A weight of L tons is suspended by n cables which share the load equally. If k, for 1 ≤ k ≤ n − 1, of the cables have broken, then the remaining (n − k) cables share the load equally. As soon as (n − 1) cables have failed, new cables will be installed instantly to restore the number of cables to n. Let X(t) be the number of unbroken cables at time t. The instantaneous failure rate of a single cable which carries M tons is αM, where α > 0 is a constant. The remaining time to failure of any cable operating at time t is independent of the past history of the process, conditional on X(t).
(a) Specify an appropriate state space for (X(t) : t ≥ 0), and explain why this stochastic process satisfies the Markov property.
(b) Give the generator matrix for (X(t)).
(c) Suppose that individual cables have a failure rate of 0.2 per year per ton of load.
(i) If 4 such cables are used to support 20 tons, find the probability that the system lasts for at least 2 years before reinstalling the cables. Hint: determine the number of states that the process must go through to get to the first reinstallation, and then express the total duration as the sum of the hold times in each of the states to be visited.
(ii) How many cables should be used to ensure with probability 0.999 that the system lasts for at least 2 years before reinstalling the cables?
Note: The sum of n independent Exp(µ) random variables has a Gamma distribution with shape parameter n and rate parameter µ.
My effort/wok so far:
My effort so far:
a-) An appropriate state space is: S = {0,1} where 0 means the cable is broken, and 1 means the cable is unbroken
Now, this stochastic process satisfies a markov property because it independent of the previous history of states, except being conditionally indpendent on the latest state only. That is:
P{X(t+u) = j | H(u), X(u) = i} = P(x(t+u) = j | x(u) = i)
Where it was given that the remaining time to failre of any cable operating at time (t) is independent of he history of the process, conditional on X(t). Thus, our markov process satisfies the markov property.
b-) The generator matrix (Q) has these elements:
qij = {(ri)*(pij) if i not equal to j ; or it is -ri if i = j
and since it is given that the instantaneous failure rate of a single cable which carries M tons is αM, where α > 0 is a constant, then we have r0 = αM
Now, could you please correct any of my mistakes?
Also, I don't know to continue from here? How to derive the values of pij and r1, etc. for Q matrix?
Thank you very much in advance for your help.
(a) Specify an appropriate state space for (X(t) : t ≥ 0), and explain why this stochastic process satisfies the Markov property.
(b) Give the generator matrix for (X(t)).
(c) Suppose that individual cables have a failure rate of 0.2 per year per ton of load.
(i) If 4 such cables are used to support 20 tons, find the probability that the system lasts for at least 2 years before reinstalling the cables. Hint: determine the number of states that the process must go through to get to the first reinstallation, and then express the total duration as the sum of the hold times in each of the states to be visited.
(ii) How many cables should be used to ensure with probability 0.999 that the system lasts for at least 2 years before reinstalling the cables?
Note: The sum of n independent Exp(µ) random variables has a Gamma distribution with shape parameter n and rate parameter µ.
My effort/wok so far:
My effort so far:
a-) An appropriate state space is: S = {0,1} where 0 means the cable is broken, and 1 means the cable is unbroken
Now, this stochastic process satisfies a markov property because it independent of the previous history of states, except being conditionally indpendent on the latest state only. That is:
P{X(t+u) = j | H(u), X(u) = i} = P(x(t+u) = j | x(u) = i)
Where it was given that the remaining time to failre of any cable operating at time (t) is independent of he history of the process, conditional on X(t). Thus, our markov process satisfies the markov property.
b-) The generator matrix (Q) has these elements:
qij = {(ri)*(pij) if i not equal to j ; or it is -ri if i = j
and since it is given that the instantaneous failure rate of a single cable which carries M tons is αM, where α > 0 is a constant, then we have r0 = αM
Now, could you please correct any of my mistakes?
Also, I don't know to continue from here? How to derive the values of pij and r1, etc. for Q matrix?
Thank you very much in advance for your help.
