Question

A job shop consists of three machines and two repair-persons. The amount of time a machine operates before breaking down is exponentially distributed with mean 8 hours. The amount of time it takes to fix a machine is exponential with mean 5 hours (the repair-persons never work together on the same machine). The number of functional machines can be modeled as birth and death process. ⦁ The birth rates are λ0 = λ1 = _____, λ2 = _____, λ3 = λ4 = … = 0.

⦁ The death rates are μ1 = _____, μ2 = _____, μ3 = _____. Note: μ0 = 0.

⦁ Find the limiting probabilities P0, P1, P2, P3.

⦁ On average, how many machines are functional?

⦁ What proportion of the time are both repair-persons busy?

Answer 1

Answer:

Given,

0 = 3/8

1 = 2/8

2 = 1/8

i = 0 , i>=3

1 = 1/5

2 = 2/5

3 = 2/5

Now consider,

p1 = o/1 * po

= 3/8 / 1/8 po

= 3/8 * 8/1

= 3 po

p2 = 1/2 * p1

= 2/8 / 2/5 p1

= 0.625p1

= 0.625*3po

p2 = 1.875 po

p3 =  2/3 * p2

= 1/8 / 2/5 p2

= 0.3125*1.875po

p3 = 0.586 po

Now consider,

po = 1

po = (1 + 3 + 1.875 + 0.586)^-1

= 0.1548

a)

p1 + 2p2 + 3p3 = 3 + 1.875*2 + 0.586*3

= 8.508

b)

Proportion of time are both repair-persons busy

= p2 + p3

= 1.875 + 0.586

= 2.461