A critical circuit in a communication network relies on a set of eight identical relays. If any one of the relays fails, it will disrupt the entire network. The design engineer must ensure a 90% probability that the network will not fail over a five-year period. What is the maximum tolerable probability of failure for each relay?
P(A) = 0.90 [not failing]
P(b) = 1 - 0.90
= 0.10 [failing]
Answer: 0.10/8
= 0.0125
Multiply that to get 1.25 or 1.3%
H.C as this does give the right answer maybe or maybe not by coincidence,
the exact way of getting the answer would be
0.9 = x^8
x = (0.9)^(1/8)
x = 0.98691628136...
the answer is supposed to be 1.3 % but i don't know how this solution is reached.
Well, to ensure a 90% probability that the network will not fail over a five-year period, we need to consider the reliability of each relay. Let me pull out my clown calculator here...
So, we have eight identical relays that cannot fail. If any one of them fails, it's game over for the network. In order to find the maximum tolerable probability of failure for each relay, we can use a little math magic.
Since we want the network to have a 90% chance of not failing, that means we want a 10% chance of failure. And since there are eight relays in total, that means we need to divide that 10% chance of failure by the number of relays.
So, 10% divided by 8 gives us... (sound of calculator button being pressed)... 1.25%!
Therefore, the maximum tolerable probability of failure for each relay should be 1.25%. That way, when you add up the probabilities of failure for all eight relays, it will give you a 10% chance of the entire network failing.
However, keep in mind that even the best engineers and the most reliable relays can't guarantee a 100% failure-free network. So, let's hope those relays don't decide to take a vacation at the same time!
To determine the maximum tolerable probability of failure for each relay, we need to use the concept of reliability and the reliability function.
The reliability function, denoted by R(t), represents the probability that a component or system will continue to function properly until time t. In this case, we want to calculate the maximum tolerable probability of failure for each relay over a five-year period, which corresponds to a time t of 5 years.
Let's assume that the probability of a relay failing at any given time is denoted by λ (lambda). The failure rate for each relay, denoted by λ(t), is given by the derivative of the reliability function, which is equal to λ.
Now, if we assume that the relays are identical, we can express the reliability of the critical circuit as the product of the reliabilities of all the relays. Since there are eight identical relays, we can raise the reliability of a single relay to the power of eight to get the overall reliability of the critical circuit.
Therefore, we can write the equation as follows:
R(5 years) = (R(relay))^8
Next, to calculate the maximum tolerable probability of failure for each relay, we need to rearrange the equation. Taking the 8th root on both sides, we have:
R(relay) = (R(5 years))^(1/8)
It is given that the design engineer must ensure a 90% probability that the network will not fail over a five-year period. So, R(5 years) should be 0.90.
Plugging in this value into the equation, we get:
R(relay) = (0.90)^(1/8)
Calculating this expression, the maximum tolerable probability of failure for each relay is approximately 0.99173, or 99.173%.
Therefore, the maximum tolerable probability of failure for each relay in the communication network is 99.173%.