Barra empotrada con un extremo libre

As stated before, reliability is measured mathematically by the probability of a system functioning without encountering a failure for a specified period of time. Hence, the reliability function 𝑅𝑅(𝑡𝑡) is defined as the probability that the system has operated over the time interval 0 to 𝑡𝑡. In contrast, the availability is the fraction of time the system is up, and hence, the availability function 𝐴𝐴(𝑡𝑡) is defined to be the probability that the system is operating at time 𝑡𝑡. However, 𝐴𝐴(𝑡𝑡) has no information on how many failure-repair cycles in the interval 0 to 𝑡𝑡. In the case of irreparable systems, 𝐴𝐴(𝑡𝑡) = 𝑅𝑅(𝑡𝑡), and with repairable systems, 𝑅𝑅(𝑡𝑡) ≤ 𝐴𝐴(𝑡𝑡) [48], [49], [80], [113], [114], [120], [121]. Both reliability and availability represent two different but related operational measures of system performance. Availability is a combined measure of maintainability and reliability, and it has been widely used as a measure of system effectiveness [122]. Yet, availability calculations require more information about maintenance and logistical support of the system, and so, it is generally more difficult than reliability to attain. For critical applications, however, a reliability measure is the most important and stringent measure [115]. Since we are dealing with security, the reliability measure is the measure of choice in the proposed extensions from dependability theory.

There are two general approaches to studying reliability: i) reliability based on random events and ii) reliability based on random variables. For completeness, we show both approaches.

Reliability modeling based on random events: To demonstrate reliability based on random events,

let S denote the event that a system of 𝑛𝑛 components will be functioning, and the event that component 𝑖𝑖 is working is denoted by 𝑥𝑥𝑖𝑖, 𝑖𝑖 ∈ {1, … , 𝑛𝑛}; thus 𝑥𝑥𝑖𝑖 represents the failure of component 𝑖𝑖

(the event when component 𝑖𝑖 is not working).

Now, if the system components are logically arranged in series, the system will be working if and only if all 𝑛𝑛 components are working. Hence, event S will be the intersection of all 𝑥𝑥𝑖𝑖′𝑆𝑆, 𝑖𝑖 = 1,2, … , 𝑛𝑛,

events as follows.

𝑆𝑆 = 𝑥𝑥1∩ 𝑥𝑥2∩ … ∩ 𝑥𝑥𝑛𝑛 (2-14)

And the probability of the system working, i.e., reliability of the system, is given by 𝑅𝑅𝑠𝑠= 𝑃𝑃(𝑆𝑆) = 𝑃𝑃(𝑥𝑥1∩ 𝑥𝑥2∩ … ∩ 𝑥𝑥𝑛𝑛) = 𝑃𝑃(𝑥𝑥1𝑥𝑥2… 𝑥𝑥𝑛𝑛)

If failure events of the system are not independent (all the random events 𝑥𝑥𝑖𝑖′𝑆𝑆, 𝑖𝑖 = 1, … , 𝑛𝑛, are

dependent), we write

𝑅𝑅𝑠𝑠= 𝑃𝑃(𝑆𝑆) = 𝑃𝑃(𝑥𝑥1) 𝑃𝑃(𝑥𝑥2⁄ ) 𝑃𝑃(𝑥𝑥𝑥𝑥1 3⁄𝑥𝑥1𝑥𝑥2) … 𝑃𝑃(𝑥𝑥𝑛𝑛⁄𝑥𝑥1𝑥𝑥2… 𝑥𝑥𝑛𝑛−1) (2-15)

And when failure events are independent (all the random events 𝑥𝑥𝑖𝑖′𝑆𝑆, 𝑖𝑖 = 1, … , 𝑛𝑛, are independent),

we write 𝑅𝑅𝑠𝑠= 𝑃𝑃(𝑆𝑆) = 𝑃𝑃(𝑥𝑥1) 𝑃𝑃(𝑥𝑥2) 𝑃𝑃(𝑥𝑥3) … 𝑃𝑃(𝑥𝑥𝑛𝑛) (2-16) So, 𝑅𝑅𝑠𝑠= � 𝑃𝑃(𝑥𝑥𝑖𝑖 𝑛𝑛 𝑖𝑖=1 ) = � 𝑅𝑅𝑖𝑖 𝑛𝑛 𝑖𝑖=1 , 𝑤ℎ𝑒𝑒𝑟𝑟𝑒𝑒 𝑅𝑅𝑖𝑖 = 𝑃𝑃(𝑥𝑥𝑖𝑖) (2-17)

