Question

People arrive according to a Poisson process with rate λ, with each person independently being equally likely to be either a man or a woman. If a woman (man) arrives when there is at least one man (woman) waiting, then the woman (man) departs with one of the waiting men (women). If there is no member of the opposite sex waiting upon a person’s arrival, then that person waits. Let X(t) denote the number waiting at time t. Argue that E[X(t)] ≈ .78√2λt when t is large.

Answer 1

total probability for x

= sum(m= 0 to infinity)

(2m+x)Combin(m) * (0.5) ^ (2m+x) * (lambda*t)^(2m+x) * exp(-lambda *t )/ (2m+x)!

First part is the probability of having m women and m men in total of (2m+x) people

Second part is the regular poisson probability

Sum over m = 0 to infinity should give the probability for having x people standing at time t (may have to multiply by factor of 2 to account for men or woman at the end).

Then get the expected value by taking expectation of this for x ranging from 0 to infinity. This should give the required value.