Operational Research in Engineering Sciences: Theory and Applications Vol. 4, Issue 2, 2021, pp. 124-139 ISSN: 2620-1607 eISSN: 2620-1747 DOI: https://doi.org/10.31181/oresta20402124r * Corresponding author. mangeyram@geu.ac.in (M. Ram), vash.tyagi@gmail.com (V. Tyagi) RELIABILITY CHARACTERISTICS OF RAILWAY COMMUNICATION SYSTEM SUBJECT TO SWITCH FAILURE Mangey Ram 1, Vaishali Tyagi2* 1 Department of Mathematics, Computer Science and Engineering, Graphic Era, Dehradun, Uttarakhand, India and Institute of Advanced Manufacturing Technologies, Peter the Great St. Petersburg Polytechnic University, Saint Petersburg, Russia 2 Department of Mathematics and Statistics, Kanya Gurukul Campus, Gurukul Kangri, Haridwar, Uttarakhand, India Received: 20 March 2021 Accepted: 23 June 2021 First online: 08 July 2021 Research paper Abstract. In the present study, a railway communication system (RCS) reliability model is developed based on system failure. The proposed RCS has control centre and stations which are arranged in such a manner that failure of control centre or a single station stops the working of overall system i.e., all switches must be working for communication to be available. To improve the reliability of the proposed communication system, a ring architecture is employed. In this architecture one additional communication path is connected in parallel configuration. Provision of two path of communication ensures that failure of one path will not cause a communication failure and communication will be available through additional path. All failures of RCS are exponentially distributed. Mathematical modelling of the system is carried out using Markov process by which the differential equations are generated. These differential equations are further used to evaluate the reliability measures like availability, reliability, mean time to failure of the proposed RCS. Likewise, sensitivity analysis is done to determine the impact of failures on RCS’s performance measures. The proposed Markov process-based model gives the information about the failure and working of the multi- state railway communication system. Finally, numerical results are provided with graphs to demonstrates the usefulness of the findings. Key words: Railway communication system; Reliability; Mean time to failure; Markov process; Sensitivity 1. Introduction In the present day’s society demands inexpensive, more secure, and timesaving public transport. Railway transportation systems attracts a lot of passengers because Reliability Characteristics Of Railway Communication System Subject To Switch Failure 125 of their capacity of transporting the people with high luxury, great comfort and large get-up-and-go efficacy (Ai et al., 2014). A lot of people choose trains for travelling because of the easy understanding, experience and comfort that the rail transport gives. Railway communications systems are needed to develop the communication between train and path equipment for traffic management and dealing with continuous high-data-rate traveler services, hypermedia dispatching video transmissions, railway mobile ticketing, and the internet of things (IOT) for railways (Ai et al., 2015; Guan et al., 2017). The security of railway’s employees, passengers and of the general public are the first requirements and is of specific significance in the railway industry. Railway industry looking for many aspects to improve the security / safety and the reliability of the railway systems. A railway system is a very large and complex stochastic dynamic system. This system is already interesting by itself. Large, stochastic complex systems are generally examined with deterministic methods. Many authors have written about deterministic optimization of railway systems over the last twenty years. A lot of research has been done in the context of the different techniques to increase the reliability