On the other side, if system components are logically arranged in parallel, the system will be working if at least one of the 𝑛𝑛 components is working. This arrangement is sometimes called a redundant configuration. Consequently, event S will be the union of all 𝑥𝑥𝑖𝑖′𝑆𝑆, 𝑖𝑖 = 1,2, … , 𝑛𝑛, events,

𝑆𝑆 = 𝑥𝑥1∪ 𝑥𝑥2∪ … ∪ 𝑥𝑥𝑛𝑛 (2-18)

𝑅𝑅𝑝= 𝑃𝑃(𝑆𝑆) = 𝑃𝑃(𝑥𝑥1∪ 𝑥𝑥2∪ … ∪ 𝑥𝑥𝑛𝑛) = 𝑃𝑃(𝑥𝑥1+ 𝑥𝑥2+ ⋯ + 𝑥𝑥𝑛𝑛)

If failure events of the system are not independent (all the random events 𝑥𝑥𝑖𝑖′𝑆𝑆, 𝑖𝑖 = 1, … , 𝑛𝑛, are

dependent), then we write

𝑅𝑅𝑝= 𝑃𝑃(𝑆𝑆) = [𝑃𝑃(𝑥𝑥1) +𝑃𝑃(𝑥𝑥2) + 𝑃𝑃(𝑥𝑥3) + ⋯ + 𝑃𝑃(𝑥𝑥𝑛𝑛)]

− �𝑃𝑃(𝑥𝑥1𝑥𝑥2) +𝑃𝑃(𝑥𝑥1𝑥𝑥3) + ⋯ + 𝑃𝑃�𝑥𝑥𝑖𝑖𝑥𝑥𝑗�_{𝑖𝑖≠𝑗}�

+ ⋯ + (−1)𝑛𝑛−1_{𝑃𝑃(𝑥𝑥}

1𝑥𝑥2… 𝑥𝑥𝑛𝑛)

(2-19)

The formula above can be written in a simpler way if one deals with the probability of system failure instead. Parallel system failure occurs if all components fail, yielding the intersection operation of all the events of components failures

𝑅𝑅𝑝= 1 − 𝑃𝑃(𝑥𝑥1𝑥𝑥2… 𝑥𝑥𝑛𝑛) (2-20)

And when failure events are independent (all the random events 𝑥𝑥𝑖𝑖′𝑆𝑆, 𝑖𝑖 = 1, … , 𝑛𝑛, are independent),

we write 𝑅𝑅𝑝= 𝑃𝑃(𝑆𝑆) = 1 − 𝑃𝑃(𝑥𝑥1) 𝑃𝑃(𝑥𝑥2) 𝑃𝑃(𝑥𝑥3) … 𝑃𝑃(𝑥𝑥𝑛𝑛) (2-21) So, 𝑅𝑅𝑝= 1 − � 𝑃𝑃(𝑥𝑥𝑖𝑖 𝑛𝑛 𝑖𝑖=1 ) = 1 − � 𝑅𝑅𝑖𝑖 𝑛𝑛 𝑖𝑖=1 , 𝑤ℎ𝑒𝑒𝑟𝑟𝑒𝑒 𝑅𝑅𝑖𝑖 = 𝑃𝑃(𝑥𝑥𝑖𝑖) (2-22) Reliability modeling based on random variables: To demonstrate reliability based on random

variables, we suppose that X𝑖𝑖, the state of the ith component, is a random variable. So,

𝑅𝑅𝑖𝑖 = 𝑃𝑃{X𝑖𝑖 = 1} = 1 − 𝑃𝑃{X𝑖𝑖 = 0} = 𝑝𝑖𝑖 (2-23)

and is called the reliability of the ith component. Also, 𝑅𝑅 = 𝑃𝑃{∅(𝐱) = 1} = 1 − 𝑃𝑃{∅(𝐱) = 0},

𝑤ℎ𝑒𝑒𝑟𝑟𝑒𝑒 𝐱 = (𝑋𝑋1, 𝑋𝑋2, … , 𝑋𝑋𝑛𝑛)

