A job circling five servers

Question

Five servers are arranged in a ring, numbered $0$ through $4$ . A job at server $i$ moves each second to server $i+1$ (next one clockwise) with probability $0.7$ , or to server $i-1$ (counterclockwise) with probability $0.3$ , wrapping around the ring.

In the long run, what fraction of time does the job sit on any one chosen server?

Accepted Answer

The forward and backward probabilities are unequal, which might suggest the busy direction gets favored. But look at the arrivals into a single server j. It receives the job from server j-1 moving forward (probability 0.7) and from server j+1 moving backward (probability 0.3). The total chance of arriving at server j is 0.7 + 0.3 = 1, the same for every server. Rows sum to 1 (the job always moves) and columns sum to 1 too, so the chain is doubly stochastic, which forces an even long-run split. With five servers, \pi = 	frac15 	ext{ per server}. So the job spends one fifth of its time on any…

A job circling five servers

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