A single transmit antenna broadcast channel (BC) with large (n) number of users in a Rayleigh fading environment is considered. It is assumed each user either receives the minimum rate constraint of R_min or remains silent. Recently, it is shown that by applying a proper power allocation strategy, the maximum number of active users in such channel scales as $\\frac{\\log(\\log(n))}{R_{min}}$ in the asymptotic case of n → ∞. The main contribution of this paper is to derive the expected delay of the optimum power allocation strategy in terms of fairness maximization. Accordingly, it is shown the expected delay of this strategy behaves like $R_{min}\\frac{{n}\\log({n})}{\\log(\\log(n))}+\\omega\\big(\\frac{n}{\\log(\\log(n))}\\big)$, where the expected delay is defined as the minimum number of channel uses which guarantees each user receives at least one packet.