1) Consider a machine shop that has two machines which are independent, identica

Question

1) Consider a machine shop that has two machines which are independent, identical. Each machine fails after an amount of time that is Expo(H) distributed: if it is down it is repaired after an Expo(a) amount of time. There is only one repairman and the machines are repaired in the order in which they fail, where the repair times are iid. Let X (t) be the number of machines at time t. a) Model (X(t), t 2 0) as a CTMC, and give its generator matrix. b) Suppose that the mean life time of a machine is 10 days, while the mean repair time is one day. Determine the long-run equilibrium distribution of the CTMC. For what fraction of time are both machines working?

Explanation / Answer

a)Now first of all at time "t" there can be 3 possibiities.Either there can be 2 machines presnt ,1 machine present or no machine present.So the possible states od the markov chain X(t) are 0,1,2.

Also the point to be noted is that only One machine can be repaired at a single time as there is only 1 repairman.

so PROB(of going from 0 to 2 machines)=0

PROB(2 to 0 machines)=0

PROB(of going from 0 to 1 machine) means that 1 machine is being repaired

and if one 1 machine repairs at rate "l    " so the probability of going from 0 to 1 machine is “(l)/(m+l)”

similarly PROB(1 to 0 machine) means that 1 machine has failed and the rate of failure of machine is “m”   so the PROB(1-0)=m/m+l

The possible transitions from state 0 is to state 1 only with prob=(l)/(m+l)”

Prob (0-2)=0

So prob(0-0)= -(l)/(m+l) (as the sum of rows of a generator matrix =0)

So the continuous time markov chain is ,

And the generator matrix is

states

0

1

2

0

-(l)/(m+l)

(l)/(m+l)

0

1

m/m+l

-(m+l)/(m+l)=-1

(l)/(m+l)

2

0

m/m+l

-m/m+l

where m represent "mu" and

l represents 'lambda"

states

0

1

2

0

-(l)/(m+l)

(l)/(m+l)

0

1

m/m+l

-(m+l)/(m+l)=-1

(l)/(m+l)

2

0

m/m+l

-m/m+l

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