  Consider an M/M/$s$ queue with the additional feature that the arrival rate
is a random variable of which only the mean, variance, and range are known.
Using semi-infinite linear programming and duality theory for moment problems,
we establish for this setting tight bounds for the expected waiting time. These
bounds correspond to an arrival rate that takes only two values. The proofs
crucially depend on the fact that the expected waiting time, as function of the
arrival rate, has a convex derivative. We apply the novel tight bounds to a
rational queueing model, where arriving individuals decide to join or balk
based on expected utility and only have partial knowledge about the market
size.
