Problem $4($Packet Communication$)$: $(15$ points$)$ A packet communication system consists

of a buffer that stores packets from some source and a communication line that retrieves packets

from the buffer and transmits them to a receiver. The system operates in time$-$slot pairs. In the

first slot, the system stores a number of packets that are generated by the source according to a

Poisson PMF with parameter $\backslash$lambda ; however, the maximum number of packets that can be stored is

given integer b$,$ and packets arriving to a full buffer are discarded. In the second slot, the system

transmits either all the stored packets or c packets $($which ever is less$).$ Here, c is given integer with

$0<$ c $<$ b$.$

$($a$)$ Assuming that at the beginning of the first slot the buffer is empty, find the PMF of the

number of packets stored at the end of the first slot and at the end of the second slot.

$($b$)$ What is the probability that some packets get discarded during the first slot?

$1$