Long-run share of airport trips

Question

A taxi finishes each trip at one of two places, Downtown or the Airport, and where the next trip ends depends only on where the current one ended:

After a Downtown drop-off, the next trip ends at the Airport with probability $0.25$ (and Downtown with probability $0.75$ ).
After an Airport drop-off, the next trip ends Downtown with probability $0.5$ (and Airport with probability $0.5$ ).

Over a very long shift, what fraction of trips end at the Airport?

Accepted Answer

We want the stationary distribution, the steady long-run shares (\piD, \piA) that stop changing from one trip to the next. In steady state, the flow of probability out of "Airport" must equal the flow into it. Trips leaving the Airport happen at rate \piA 	imes 0.5 (Airport drop-offs that go Downtown next). Trips entering the Airport happen at rate \piD 	imes 0.25 (Downtown drop-offs that head to the Airport next). Balancing the two: \piD 	imes 0.25 = \piA 	imes 0.5 \implies \piD = 2\,\piA. Since the shares add to 1, substitute \piD = 2\piA: 2\piA + \piA = 1 \implies \piA = 	frac13. So i…

Long-run share of airport trips

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