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

Abstract

Consider a broadcast channel with $n$ users, where different users receive different messages, and suppose that each user has to receive $m$ packets. A quantity 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 of channel uses required to guarantee that all users will receive $m$ packets. For the case of a \emph{homogeneous} network, where in each channel use the transmitter chooses a user at random, i.e. with probability $1/n$, and sends him$/$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 the delay $D_{m,n}$, is to determine its behavior as $n$ and $m$ grow large. Regarding this problem, Sharif and Hassibi \cite{Sh}, \cite{S-H} managed to calculated 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, raised the question of the determination of the asymptotic profile of $D_{m,n}$ for large $m$ 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) by Ivchenko \cite{I1} and \cite{I2}. In this article we determine the asymptotics of the moments of $D_{m,n}$ for large $m$ and $n$. We also derive its limiting distribution in the "supercritical case" where $m$ grows faster than $\ln n$ and in the "critical case" $m \sim \beta \ln n$, by an approach which is different from the one used by Ivchenko.