$1.(16$ pts$)$ Consider $8$ stations that are attached to a shared bus. The stations generate packets of the same size, and the time axis is divided into slots equal to the packet transmission duration. The collision control protocol requires that when a station has a packet ready, it transmits in the immediate next time slot with a probability p$,$ where $0<$ p $<1\mathrm{.}0.$ If it does not transmit in that slot, it tries in the next slot with probability p$.$ This procedure continues until the packet is successfully transmitted. Determine the value of p that maximizes the rate at which packets are successfully transmitted. If the channel capacity is $30$ mbps$,$ what is the maximum achievable transfer rate$1?$

Hint: Given any time$-$slot z$,$ the probability that a ready station chooses not to transmit its packet in the slot z is $(1$ p$).$ You then find the probability that exactly one station transmits in slot z$.$