 # Markov Method

This is a widely used method for performing reliability-related analysis of engineering systems and is named after a Russian mathematician, Andrei A. Markov (1856-1922). FIGURE 4.3 A fault tree with the given and calculated fault event occurrence probability values.

The method is used quite commonly for modeling repairable engineering systems with constant failure and repair rates and is subject to the following assumptions :

• • The transitional probability from one system state to another in the finite time interval At is given by XAt, where A is the transition rate (e.g., system failure or repair rate) from one system state to another.
• • All occurrences are independent of each other.
• • The probability of more than one transition occurrence in the finite time interval At from one system state to another is negligible (e.g., (АДг)(АД?) -» 0).

The application of this method is demonstrated by solving the example shown below.

Example 4.3

Assume that a system can be either in an operating or a failed state. The system constant failure and repair rates are A. and fis, respectively. The system state space diagram is shown in Figure 4.4. The numerals in box and diamond denote the system states. Develop equations for the system time-dependent and steady-state availabilities and unavailabilities, reliability, and mean time to failure by using the Markov method. FIGURE 4.4 System state space diagram.

By using the Markov method, we write down the following equations for the system states 0 and 1 shown in Figure 4.4, respectively. where

t is the time.

As At is the probability of system failure in finite time interval At.

HsAt is the probability of system repair in finite time interval At.

(1-Ал.Д?) is the probability of no failure in finite time interval At.

|l - [is An is the probability of no repair in finite time interval At.

Pj(t) is the probability that the system is in state j at time t, for j = 0, 1.

P0(f + A?) is the probability of the system being in operating state 0 at time (/ +ДГ).

P1 (? + At) is the probability of the system being in failed state 1 at time (t + At). From Equation (4.3), we get From Equation (4.5), we write Thus, from Equation (4.6), we obtain Similarly, using Equation (4.4), we get At time t = 0. P0 (0) = 1 and P{ (0) = 0.

By solving Equations (4.7) and (4.8), we obtain the following equations : Thus, the system time-dependent availability and unavailability, respectively, are and where

A Vs (?) is the system time-dependent availability.

UAVS (?) is the system time-dependent unavailability.

By letting time t go to infinity in Equations (4.11) and (4.12), we get and where

A Vs is the system steady-state availability.

UAVS is the system steady-state unavailability.

For i-is = 0, from Equation (4.9), we get By integrating Equation (4.15) over the time interval [0,oo], we get the following equation for the system mean time to failure : where

MTTFS is the system mean time to failure.

Thus, the system time-dependent and steady-state availabilities and unavailabilities, reliability, and mean time to failure are given by Equations (4.11), (4.13), (4.12), (4.14), (4.15), and (4.16), respectively.

Example 4.4

Assume that the constant failure and repair rates of a system used in industry are 0.0004 failures per hour and 0.0008 repairs per hour, respectively. Calculate the system steady-state availability and availability during a 100-hour mission.

By substituting the given data into Equations (4.13) and (4.11), we get and Thus, the system steady-state availability and availability during a 100-hour mission are 0.6666 and 0.9623, respectively.

# Network Reduction Approach

This is probably the simplest approach for determining the reliability of systems composed of independent series and parallel systems. However, the subsystems forming bridge networks/configurations can also be handled by first using the delta- star method . Nonetheless, the network reduction approach sequentially reduces the series and parallel subsystems to equivalent hypothetical single units until the whole system under consideration itself becomes a single hypothetical unit. The example presented below demonstrates this approach.

Example 4.5

An independent unit network representing a system is shown in Figure 4.5 (i). The reliability R; of unit j for j = 1,2, 3, ..., 7 is given. Calculate the network reliability by utilizing the network reduction approach.

First, we have highlighted subsystems А, В, C, and D of the network as shown in Figure 4.5 (i). The subsystems В and C have their units in series; thus, we reduce them to single hypothetical units as follows: and where

RB is the subsystem В reliability.

Rc is the subsystem C reliability.

The reduced network is shown in Figure 4.5 (ii). Now, the network is made up of two parallel subsystems A and D. Thus, we reduce both these subsystems to single hypothetical units as follows: and where

Ra is the subsystem A reliability.

Rd is the subsystem D reliability.

Figure 4.5 (iii) shows the reduced network with the above calculated values. This resulting network is a two-unit series system and its reliability is given by The single hypothetical unit shown in Figure 4.5 (iv) represents the reliability of the whole network shown in Figure 4.5 (i). More clearly, the whole network is reduced to a single hypothetical unit. Thus, the whole network reliability, Rs, is 0.5036.