(2-24) and is called the reliability of the system. Since ∅(𝐱) is a Bernoulli random variable, the reliability of the system 𝑅𝑅 can be computed by taking the expectation, that is,

For systems connected in series, when all the random variables 𝑋𝑋𝑖𝑖′𝑆𝑆, 𝑖𝑖 = 1, … , 𝑛𝑛, are independent, we can write 𝑅𝑅𝑠𝑠= 𝑃𝑃{X1= 1}𝑃𝑃{X2= 1} … 𝑃𝑃{X𝑛𝑛= 1} = � 𝑝𝑖𝑖 𝑛𝑛 𝑖𝑖=1 (2-26) And when all the random variables 𝑋𝑋𝑖𝑖′𝑆𝑆, 𝑖𝑖 = 1, … , 𝑛𝑛, are dependent we write

𝑅𝑅𝑠𝑠= 𝑃𝑃{𝑋𝑋1= 1}𝑃𝑃{𝑋𝑋2= 1/𝑋𝑋1= 1} … 𝑃𝑃{𝑋𝑋𝑛𝑛 = 1 𝑋𝑋⁄ = 1, 𝑋𝑋1 2= 1, … , 𝑋𝑋𝑛𝑛−1= 1} (2-27)

Also, for systems connected in parallel, when all the random variables 𝑋𝑋𝑖𝑖′𝑆𝑆, 𝑖𝑖 = 1, … , 𝑛𝑛, are

independent, we can write

𝑅𝑅𝑝= 1 − 𝑃𝑃{𝑋𝑋1= 0}𝑃𝑃{𝑋𝑋2= 0} … 𝑃𝑃{𝑋𝑋𝑛𝑛 = 0}

= 1 − �(1 − 𝑝𝑖𝑖) 𝑛𝑛

𝑖𝑖=1

(2-28) And when all the random variables 𝑋𝑋𝑖𝑖′𝑆𝑆, 𝑖𝑖 = 1, … , 𝑛𝑛, are dependent we write

𝑅𝑅𝑝= 1 − 𝑃𝑃{𝑋𝑋1= 0}𝑃𝑃{𝑋𝑋2= 0/𝑋𝑋1= 0} … 𝑃𝑃{𝑋𝑋𝑛𝑛= 0 𝑋𝑋⁄ = 0, 𝑋𝑋1 2 = 0, … , 𝑋𝑋𝑛𝑛−1= 0}

And the same logic applies to mixed structure systems, i.e., systems that contain parallel and series components together.

Above we have considered the reliability of the system in terms of static probabilities, i.e., probabilities are considered as constants. Systems, however, operate and hence fail over time. As such, it is commonly accepted to model the reliability of systems as a function of time. To show that, let 𝑀𝑀 be a random variable that represents the time to failure of the system. The reliability can then be defined in terms of probability of failure as a function of time, as illustrated in Figure 2-6, and written as

Figure 2-6: The failure process in reliability context

This function means the system will function for time 𝑡𝑡 or greater if and only if it is still functioning at time 𝑡𝑡. 𝑅𝑅(𝑡𝑡) is a monotonic non-increasing function of 𝑡𝑡 with unity at the start of life: 𝑅𝑅(0) = 1 and 𝑅𝑅(∞) = 0 [49]. 𝐹𝐹(𝑡𝑡), failure distribution, however represents the probability that the system will fail before time 𝑡𝑡, and equals

𝐹𝐹(𝑡𝑡) = 𝑃𝑃(𝑀𝑀 ≤ 𝑡𝑡) = 1 − 𝑅𝑅(𝑡𝑡)

Nevertheless, time-dependent reliability is the technique of decomposing the system into a set of subsystems, or components, where their reliabilities are known or to be computed with respect to time, so

𝑅𝑅(𝑡𝑡) = 𝑅𝑅(𝑃𝑃1(𝑡𝑡), 𝑃𝑃2(𝑡𝑡), … , 𝑃𝑃𝑛𝑛(𝑡𝑡)) (2-30)

where,

𝑃𝑃𝑖𝑖(𝑡𝑡) = 𝑃𝑃{𝑙𝑖𝑖𝑓𝑓𝑒𝑒𝑡𝑡𝑖𝑖𝑆𝑆𝑒𝑒 𝑚𝑚𝑓𝑓 𝑠𝑠𝑚𝑚𝑆𝑆𝑝𝑚𝑚𝑛𝑛𝑒𝑒𝑛𝑛𝑡𝑡 𝑖𝑖 > 𝑡𝑡} = 1 − 𝐹𝐹𝑖𝑖(𝑡𝑡)

