The length of a queue outside a shop changes by customer each minute
PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Probability, statistics and stochastic methods · Probability, statistics and stochastic methods
Many Parts, One Behaviour, Ch Heavy tails: power laws and how to measure them
Problem
The length of a queue outside a shop changes by ±1 customer each minute, up or down with equal probability 21, independently of the past. Treat the length as an unrestricted symmetric random walk on the integers, ignoring that a real queue cannot go negative, so that the standard first-return analysis applies. At time 0 the queue length equals its long-run average value L0. (a) Starting from u2n=(n2n)4−n (the probability of being back at L0 after 2n minutes) and f2n=u2n/(2n−1) (the probability of first return exactly at minute 2n), use Stirling's approximation n!≈2πn(n/e)n to derive u2n≈(πn)−1/2 and hence f2n≈(2π)−1n−3/2. (b) Evaluate f2n exactly for n=8 (first return after 16 minutes), compare with the large-n approximation from part (a), and state the percentage difference. (c) State whether the mean time for the queue length to first return to L0 is finite, with a one-sentence reason based on the tail found in part (a).