Consider a scenario in which $\textbf{n}$ hosts share the same medium. The medium has bandwidth (max. data rate) $c$. Time is divided into slots of duration $T$ seconds ($\textit{i.e.} \left [ lT,(l+1)T \right ],l = \dots,-1,0,1,\dots $). Each host transmits a frame of size $cT$ bits at bit rate $c$ during any given time slot with probability $p$ independent of all other hosts. Such a frame transmission begins at the start of any slot. If only one frame is transmitted in a time-slot then we say the transmission was successful. If more than one frame is transmitted then we say that a “collision” has occurred and none of the transmissions in that slot were successful.
(a) Derive an expression for the expected number of successful transmission in a time-slot in terms of $n$ and $p$.
(b) For a fixed number of hosts $n$, what is the largest possible expected bit rate of successful transmissions (i.e. maximum throughput found by varying $p$)? Interpret your results.
(c) What is the limit of maximum throughput as $n \rightarrow \infty$ ? Interpret your results.
