1. 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 (d).
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) to (d) for . Pl = Pb =Pr = 10 ^-2
He did this problem in the "A Problem" video just with different numbers. ;)
To derive the expressions for parts (a) to (d), we need to use the probabilities of link failure (Pl), repeater failure (Pr), and bridge failure (Pb) provided in the question.
a) Probability of failure of system A:
In system A, there is a single ring with 100 stations. The probability of the entire system failing is the probability that at least one link or repeater fails. Since the stations are connected by links, and each station has a repeater, we need to consider both link and repeater failures. Therefore, the probability of failure of system A can be calculated as follows:
Probability of failure of system A = 1 - (1 - Pl)^100 * (1 - Pr)^100
b) Probability of complete failure of system B:
In system B, there are four rings, each with 25 stations. The probability of the entire system failing is the probability that all links, repeaters, and bridges fail. So, the probability of complete failure of system B can be calculated as follows:
Probability of complete failure of system B = (Pl^25 * Pr^25 * Pb) ^ 4
c) Probability that a particular station will find the network unavailable, for systems A and B:
For system A, since it is a single ring, any station will only experience network unavailability if either its link or its repeater fails. Therefore, the probability that a particular station in system A will find the network unavailable is:
Probability of network unavailability for system A = Pr + Pl - (Pr * Pl)
For system B, each station is part of one of the four rings. A station will find the network unavailable if any link, repeater, or bridge in its ring fails. Therefore, the probability that a particular station in system B will find the network unavailable is:
Probability of network unavailability for system B = Pr + Pl + Pb - (Pr * Pl * Pb)
d) Probability that any two stations selected at random will be unable to communicate, for systems A and B:
In both system A and system B, any two stations can communicate if and only if there is no failure in the links and repeaters between them. Therefore, the probability that any two stations selected at random will be unable to communicate is the same as the probability of a link or repeater failure. Thus, for both systems, the probability is given by:
Probability of two stations being unable to communicate = Pl
e) To compare the values of the expressions in parts (a) to (d) for Pl = Pb = Pr = 10^-2, substitute these values into each expression and evaluate them separately.