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

A manufacturing facility has a repair shop with two repairmen
who repair failed machines on a first-fail-first-serve basis. They
work together on the machine if there is one machine down (the
repair still takes the same amount of time), and otherwise, each
works on a separate machine. Thus, if there are more than two
machines down, new failures simply wait for their turn to be
repaired. Assume that machine failures arrive in a combined failure
stream, so that we do...

Problem 2
There are three machines and two mechanics in a factory. The
break time of each machine is exponentially distributed with λ = 1
(per day). The repair time of a broken machine is also
exponentially distributed with a mean of 3 hours. (Mechanics work
separately).
(1). Construct the rate diagram for this queueing system. (be
careful about the arrival rate λn)
(2). Set up the rate balance equations, then solve for pn’s.
(3). Compute L.
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please post solutions using R
a manufacturing facility has a repair shop with two repairmen
who repair failed machines on a first-fail-first-serve basis. They
work together on the machine if there is one machine down (the
repair still takes the same amount of time), and otherwise, each
works on a separate machine. Thus, if there are more than two
machines down, new failures simply wait for their turn to be
repaired. Assume that machine failures arrive in a combined failure...

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