People arrive according to a Poisson process with rate , with each person independently being equally likely

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)] ≈ 0.78

√

2λt when t is large.

Hint: If Z is a standard normal random variable, then E[|Z|] =

√

2/π ≈

0.78.