As in [123], for a series system with 𝑛𝑛 independent components, we can write 𝑅𝑅(𝑡𝑡) = 𝑅𝑅1(𝑡𝑡) × 𝑅𝑅2(𝑡𝑡) × … × 𝑅𝑅𝑛𝑛(𝑡𝑡)

= � 𝑅𝑅𝑖𝑖(𝑡𝑡) 𝑛𝑛 𝑖𝑖=1

(2-31) And, for a parallel system with 𝑛𝑛 independent components, we can write

𝑅𝑅(𝑡𝑡) = 1 − [(1 − 𝑅𝑅1(𝑡𝑡)) × (1 − 𝑅𝑅2(𝑡𝑡)) × … × (1 − 𝑅𝑅𝑛𝑛(𝑡𝑡))]

= 1 − �(1 − 𝑅𝑅𝑖𝑖(t)) 𝑛𝑛

𝑖𝑖=1

(2-32) Yet, one must note that 𝑅𝑅 can be expressed as a function of components’ reliabilities, 𝑅𝑅 = 𝑅𝑅(𝐩), that is, a monotonic increasing function of 𝐩, where 𝐩 = (𝑝1, 𝑝2, … , 𝑝𝑛𝑛) [48].

time t failure

0 _{time interval t} time to failure T (r.v.)

Regardless of the modelling approaches above, a common structure in reliability evaluation is a k- out-of-n system with identical and independent components [47], [48], [104], [113], [115]. Considering this system as an application of the binomial distribution, its reliability function is given by 𝑅𝑅(𝐩) = � �𝑛𝑛 𝑖𝑖 � 𝑛𝑛 𝑖𝑖=𝑘 𝑝𝑖𝑖_{(1 − 𝑝)}𝑛𝑛−𝑖𝑖 where 𝑝 = 𝑝1 = 𝑝2= ⋯ = 𝑝𝑛𝑛 (2-33) Note that calculating the reliability of a k-out-of-n system when its components are not identical can be very complicated procedure, as the state enumeration approach is used to sum up all the probabilities of possible system realisations with the number of the working components is not less than 𝑘.

In light of the above, two important conclusions stated in [49], [113], [120] must be taken into account for series and parallel arrangements. Firstly, for a series arrangement system, the larger (lower) the number of components connected in series, the lower (larger) is the reliability of the system. Therefore, the reliability of the (𝑛𝑛 + 1) components series system is upper-bounded by the reliability of the same system having (𝑛𝑛) components. So, adding an extra component (𝑛𝑛 + 1) for a series arrangement of 𝑛𝑛 components, say 𝑅𝑅𝑛𝑛+1(𝑡𝑡), 𝑤ℎ𝑒𝑒𝑟𝑟𝑒𝑒 𝑅𝑅𝑛𝑛+1(𝑡𝑡) < 1, we write

𝑅𝑅(𝑡𝑡) = 𝑅𝑅1(𝑡𝑡) × 𝑅𝑅2(𝑡𝑡) × … × 𝑅𝑅𝑛𝑛(𝑡𝑡) = � 𝑅𝑅𝑖𝑖(𝑡𝑡) 𝑛𝑛 𝑖𝑖=1 > 𝑅𝑅1(𝑡𝑡) × 𝑅𝑅2(𝑡𝑡) × … × 𝑅𝑅𝑛𝑛(𝑡𝑡) × 𝑅𝑅𝑛𝑛+1(𝑡𝑡) = � 𝑅𝑅𝑖𝑖(𝑡𝑡) 𝑛𝑛+1 𝑖𝑖=1 (2-34)

