$(77$ points$)$ A communication channel of capacity $C$ Mbps $($megabits per second$)$ is shared by a group of $N$ users. Here, the channel capacity is the rate at which information can be transmitted over the channel. Each user, User $i,$ either transmits at a rate of $T_{i}Mbps$ or becomes idle with no transmission, where $i=1,2,3,$cdots,$N.$ Unless stated otherwise, the transmission activity of a user is independent with that of another. In a certain time period, the probability that User i transmits is $p_i,$ where $AS0.$ The channel $is$ saturated when the total transmission rate $of$ all users equals $to$ the channel capacity $or$ higher.

Let A and $Sbe$ the number $of$ users who are transmitting concurrently and the total transmission rate $of$ all users, respectively. There $isno$ overhead $in$ sharing the channel.