter 1 and  2 and the standby supporting device is switched to operation. System failure occurs when the unit has failed with parameter  and it is sent for repair with parameter  or the failure of all copies of type I or type II supporting devices. 0 4 0 0 1 1 0 1 1   1 0 6 2 0 7 0 2 2 0 3 0 2 5 2 8 Figure 1. Transition diagram of System S0: Initial state, main unit and type I copy I supporting device are working, type I copy II supporting device, type II copy I and II supporting devices are on standby. The system is operative. S1: Type I copy I supporting device has failed and is under repair, main unit and type I copy II supporting device are working, type II copy I and II supporting devices are on standby. The system is operative. S2: Type I copy II supporting device has failed and is under repair, the unit and type II copy I supporting device are working, type II copy II supporting device is on standby. The system is operative. S3: Type II copy I supporting device has failed and is under repair, the unit and type II copy II supporting device are working. The system is operative. S4: Main unit has failed and is under repair, type I copy I supporting device is idle, type I copy II supporting device, type II copy I and II supporting devices are on standby. The system is inoperative. S5: Type II copy II supporting device has failed and is under repair, the unit is idle. The system is inoperative. S6: Main unit has failed and is under repair, type I copy II supporting device is idle, type II copy I and II supporting devices are on standby. The system is inoperative. S7: Main unit has failed and is under repair, type II copy I supporting device is idle, type II copy II supporting device is on standby. The system is inoperative. S8: Main unit has failed and is under repair, type II copy II supporting device is idle. The system is inoperative. 806 Ibrahim Yusuf and Nura Jibrin Fagge 3. Formulation of the Models Define Pi (t ) to be the probability that the system at time t is in state i, i  0,1, 2, 3,...,8 . The corresponding differential difference equations associated with the transition diagram in Figure 1 are: p0 (t )  (0  1 ) p0 (t )  1 p1 (t )  0 p4 (t ) , (1) p1(t )  ( 1  1  0 ) p1 (t )  1 p0 (t )  1 p2 (t )  0 p6 (t ) , p2 (t )  ( 1  2  0 ) p2 (t )  1 p1 (t )  2 p3 (t )  0 p7 (t ) , (2) (3) p3 (t )  ( 2  2  0 ) p3 (t )  2 p2 (t )  2 p5 (t )  0 p8 (t ) , p4 (t )   0 p4 (t )  0 p0 (t ) , (4) (5) p5 (t )   2 p5 (t )  2 p3 (t ) , p6 (t )   0 p6 (t )  0 p1 (t ) , (6) (7) p7 (t )   0 p7 (t )  0 p2 (t ) , '' p8 (t )   0 p8 (t )  0 p3 (t ) '' . (8) (9) The initial condition for this problem is: 1, i  0 '' pi (0)   '' , 0, i  1, 2,3,...,8 (10) Solving the differential difference Equations (1) – (9) using (10), the state probabilities pi (t ), i  1, 2,3,...,9 , the steady-state availability, busy period due to failures of main unit, type I and type II supporting devices and profit function are:   2  2  0 1221  0 2212  0 2122 , (11) AV ()  0 1 2 D0 0 12 22  01122  012 22  0122 2 , D0     2   2  2 BP 2 ()  1 0 1 2 1 0 2 , D0 2       2 2  BP3 ()  1 2 0 2 1 2 0 , D0 BP1 ()  (12) (13) (14) where D0  0 2212  0 2122  01222  0 1221  0 12 22  20122  22012  12201  12 220 PF (  )  C0 AV (  )  C1 BP1 (  )  C 2 BP 2 (  )  C3 BP 3 (). , (15) (16) 4. Numerical Example and Discussion Numerical examples are presented to demonstrate the impact of repair and failure rates on steady-state availability and net profit of the system based on given values of the AAM: Intern. J., Vol. 12, Issue 2 (December 2017) 807 parameters. For the purpose of numerical example, the following sets of parameter values are used: C0  100, 000 , C1  1, 000 , C2  500 , C3  500 , 2  0.25 , 1  0.3 , 0  0.2, 0.4, 0.6 , 0  0.2,0.4,0.6 , 2  0.3 , 1  0.02 0.7  0=0.2  0=0.4 0.6 Availability  0=0.6 0.5 0.4 0.3 0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 Figure 2. Availability against 1 for different values of 0 (0.2, 0.4, 0.6) Simulations in Figure 2 depicted the system availability with respect to type I supporting device failure rate 1 for different values of main unit failure rate 0 . In these Figure, system availability decreases as 1 increases for different values of 0 . The gaps between the curves in Figure 2 widen as 1 increases and decreases in 0 . This sensitivity analysis implies that preventive maintenance to the main unit and supporting devices should be invoked to lower their failure rate, to reduce maintenance cost of system failure, to improve and maximize the system availability as well as production output. 