of a railway communication system. Aggarwal (1975) obtained reliability expression for communication system. Marquez et al. (2003) discussed about the improvement of a way to deal the use of remote monitoring to the reliability centred maintenance of railway attendances. Tao et al. (2007) used fault tree analysis method for RCS, in which main factors affecting the failure of RCS are determined by minimum cut set analysis. De Felice and Petrillo (2011) proposed a methodological approach based on human reliability analysis (HRA) and failures modes, effects criticality analysis (FMECA) to calculate the reliability of railway transportation system. HRA gives a logical analysis of factors affecting human performance, prompts suggestions for improvement. Lin (2015) proposed an advanced finite state Markov chain channel model for high-speed railway fading channels and derived the expression of state transition probabilities under different speed modes. Unterhuber et al. (2016) provided a summary of communication systems in trains and discussed about possible direction for future wireless network. Also, authors identified the gaps for station classification in railway environment, total velocity, and relative velocity for radio broadcast measurement. He et al. (2017) studied the propagation characteristics for rapid railway communication system including metropolitan, rural and tunnel with straight and curved route. Zhang et al. (2018) proposed a Markov model for railway communication system and used multi-link transmission communication technique to improve the capacity of RCS. Kumar and Kumar (2019) evaluated the reliability, and mean time to failure of the wireless communication system regarding its component failure. Authors also identified the critical component by sensitivity analysis. Song (2019) described a communication-based train control system and discussed the different constraints that affects the working of communication system. Authors also evaluated system availability and performance by applied stochastic petri nets. In this research article, a railway communication system with control centre and stations is considered. The considered multi-state repairable system having control centre and stations in such a manner that for communication, all switches must be in working condition. Failure in control centre and any of one station arises a communication failure. To improve the reliability of the proposed communication Ram and Tyagi/Oper. Res. Eng. Sci. Theor. Appl. 4 (2) (2021) 124-139 126 system, an additional path is connected in parallel configuration. In the proposed model of RCS, probability of each transition states is obtained and reliability measures such as system availability, system reliability, mean time to failure (MTTF), and sensitivity of reliability have been computed. The rest of the paper is categorised as follows. The description of the proposed RCS with requisite assumptions and notations are presented in section 2. In section 3, the set of differential equations is constructed based on Markov process and also probability of each state is calculated using Laplace transformation. In section 4, the numerical calculations to compute the reliability measures of RCS such as availability, reliability, mean time to failure, and sensitivity analysis is given. In section 5, the behaviour of the reliability measures is discussed with the help of tables and graphs. Finally, section 6 gives the concluding remark to highlight the significant and some future prospects of the present work. 2. System modelling In this study, a railway communication system with control centre and stations is considered. In this system a communication which is provided in the control centre as well as in each of the stations which are connected in a series configuration. It can be seen in the diagram that all switches must be working for