## 3 Markov Model  
One important step to compare the reliability of power converter is to use a quantitative reliability evaluation which is executed by employing some suitable models. The Markov chain is one of the well-known reliability models which is used to evaluate the reliability of different power converters. In order to obtain the Markov model of a power converter, the failure rates of the utilized components must be calculated in different operation modes. Since power switch failure is more probable than that of the other devices [32; 33], only the failure of the switches is considered in this paper. According to MIL-HDBK-217F, the failure rate of a semiconductor switch is given as:  
\[\lambda_{fail}=\lambda_{b}\pi_{q}\pi_{app}\pi_{even}\pi_{th} \tag{1}\]  
where, \(\lambda_{b}\), \(\pi_{q}\), \(\pi_{app}\), \(\pi_{env}\),and \(\pi_{th}\) are the basic failure rate, the quality of configuration factor, application factor, environment factor, and temperature factor, respectively. The basic failure rate can be obtained by referring to historical reports of power switch failure rates under normal and standard conditions. The other factors are accelerated factor that increase the failure rates of the power switch. For different manufacturing technology, environmental conditions, application type, \(\pi_{q}\), \(\pi_{app}\), and \(\pi_{env}\), can be considered as fixed values. The temperature factor of semiconductor devices (\(\pi_{th}\)) can be calculated as follow:  
\[\pi_{t}=e^{-a(\frac{1}{I_{j}+b}-\frac{1}{c})} \tag{2}\]  
\[T_{j}=T_{a}+(R_{th})P_{loss} \tag{3}\]  
where, \(a\), \(b\), and \(c\) are constant values [33; 34], and \(T_{j}\) is the junction temperature of the power switch. In order to compare the MTTF of the switches, the power losses should be obtained. The total power loss of a typical switch is as follows:  
\[P_{Loss}=\frac{1}{2}I_{avg}V_{s}f_{sw}(t_{off}+t_{on})+\frac{1}{2}f_{sw}C_{oss }V_{s}^{2}+R_{s}I_{rms}^{2} \tag{4}\]  
Figure 1: The studied converter and redundant switch configurations, (**a**) general configuration of an n-phase interleaved boost converter, (**b**) parallel-switch configuration equipped with two individual relays, and (**c**) standby-switch configuration equipped with one relay.  
where \(I_{avg}\), \(V_{s}\), \(f_{sw}\), and \(C_{oss}\) are average current of switch, voltage across the switch, switching frequency, and parasitic drain-source capacitor, respectively. Also, \(t_{on}\), \(t_{off}\), \(I_{rms}\), and \(R_{s}\) are the turn _on_ and _off_ switching times of the switch, Root Mean Square (RMS) current passing through the switch and on-state resistance of the switch, respectively. Having the failure rates (\(\lambda\)) in different modes, the Markov chain is used to determine the total MTTF of the system. Figure 2 exemplifies a typical Markov chain, which is a stochastic process of \(\lambda(t)\).  
According to Figure 2, the probability that the system is in the \(m\)th state at t moment (P\({}_{m}\)(t)), is given by the following equation:  
\[\frac{\mathrm{d}\mathrm{P}_{m}(t)}{\mathrm{dt}}=-\sum_{m\
eq j}\lambda_{mj}P_{ i}(t)+\sum_{j\
eq m}\lambda_{jm}P_{j}(t) \tag{5}\]  
As stated in reference [35], the reliability is the probability of a device performing its purpose adequately for the period of time intended under the operating conditions encountered. The MTTF is the average time during which the system successfully operates before it fails. This index is given by:  
\[MTTF=\int\limits_{0}^{\infty}R(t)dt \tag{6}\]  
where \(R(t)\) is the reliability of the system.