4 6 x 10  0=0.2  0=0.4 Profit 5  0=0.6 4 3 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1 Figure 3. Profit against 1 for different values of 0 (0.2, 0.4, 0.6) 0.9 1 808 Ibrahim Yusuf and Nura Jibrin Fagge Figure 3 depicts the profit with respect to type I supporting device failure rate 1 for different values of main unit failure rate 0 . Here, the profit decreases as 1 increases for different values of 0 . However, the gaps between the curves in this Figure widen as 1 increases and decreases in 0 . This implies that maintenance to the entire system is vital to lower the failure rate, reduce maintenance cost of system failure, improve and maximize production output as well as the profit. 0.7 Availability 0.6  0=0.2  0=0.4 0.5  0=0.6 0.4 0.3 0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 Figure 4. Availability against 1 for different values of 0 (0.2, 0.4, 0.6) Result of availability with respect to type I supporting device repair rate 1 is depicted in Figure 4 above for different values of 0 . From the Figure, system availability increases as 1 increases for different values of 0 . The gaps between the curves in this Figure widen as 1 increases and decreases in 0 . 4 6 x 10  0=0.2  0=0.4 Profit 5  0=0.6 4 3 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1 Figure 5. Profit against 1 for different values of 0 (0.2, 0.4, 0.6) 0.9 1 AAM: Intern. J., Vol. 12, Issue 2 (December 2017) 809 Results in Figure 5 above show that profit increases as 1 increases for different values of 0 . The gaps between the curves in the Figure widen as 1 increases and decreases in 0 . 0.8 0=0.2 0=0.4 0.7 Availability 0=0.6 0.6 0.5 0.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 Figure 6. Availability against 1 for different values of 0 (0.2, 0.4, 0.6) Simulation in Figure 6 above shows that availability decreases as 1 increases for different values of  0 . The gaps between the curves in the Figure decrease as 0 increases. 4 8 x 10 0=0.2 7 0=0.4 0=0.6 Profit 6 5 4 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 Figure 7. Profit against 1 for different values of 0 (0.2, 0.4, 0.6) It can be seen in Figure 7 above that the profit tends to decrease with increase in the value of 1 for different values of  0 . However, the profit is higher when  0 is higher. 810 Ibrahim Yusuf and Nura Jibrin Fagge 0.8 0=0.2 0.7 Availability 0=0.4 0.6 0=0.6 0.5 0.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 Figure 8. Availability against 1 for different values of 0 (0.2, 0.4, 0.6) In Figure 8, the system availability increases as 1 increases for different values of  0 . It is evident from the Figure that availability is higher when 0 increases from 0.2 to 0.6. This sensitivity analysis implies that availability will be improved significantly with increase in the value of 1 and  0 . 4 8 x 10 7 Profit 6 5 0=0.2 0=0.4 4 0=0.6 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 Figure 9. Profit against 1 for different values of 0 (0.2, 0.4, 0.6) In Figure 9, the profit increases as 1 increases for different values of  0 . It is evident from Figure that the profit is higher when 0 increases from 0.2 to 0.6. This sensitivity analysis implies that availability will be improved significantly with increase in the value of 1 and  0 . AAM: Intern. J., Vol. 12, Issue 2 (December 2017) 811 5. Conclusion In this paper, we constructed a system consisting of a main unit connected to two types of supporting devices for its operation with each supporting device having a copy on standby to study the availability and profit of the system. We have developed the explicit expressions for the system availability and profit and performance comparison with respect to main unit failure and repair rates. It is evident from Figures 2 – 9 that availability and profit are higher with decrease in 0 and increase in  0 . The system can further be developed into system with multiple types of supporting devices in solving reliability and availability problems. 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