Also, the reliability of the system is less than the reliability of its least reliable component; and the system’s reliability decreases (increases) if any component’s reliability decreases (increases). Therefore, the reliability of the series system is upper-bounded by the reliability of its least reliable component. This feature is analogous to the weakest link of a security system, which is considered a measure of the security strength as discussed before. Thus, considering a series arrangement of 𝑛𝑛 components with 𝑅𝑅𝑠𝑠(𝑡𝑡), where 𝑆𝑆 ∈ {1, … , 𝑛𝑛}, denoting the least reliable component, we write

𝑅𝑅(𝑡𝑡) = 𝑅𝑅1(𝑡𝑡) × 𝑅𝑅2(𝑡𝑡) × … × 𝑅𝑅𝑛𝑛(𝑡𝑡) = � 𝑅𝑅𝑖𝑖(𝑡𝑡) 𝑛𝑛 𝑖𝑖=1

< 𝑅𝑅𝑠𝑠(𝑡𝑡) (2-35)

Secondly, for a parallel arrangement system, the larger the number of components in parallel, the larger is the reliability of the system. Therefore, the reliability of the (𝑛𝑛 + 1) components parallel system is lower-bounded by the reliability of the same system having (𝑛𝑛) components. So, adding an extra component (𝑛𝑛 + 1) for a parallel arrangement of 𝑛𝑛 components, say 𝑅𝑅𝑛𝑛+1(𝑡𝑡), 𝑤ℎ𝑒𝑒𝑟𝑟𝑒𝑒 𝑅𝑅𝑛𝑛+1(𝑡𝑡) <

1, we write 𝑅𝑅(𝑡𝑡) = 1 − [(1 − 𝑅𝑅1(𝑡𝑡)) × (1 − 𝑅𝑅2(𝑡𝑡)) × … × (1 − 𝑅𝑅𝑛𝑛(t))] = 1 − ��1 − 𝑅𝑅𝑖𝑖(t)� 𝑛𝑛 𝑖𝑖=1 < 1 − [(1 − 𝑅𝑅1(𝑡𝑡)) × (1 − 𝑅𝑅2(𝑡𝑡)) × … × (1 − 𝑅𝑅𝑛𝑛(t)) × (1 − 𝑅𝑅𝑛𝑛+1(t))] = 1 − �(1 − 𝑅𝑅𝑖𝑖(t)) 𝑛𝑛+1 𝑖𝑖=1 (2-36)

Furthermore, the reliability of the system is larger than the reliability of its most reliable component. Therefore, the reliability of the parallel system is lower-bounded by the reliability of its most reliable component. Considering a parallel arrangement of 𝑛𝑛 components with 𝑅𝑅𝑙(𝑡𝑡), 𝑤ℎ𝑒𝑒𝑟𝑟𝑒𝑒 𝑙 ∈

{1, … , 𝑛𝑛}, as the most reliable component, we write

𝑅𝑅(𝑡𝑡) = 1 − [(1 − 𝑅𝑅1(𝑡𝑡)) × (1 − 𝑅𝑅2(𝑡𝑡)) × … × (1 − 𝑅𝑅𝑛𝑛(t))] = 1 − �(1 − 𝑅𝑅𝑖𝑖(t)) 𝑛𝑛 𝑖𝑖=1 > 𝑅𝑅𝑙(𝑡𝑡) (2-37) We conclude this section by reviewing four important functions related to reliability evaluation: (i) the failure density function, which describes how the failure probability is spread over time and denoted by 𝑓𝑓(𝑡𝑡). 𝑓𝑓(𝑡𝑡) is always non-negative and the total area beneath it is always equal to one as it is basically a probability distribution function, so

� 𝑓𝑓(𝑡𝑡)𝑑𝑡𝑡 = 1∞

0 (2-38)

(ii) The failure distribution function 𝐹𝐹(𝑡𝑡), which is the cumulative distribution function of the failure density function 𝑓𝑓(𝑡𝑡). As mentioned earlier, 𝐹𝐹(𝑡𝑡) represents the probability that the system will fail before time 𝑡𝑡 and gives the area beneath the failure density function until time 𝑡𝑡, and equals

𝐹𝐹(𝑡𝑡) = 𝑃𝑃(𝑀𝑀 ≤ 𝑡𝑡) = � 𝑓𝑓(𝑣)𝑑𝑣𝜆𝜆

0 , 𝑤ℎ𝑒𝑒𝑟𝑟𝑒𝑒 𝑣 𝑖𝑖𝑆𝑆 𝑎𝑎 𝑑𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 𝑣𝑎𝑎𝑟𝑟𝑖𝑖𝑎𝑎𝑏𝑏𝑙𝑒𝑒

