Answer :
The proportion of time that each server is idle will be:
Server 1 = 0.467
Server 2 = 0.2
Server 3 = 0.333
What proportion of time is each server idle?
Let the state be the idle server. The balance equations are:
Rate Leave = Rate Enter
(μ2 + μ3)P1 = μ1( P2 + P3)
(μ1 + μ3)P2 = μ2(P1+ P3)
(μ1 + μ2)P3 = μ3(P1+ P3)
P1 + P2 + P3 = 1
These are to be solved and the quantity Pi represents the proportion of time that server i is idle.
Server 1 will be:
= (μ1² + μ1μ2 + μ1μ3) / (μ1 + μ2 + μ3)
= (7² + 7.3 + 7.5) / (7+3+5)²
= 0.467
Server 2 will be:
μ2² + μ1μ2 + μ2μ3) / (μ1 + μ2 + μ3)
= 0.2
Server 3 will be:
μ3² + μ1μ3 + μ2μ3) / (μ1 + μ2 + μ3)
= 0.333
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