Probability with Poisson PMF

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 λ; however, the maximum number of packets that can be stored is a 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). Hence, c is a given integer with 0 < c < b.

(a) Assuming that at the beginning of the first slot the buffer is empty, find the PMF(probability mass function) of the number of packets stored at the end of the first slot and at the end of the second slot. hints: In the first slot, the total number of packets is less than or equal to b and in the second slot, the

total number of slots is X − min{X, c}.

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