the communication system to be function. If there is a failure in switch at any station, there will be a communication failure at those stations as well as at all stations beyond that point i.e., all switches must be working to continue communication. Further one additional communication path is connected in parallel configuration to improve the reliability of proposed system. Adding an additional path means that the communication will be available after failure of any one of the paths. On failure of both path’s switches, the communication system will be failed. Figure 1. Reliability block diagram of railway communication system To study and formulation of the system, following assumptions and notations are made (Table 1): • The communication system consists of two paths namely path 1 and path 2, in which each path has a control centre and 3 stations. • The proposed communication system has main three states full operation, degradation states and failure state. • At time t = 0, control centre and stations are in fully operation state and communication system is available. Reliability Characteristics Of Railway Communication System Subject To Switch Failure 127 • This study assumes that the failure rates of path 1 units and path 2 units are statistically independent, constant and are exponentially distributed with failure rates λc1, λ1, and λc2, λ2. • Failure of one path will not cause a communication failure. • Repair service is always available to repair the failed unit, i.e., as soon as an operating unit fail, it is instantaneously detected and sent for repair. • When a failed unit is repaired it considered to be a new one. Table 1. Notations t Time variable λc1 Failure rate of path 1 control centre λc2 Failure rate of path 2 control centre λ1 Failure rate of all stations of path 1 λ2 Failure rate of all stations of path 2 Pi(t) The state probability of the system at instant ‘t’ for i = 0 to 7 Pi (x, t) The failed state probability of the system at instant ‘t’ and an elapsed repair time x for i = 8 to 21 x Elapsed repair time μ (x) Repair rate for repaired state Figure 2. Transition diagram Ram and Tyagi/Oper. Res. Eng. Sci. Theor. Appl. 4 (2) (2021) 124-139 128 3. Governing Equations The following set of equations have been derived for the proposed communication system by using Markov process. Original State 21 1 2 1 2 0 80 ( ) ( , ) i i c c P t μP x t dx t  =   + + + + =    (1) Degraded States ( )1 2 1 2 1 1 2 0( ) ( )c c P t c c P t t   +  +  +  +  =  +    (2) 1 1 2 2 1 ( ) ( )c P t P t t   +  +  =    (3) 2 2 3 1 1 ( ) ( )c P t P t t   +  +  =    (4) 1 2 1 4 2 0 ( ) ( )c c P t P t t   +  +  +  =    (5) 1 1 5 2 4 ( ) ( )c P t c P t t   +  +  =    (6) 1 2 2 6 1 0 ( ) ( )c c P t P t t   +  +  +  =    (7) 2 2 7 1 6 ( ) ( )c P t c P t t   +  +  =    (8) Failed States ( , ) 0, 8, 9,...20, 21 i μ P x t i t x    + + = =    (9) Boundary Conditions 8 2 1 (0, ) ( )P t c P t=  (10) 9 1 1 (0, ) ( )P t c P t=  (11) 10 1 2 (0, ) ( )P t P t=  (12) 11 1 2 (0, ) ( )P t c P t=  (13) 12 2 3 (0, ) ( )P t P t=  (14) 13 2 3 (0, ) ( )P t c P t=  (15) 14 1 4 (0, ) ( )P t P t=  (16) Reliability Characteristics Of Railway Communication System Subject To Switch Failure 129 15 1 4 (0, ) ( )P t c P t=  (17) 16 1 5 (0, ) ( )P t P t=  (18) 17 1 5 (0, ) ( )P t c P t=  (19) 18 2 6 (0, ) ( )P t P t=  (20) 19 2 6 (0, ) ( )P t c P t=  (21) 20 2 7 (0, ) ( )P t P t=  (22) 21 2 7 (0, ) ( )P t c P t=  (23) Initial Condition 0 (0) 1 (0) 0, 1, 2,..., 21 i P P i = = = (24) After taking the Laplace transform from Equations (1) to (23), one can get the following set of equations:   21 1 2 1 2 0 80 ( ) ( , ) i i s c c P s μP x s dx  = + + + + =  (25)   ( )1 2 1 2 1 1 2 0( ) ( )s c c P s c c P s+ + + + =  + (26)  1 1 2 2 1( ) ( )s c P s P s+ + =  (27)  2 2 3 1 1( ) ( )s c P s P s+ + =  (28)  1 2 1 4 2 0( ) ( )s c c P s P s+ + + =  (29)  1 1 5 2 4( ) ( )s c P s c P s+ + =  (30)  1 2 2 6 1 0( ) ( )s c c P s P s+ + + =  (31)  2 2 7 1 6( ) ( )s c P s c P s+ + =  (32) ( , ) 0, 8, 9,...20, 21 i s μ P x s i x   + + = =   (33) 8 2 1 (0, ) ( )P s c P s=  (34) 9 1 1 (0, ) ( )P s c P s=  (35) 10 1 2 (0, ) ( )P s P s=  (36) 11 1 2 (0, ) ( )P s c P s=  (37) 12 2 3 (0, ) ( )P s P s=  (38) Ram and Tyagi/Oper. Res. Eng. Sci. Theor. Appl. 4 (2) (2021) 124-139 130 13 2 3 (0, ) ( )P s c P s=  (39) 14 1 4 (0, ) ( )P s P s=  (40) 15 1 4 (0, ) ( )P s c P s=  (41) 16 1 5 (0, ) ( )P s P s=  (42) 17 1 5 (0, ) ( )P s c P s=  (43) 18 2 6 (0, ) ( )P s P s=  (44) 19 2 6 (0, ) ( )P s c P s=  (45) 20 2 7 (0, ) ( )P s P s=  (46) 21 2 7 (0, ) ( )P s c P s=  (47) Now, solving Equation (25) - (33) with the help of (34) - (47), the following state transition probabilities are obtained: 0 1 2 1 2 1 ( ) ( ) ( ) ( ) ( ) ( ) P s s c c S s U s V s W s =  + + + + − + +  (48) 1 2 1 0 1 2 1 2 ( ) ( ) ( ) c c P s P s s c c  + = + + + + (49) 2 1 2 2 0 1 1 1 2 1 2 ( ) ( ) ( ) ( )( ) c c P s P s s c s c c   + = + + + + + + (50) 1 1 2 3 0 2 2 1 2 1 2 ( ) ( ) ( ) ( )( ) c c P s P s s c s c c   + = + + + + + + (51) 2 4 0 1 2 1 ( ) ( )P s P s s c c  = + + + (52) 2 2 5 0 1 1 1 2 1 ( ) ( ) ( )( ) c P s P s s c s c c   = + + + + + (53) 1 6 0 1 2 2 ( ) ( )P s P s s c c  = + + + (54) 1 1 7 0 2 2 1 2 2 ( ) ( ) ( )( ) c P s P s s c s c c   = + + + + + (55) 2 1 2 8 0 1 2 1 2 ( ) 1 ( ) ( ) ( ) ( ) c c c S s P s P s s c c s    + − =   + + + +   (56) 1 1 2 9 0 1 2 1 2 ( ) 1 ( ) ( ) ( ) ( ) c c c S s P s P s s c c s    + − =   + + + +   (57) Reliability Characteristics Of Railway Communication System Subject To Switch Failure 131 1 2 1 2 10 0 1 1 1 2 1 2 ( ) 1 ( ) ( ) ( ) ( )( ) c c S s P s P s s c s c c s     + − =   + + + + + +   (58) 1 2 1 2 11 0 1 1 1 2 1 2 ( ) 1 ( ) ( ) ( ) ( )( ) c c c S s P s P s s c s c c s     + − =   + + + + + +   (59) 1 2 1 2 12 0 2 2 1 2 1 2 ( ) 1 ( ) ( ) ( ) ( )( ) c c S s P s P s s c s c c s     + − =   + + + + + +   (60) 2 1 1 2 13 0 2 2 1 2 1 2 ( ) 1 ( ) ( ) ( ) ( )( ) c c c S s P s P s s c s c c s     + − =   + + + + + +   (61) 1 2 14 0 1 2 1 1 ( ) ( ) ( ) ( ) S s P s P s s c c s    − =   + + +   (62) 1 2 15 0 1 2 1 1 ( ) ( ) ( ) ( ) S s P s P s s c c s    − =   + + +   (63) 2 1 2 16 0 1 1 1 2 1 1 ( ) ( ) ( ) ( )( ) c S s P s P s s c s c c s     − =   + + + + +   (64) 2 1 2 17 0 1 1 1 2 1 1 ( ) ( ) ( ) ( )( ) c c S s P s P s s c s c c s     − =   + + + + +   (65) 1 2 18 0 1 1 1 2 1 1 ( ) ( ) ( ) ( )( ) S s P s P s s c s c c s    − =   + + + + +   (66) 1 2 19 0 1 2 2 1 ( ) ( ) ( ) ( ) c S s P s P s s c c s    − =   + + +   (67) 1 1 2 20 0 2 2 1 2 2 1 ( ) ( ) ( ) ( )( ) c S s P s P s s c s c c s     − =   + + + + +   (68) 2 1 1 21 0 2 2 1 2 2 1 ( ) ( ) ( ) ( )( ) c c S s P s P s s c s c c s     − =   + + + + +   (69) The probabilities of up and down states are as follows: 1 2 2 1 2 1 1 2 2 1 1 2 2 1 1 2 0 2 1 1 1 1 2 1 2 2 2 1 1 ( ) ( ) ( ) ( ) ( ) ( ) 1 1 ( ) ( ) ( ) up c c s c c s c s c s c c P s P s c c s c s c c s c    +    + + + +   + + + + + + + + + + +   =          + + +    + + + + + + +      (70) Ram and Tyagi/Oper. Res. Eng. Sci. Theor. Appl. 4 (2) (2021) 124-139 132 1 2 1 2 1 2 1 1 1 11 2 1 2 2 11 1 2 2 2 2 2 2 0 2 2 1 1 2 1 1 1 1 2 1 1 1 1 1 ( ) ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) ( ) ( ) ( down c c c s c s cc c cs c c s c s c S s P s P s c c c cs s c c s c s c       +  + + + +  +  +  +  +    +     +  +  +  +   +  +  +  +  +    −  =        +  + + +   +  +  +  +  +  +  +    1 1 1 2 2 2 2 1 2 2 2 2 2 ) ( ) ( ) c c c c s c c s c s c                            +  + +  +  +  +  +  +  +  +        (71) where, 1 2 1 2 1 2 1 1 1 11 2 1 2 2 11 2 1 2 2 2 2 2 ( ) ( )( ) ( ) ( ) ( ) ( ) c c c s c s cc c U s cs c c s c s c       +  + +  +  +  +  +  +    =      +  +  +  +   + +  +  +  +  +   2 1 2 1 2 1 1 1 2 1 1 1 1 1 ( ) ( ) ( ) ( ) c c c V s c s c c s c s c       =  + + +  + + + + + + +  1 2 1 1 2 2 2 1 2 2 2 2 2 2 ( ) ( ) ( ) ( ) c c c W s c s c c s c s c       =  + + +  + + + + + + +  4. Numerical Calculations For computing the reliability measures of the proposed railway communication system, following failure and repair rates will be assumed as given by Table 2. Table 2. Assumed failure and repair rate of proposed railway communication system. Failure and repair rate/per hour λ1 = 0.009 λc1 = 0.06 λ2 = 0.007 λc2 = 0.03 μ = 1 4.1. Availability The availability of the proposed system is computed by substituted the values of failure and repair rates as given in Table 2, in Equation (70), after putting these values, availability of the proposed RCS in terms of time t is given as follows: ( 0.9887 ) ( 0.2075 ) ( 0.0779 ) ( 0.0439 ) ( ) - 0.01158 0.05608 0.00085 0.00277 0.95911 t t t t A t = e e e e − − − − + − − + (72) Now after varying time from 0 to 50 with the interval of 5 units of time, one can get the numerical values of availability of the proposed system which are demonstrates by table 3. Reliability Characteristics Of Railway Communication System Subject To Switch Failure 133 Table 3. Availability of the proposed system Time (t) Availability 0 1.00000 5 0.97611 10 0.96398 15 0.95991 20 0.95867 25 0.95838 30 0.95829 35 0.95815 40 0.95811 45 0.95808 50 0.95801 0 10 20 30 40 50 0.95 0.96 0.97 0.98 0.99 1.00 1.01 A v a il a b il it y Time (t) Figure 3. Availability of the proposed system as time vary from 0 to 50 units 4.2. Reliability For the proposed system, reliability is calculated by substitute the values of failures as given in Table 2 and repair rate equal to zero in Equation (70), after substitute these failures and repairs values, the reliability function in terms of t is given by ( 0.0370 ) ( 0.6900 ) ( 0.1060 ) ( ) 0.30057 0.64938 (0.06123 0.05005) t t t R t e e t e − − − = + + + (73) Now varying time t from 0 to 10 unit with an interval of 1 unit, one can analyzed the reliability behavior of proposed system as tabulated in Table 4 and shown in Figure 4. Ram and Tyagi/Oper. Res. Eng. Sci. Theor. Appl. 4 (2) (2021) 124-139 134 Table 4. Reliability of the system Time (t) Reliability 0 1.00000 1 0.91938 2 0.76281 3 0.60074 4 0.45978 5 0.34658 6 0.25947 7 0.19404 8 0.14553 9 0.10976 10 0.08341 0 2 4 6 8 10 0.0 0.2 0.4 0.6 0.8 1.0 R e li a b il it y Time (t) Figure 4. Reliability of the system w.r.t. time 4.3. Mean time to system Failure The mean time to failure is calculated by taking μ = 0 and limit 0→s (Tyagi et al., 2021) in Equation (70). So, the MTTF of the proposed system in terms of failure rate is given by Equation (74). 1 2 2 1 2 1 1 2 2 1 1 2 2 1 1 2 1 2 1 2 2 1 1 1 1 2 1 2 2 2 1 1 ( ) ( ) ( ) ( )1 ( ) 1 1 ( ) ( ) ( ) c c c c c c c c MTTF c c c c c c c c    +    + + + +    + + +  +  +  + +   =   + + +        + + +     +  + +  +      (74) Now setting all failure rates values as given by Table 2 and vary each failure rate one by one from 0.001 to 0.04 in Equation (74) to get the MTTF of the proposed system with respect to variation in the failure rates. The variation in MTTF can be Reliability Characteristics Of Railway Communication System Subject To Switch Failure 135 seen from Table 5 and corresponding Figure 5 shows the behaviour of MTTF regarding variation in failure rates. Table 5. MTTF of the system Variation in Failure rates MTTF λC1 λC2 λ1 λ2 0.001 68.6161 51.82156 22.35057 24.79149 0.003 62.38204 45.53850 21.65165 24.28984 0.005 57.63895 41.19976 21.29690 23.84834 0.007 53.85080 37.98025 21.12371 23.45672 0.009 50.71749 35.46688 20.45672 23.10686 0.02 39.53540 27.40036 19.61863 21.69336 0.04 29.23630 20.75466 19.01606 20.24722 0.06 23.45672 17.10419 17.34855 19.39883 0.08 19.65024 14.65523 17.21551 18.81630 0.1 16.92906 12.85991 16.70574 18.37954 0.02 0.04 0.06 0.08 0.10 10 20 30 40 50 60 70  C1  C2  1  2 M T T F Variation in failure rates Figure 5. MTTF with respect to variation in failure rates 4.4. Sensitivity