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Consider a broadcast channel with $n$ users, where different users receivedifferent messages, and suppose that each user has to receive $m$ packets. Aquantity of interest here, introduced by Sharif and Hassibi (2006-7) \cite{Sh},\cite{S-H}, is the (\emph{packet}) \emph{delay} $D_{m,n}$, namely the number ofchannel uses required to guarantee that all users will receive $m$ packets. Forthe case of a \emph{homogeneous} network, where in each channel use thetransmitter chooses a user at random, i.e. with probability $1/n$, and sendshim$/$her a packet, the same quantity $D_{m,n}$ had already appeared in the\emph{coupon collector} context, in the works of Newman and Shepp (1960)\cite{N-S} and of Erd\H{o}s and R\'{e}nyi (1961) \cite{E-R}.  A problem of particular interest in wireless communications, related to thedelay $D_{m,n}$, is to determine its behavior as $n$ and $m$ grow large.Regarding this problem, Sharif and Hassibi \cite{Sh}, \cite{S-H} managed tocalculated the asymptotics of the mean value $\mathbb{E}[D_{m,n}]$, as $n \to\infty$, for the cases (a) $m = \ln n$ and (b) $m = (\ln n)^{\rho}$, $\rho >1$. It is remarkable that Erd\H{o}s and R\'{e}nyi \cite{E-R} had, also, raisedthe question of the determination of the asymptotic profile of $D_{m,n}$ forlarge $m$ and $n$ (in 1961). And in the 1970's the limiting distribution of$D_{m,n}$ for large $m$ and $n$ was determined (in great generality) byIvchenko \cite{I1} and \cite{I2}.  In this article we determine the asymptotics of the moments of $D_{m,n}$ forlarge $m$ and $n$. We also derive its limiting distribution in the"supercritical case" where $m$ grows faster than $\ln n$ and in the "criticalcase" $m \sim \beta \ln n$, by an approach which is different from the one usedby Ivchenko.

On an Old Question of Erd\H{o}s and R\'{e}nyi Arising in the Delay Analysis of Broadcast Channels