How long until the machine restocks?

Question

A vending machine is either stocked or empty at the end of each day, and tomorrow depends only on today: - If stocked today, it becomes empty tomorrow with probability 0.4 (and stays stocked with probability 0.6). - If empty today, it becomes stocked tomorrow with probability 0.8 (and stays empty with probability 0.2). The machine is stocked today. On average, how many days until it is stocked again (its expected return time to the stocked state)?

Accepted Answer

There is a clean fact for any Markov chain: the average time to return to a state equals 1 divided by that state's long-run share. Intuitively, if a state occupies a fraction \pi of all days, then on average one visit happens every 1/\pi days. Step 1: find the long-run share of stocked. For a two-state chain, the share of a state is \frac{b}{a+b}, where a is the probability of leaving it and b the probability of arriving. Leaving stocked has probability a = 0.4; arriving (from empty) has probability b = 0.8: \pi{	ext{stocked}} = \frac{0.8}{0.4 + 0.8} = \frac{0.8}{1.2} = \frac{2}{3}. Step 2: …

How long until the machine restocks?

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