How often is the queue empty?

Question

A help desk holds up to 3 people (those waiting plus the one being served). Each minute at most one thing happens: someone arrives (if there is room) with probability 0.3, or the person in service finishes and leaves (if anyone is present) with probability 0.6. These chances are the same at every level. In the long run, what fraction of time is the queue empty (zero people)?

Accepted Answer

The count moves up or down by one, a birth-death chain, so detailed balance holds: upward flow from level i equals downward flow from level i+1, \pii 	imes (	ext{arrival chance}) = \pi{i+1} 	imes (	ext{service chance}). Because the arrival and service chances are the same at every level, each step multiplies by the same ratio r = \frac{0.3}{0.6} = 	frac12: \pi1 = 	frac12\,\pi0, \quad \pi2 = 	frac14\,\pi0, \quad \pi3 = 	frac18\,\pi0. The shares form a geometric pattern 1, 	frac12, 	frac14, 	frac18 times \pi0. Normalize so they add to 1: \pi0\,\big(1 + 	frac12 + 	frac14 + 	frac18\…

How often is the queue empty?

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