We consider the maximum queue length and the maximum number of idle servers in the classical Erlang delay model and the generalization allowing customer abandonment–the M/M/n+M queue. We use strong approximations to show, under regularity conditions, that properly scaled versions of the maximum queue length and maximum number of idle servers over subintervals [0,t] in the delay models converge jointly to independent random variables with the Gumbel extreme value distribution in the quality-and-efficiency-driven (QED) and ED many-server heavy-traffic limiting regimes as n and t increase to infinity together appropriately; we require that t
n
→∞ and t
n
=o(n
1/2-ϵ
) as n→∞ for some ϵ>0.
