Question

Homework Assignment 3.5 Summer 2018 Question 3: Continuous-time Markov Chains (a) A facility has three that are identical. Each machine fails independently with an exponential distribution with a rate of 1 every day; repairs on any machine are also exponentially distributed with a rate of 1 every 12 hours. Create a continuous-time Markov chain to model this (identify the rates, and the transition probabilities) (b) Now, assume the facility above has three machines, but one of them is of Type 1 and the other two are of Type 2. The machine of Type 1 is more critical and hence, it breaks down, the crew gives up on every other task to concentrate on fixing the machine of Type 1 first. Fail- ures are still exponentially distributed with a rate of 1 every day, and repairs are exponentially distributed with a rte of 1 every 12 hours. Create a continuous-time Markov chain to model this (identify the rates, and the transition probabilities). Answer

Answer 1

(a) It is given that the facility has three machines that are identical. At any given time the number of machine which fail can be either 0,1,2,or 3. So, this becomes our state space for the process. That is, the state spaces are defined as:

0 - None of the three machine have failed

1- One of the three machine have failed

2- Two of the three machine have failed

3- Three of the three machine have failed

It is given to us that the each machine fails independently with an exponential distribution with a rate of 1 everyday, that is, 1/24 per hour; and repairs of the machine are also exponentially distributed with a rate of 1 per 12 hours, that is 1/12 per hour.

So, the transition matrix is given as:

Assuming hourly transition rates

	0	1	2	3
0	-1/24	1/24	0	0
1	1/12	-(1/12+1/24) = -1/8	1/24	0
2	0	1/12	-(1/12+1/24) = -1/8	1/24
3	0	0	1/12	-1/12

(b) Now it is given that the facility has three machines where one of the machine is of type I and two of the other machines are of Type II. At any given time the number of machine which fail can be either 0, (Type I machine failure), 1 or 2 of the Type II machine failure.

So, this becomes our state space for the process. That is, the state spaces are defined as:

0 - None of the three machine have failed

Type 1 - The machine of type I has failed

1- One of the two machines of Type II have failed

2- Both of the two machines of Type II have failed

It is given to us that the each machine fails independently with an exponential distribution with a rate of 1 everyday, that is, 1/24 per hour; and repairs of the machine are also exponentially distributed with a rate of 1 per 12 hours, that is 1/12 per hour.

So, the transition matrix is given as:

Assuming hourly transition rates

	0	Type I	1	2
0	-(1/24+1/24) = -1/12	1/24	1/24	-
Type I	1/12	-1/12	0	0
1	1/12	1/24	-(1/12+1/24+1/24) = -1/6	1/24
2	0	1/24	1/12	-(1/24+1/12)