(2-39) 𝐹𝐹(𝑡𝑡) and 𝑓𝑓(𝑡𝑡) satisfy the following relation:

𝑓𝑓(𝑡𝑡) =_{𝑑𝑡𝑡 𝐹𝐹}𝑑 (𝑡𝑡) = −_{𝑑𝑡𝑡 𝑅𝑅}𝑑 (𝑡𝑡) (2-40)

(iii) The reliability function 𝑅𝑅(𝑡𝑡), which gives the area beneath the failure density function after time 𝑡𝑡 and equals, 𝑅𝑅(𝑡𝑡) = 𝑃𝑃(𝑀𝑀 > 𝑡𝑡) = 1 − 𝐹𝐹(𝑡𝑡) = 1 − � 𝑓𝑓(𝑣)𝑑𝑣𝜆𝜆 0 = 𝑒𝑒𝑥𝑥𝑝 �− � 𝜆𝜆(𝑣)𝑑𝑣 𝜆𝜆 0 � (2-41) (iv) The hazard function 𝜆𝜆(𝑡𝑡), sometimes called instantaneous failure rate, which is defined as the limit of the failure rate as the interval length approaches zero. It then equals

𝜆𝜆(𝑡𝑡) =𝑓𝑓(𝑡𝑡)_{𝑅𝑅(𝑡𝑡) =1 − 𝐹𝐹(𝑡𝑡) = −}𝑓𝑓(𝑡𝑡) _𝑑𝑡𝑡𝑑 [ln 𝑅𝑅(𝑡𝑡)] (2-42) An important parameter to these equations is the Mean Time To Failure (𝑀𝑀𝑀𝑀𝑀𝑀𝐹𝐹). In this context, it is the expected value of the continuous random variable 𝑀𝑀 and gives the area beneath the reliability function. 𝑀𝑀𝑀𝑀𝑀𝑀𝐹𝐹 is given by

𝑀𝑀𝑀𝑀𝑀𝑀𝐹𝐹 = � 𝑡𝑡𝑓𝑓(𝑡𝑡)𝑑𝑡𝑡∞

0 = � 𝑅𝑅(𝑡𝑡)𝑑𝑡𝑡

∞

0 (2-43)

Considering the exponential distribution model as an example, the above functions are demonstrated in Figure 2-7 and Figure 2-8. Given the constant failure rate property, the hazard rate function yields,

𝜆𝜆(𝑡𝑡) = 𝜆𝜆 =1 𝜃 =

1

𝑀𝑀𝑀𝑀𝑀𝑀𝐹𝐹 , a constant The failure density is given by

𝑓𝑓(𝑡𝑡) = 𝜆𝜆𝑒𝑒−𝜆𝜆𝜆𝜆

Similarly, the failure distribution becomes 𝐹𝐹(𝑡𝑡) = 1 − 𝑒𝑒−𝜆𝜆𝜆𝜆

The reliability function is obtained by 𝑅𝑅(𝑡𝑡) = 𝑒𝑒−𝜆𝜆𝜆𝜆

And the variance is given by 𝜎2₌ 1

𝜆𝜆2

Figure 2-7: The exponential distribution model. (a) Failure density. (b) Failure distribution

Figure 2-8: The exponential distribution model. (a) Reliability function. (b) Hazard function

𝜆𝜆𝑒𝑒−𝜆𝜆𝜆𝜆 𝑡𝑡 𝜆𝜆 𝐹𝐹(𝑡𝑡) 𝑡𝑡 1.0 (𝑎𝑎) (𝑏𝑏) 𝑓𝑓(𝑡𝑡) 1 − 𝜆𝜆𝑒𝑒−𝜆𝜆𝜆𝜆 1 𝜆𝜆 1 −1_𝑒𝑒 𝑅𝑅(𝑡𝑡) 𝑡𝑡 1.0 𝑡𝑡 (𝑎𝑎) (𝑏𝑏) 𝑒𝑒−𝜆𝜆𝑡𝑡 𝜆𝜆(𝑡𝑡) λ 1 𝜆𝜆 1 𝑒𝑒