of reliability In reliability sensitivity study, the effect of failure rates on reliability has been studied. The reliability function is the dependent variable. It’s dependent because it depends on failures rates. Those failure rates are the independent variable. Sensitivity of reliability is obtained by partial differentiation of reliability function with respect to all the failure rates. Now putting all failure rates as given by Table 2 and repair rate equal to zero in these derivatives, one can get the Table 6 and corresponding Figure 6. Ram and Tyagi/Oper. Res. Eng. Sci. Theor. Appl. 4 (2) (2021) 124-139 136 Table 6. Sensitivity of reliability of the proposed communication system Time (t) Sensitivity of reliability 1 ( )R t c   2 ( )R t c   1 ( )R t  2 ( )R t  0 0 0 0 0 1 -0.08468 -0.08575 -0.01886 -0.03526 2 -0.30679 -0.31133 -0.06439 -0.12756 3 -0.62535 -0.636 -0.123 -0.25976 4 -1.00731 -1.02689 -0.18454 -0.41824 5 -1.42634 -1.45771 -0.24159 -0.59228 6 -1.86164 -1.90766 -0.28897 -0.77354 7 -2.29709 -2.36052 -0.32325 -0.95566 8 -2.72038 -2.80387 -0.34243 -1.13384 9 -3.12234 -3.22838 -0.34555 -1.3046 10 -3.49642 -3.62729 -0.33251 -1.4655 0 2 4 6 8 10 -4.0 -3.2 -2.4 -1.6 -0.8 0.0 S e n si ti v it y o f re li a b il it y Time (t)  C1  C2  1  2 Figure 6. Sensitivity of reliability with respect to time 5. Result Discussion In this paper, different reliability characteristics have been calculated and analysed for the railway communication system. Some results related to these reliability characteristics are given below: (i) From Table 3 and Figure 3, one can see the behaviour of the availability of the proposed RCS. Availability of the RCS decreases with increases in the value of time t. At initially, i.e., time t = 0, availability is 1 and after 50 units of time, Reliability Characteristics Of Railway Communication System Subject To Switch Failure 137 system availability is 0.9580. From Figure 3, one can see that the availability graph is constant for the time period 35 units to 50 units. (ii) Table 4 and corresponding Figure 4 give an idea about the behaviour of the reliability of proposed railway communication system regarding time t for various system failure rates. From Table 4 and Figure 4, it is easily seen that the reliability of the proposed system decreases rapidly as increment in time t. At initially, reliability is one and after 10 units of time, reliability is 0.08341. (iii) From Table 5, it is easily seen that the MTTF of the proposed system continuously decreases as all the failure rates λ1, λc1, λ2, λc2 increases. With respect to all stations failure rates MTTF is decreases in a uniform manner but with respect to control centre failure rates it decreases rapidly. Figure 5 demonstrates that the MTTF is high with variation in the failure rate of the first control centre and lowest regarding failure rates of path 1 stations which means failure rate of all stations of path 1 has more frequent downtime and disruption as compare to other failure rates. (iv) Further, From Table 6 and Figure 6, one can see the behaviour of the failure rates on system reliability. The reliability of the RCS is more influenced by the variation in the second control centre failure rate which means second control centre failure rate causes the stronger change in reliability of the proposed system as time increases. 6. 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Submitted for possible open access publication under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.4108/icst.valuetools.2013.254370 RELIABILITY CHARACTERISTICS OF RAILWAY COMMUNICATION SYSTEM SUBJECT TO SWITCH FAILURE Mangey Ram 1, Vaishali Tyagi2* 1. Introduction 2. System modelling 3. Governing Equations 4. Numerical Calculations 4.1. Availability 4.2. Reliability 4.3. Mean time to system Failure 4.4. Sensitivity of reliability 5. Result Discussion 6. Conclusion References