System A consists of a single ring with 100 stations, one per repeater. System B consists of four 25 stations rings linked by a bridge. If the probability of a link failure is , a repeater failure is , and a bridge failure is , derive an expression for parts (a) to (e). 50 points
a)Probability of failure of system A.
b)Probability of complete failure of system B.
c)Probability that a particular station will find the network unavailable, for systems A and B.
d)Probability that any two stations selected at random will be unable to communicate for systems A and B.
e)Compare values of parts (a) and (b) for
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To derive an expression for each part, we need to understand the probability of failure for each component and how these components are connected in the systems.
Let's define the variables:
- p: probability of link failure (for both systems A and B)
- q: probability of repeater failure (only for system A)
- r: probability of bridge failure (only for system B)
a) Probability of failure of system A:
Since system A consists of a single ring with 100 stations, the probability of failure in this system can be calculated by considering the independent probabilities of link failure and repeater failure.
The probability of failure in one station due to link failure is given by (1 - p).
The probability of failure in one station due to repeater failure is given by q.
Since all stations are independent in a ring, we can multiply these probabilities together for each station. So, the probability of failure for the entire system A is:
Probability of failure of system A = (1 - p)^100 * q
b) Probability of complete failure of system B:
System B consists of four 25-station rings linked by a bridge. In this case, we need to take into consideration the probabilities of link failure, repeater failure, and bridge failure.
For complete failure of system B, all four rings must fail. Each ring operates independently, so we can use the probabilities of failure for a single ring to calculate the complete failure probability. The probability of complete failure of one ring is:
Probability of complete failure of one ring = (1 - p)^25 * q
Since there are four rings in system B, the probability of complete failure of system B is:
Probability of complete failure of system B = [ (1 - p)^25 * q ]^4 * r
c) Probability that a particular station will find the network unavailable, for systems A and B:
For system A, each station is part of the single ring, so the probability that a particular station finds the network unavailable is the same as the probability of failure for system A:
Probability that a particular station in system A finds the network unavailable = (1 - p)^100 * q
For system B, since the four rings are linked by a bridge, if any of the rings fail, the network will be unavailable for a particular station. Therefore, the probability that a particular station in system B finds the network unavailable is the same as the probability of complete failure for system B:
Probability that a particular station in system B finds the network unavailable = [ (1 - p)^25 * q ]^4 * r
d) Probability that any two stations selected at random will be unable to communicate, for systems A and B:
To calculate this probability, we need to consider the possible combinations of station pairs and calculate the probability that each pair is unable to communicate.
For system A, since all stations are part of the same ring, any two stations can communicate unless there is a failure in a particular station or link. Therefore, the probability that any two stations selected at random will be unable to communicate in system A is the same as the probability of failure for system A:
Probability that any two stations selected at random in system A will be unable to communicate = (1 - p)^100 * q
For system B, since the four rings are linked by a bridge, any pair of stations can communicate unless there is a failure in a particular station, link, or bridge. Therefore, the probability that any two stations selected at random will be unable to communicate in system B is the same as the probability of complete failure for system B:
Probability that any two stations selected at random in system B will be unable to communicate = [ (1 - p)^25 * q ]^4 * r
e) Comparison of values for parts (a) and (b):
To compare the values for parts (a) and (b), you need to evaluate the expressions for both systems A and B using the given values of p, q, and r. Once you substitute the values, you will obtain the actual probability of failure for each system. Then, you can compare these values to determine which system has a higher probability of failure.