i) Car repair shop. Repairs are carried sequentially : T(tune up), A(air conditi

Question

i) Car repair shop. Repairs are carried sequentially : T(tune up), A(air conditioning), B(breaks) with the mean duration time of 1,2,3 hours respectively. Asuume the repair times Ri i=1,2,3 are independent and exponentially distributed. what is the probability that 5 hours later the car is in the break repair stage? That is, find P(X(5)=3)

where W=waiting time to exit state 4(the entries of infinitesimal rates matrix A are reciprocal of the corresponding mean values of exponential distributions)

Hint ; Observe that P(x(5)=3)= P(R1+R2 +R3 > 5, R1 +R2 <5) . i.e one needs to find the distribution of R1+R2 +R3

Explanation / Answer

Let m(i) be the mean of R(i) . then R(i) is exponential distributed. Now

P(R(1)+R(2)+R(3)>5, R(1)+R(2) < 5)

= P( 5-R(3)< R(1)+R(2) < 5 ,R(3) > 0)

=P((5-R(3)-R(2)) < R(1) < 5-R(2), R(2) > 0, R(3)> 0)

Now the joint distribution of (R(1),R(2),R(3) ) is

f(r1,r2,r3) = exp(-(r1+r2/2 + r3/3))/1*2*3 as they are independent.

So the required probabaility in the integration of

f (r1, r2,r3) over R(1) varying in (5-R(3)-R(2)) < R(1) < 5-R(2) and R(2) and R(3) lies in the entire postive real number. Carry out the integration and that will be the answer.

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