Problem Consider the following communications system. M

Problem
Consider the following communications system. Messages arrive in accordance with a geometric point process at a transmission node that is served by a single transmission channel. Messages are queued in the nodal buffer until they reach the head of the queue at which time the message is transmitted across the single outgoing
channel at a transmission rate of R bits/sec . The message bit-length is geometrically distributed with a mean of 500 bits/message . The slot length is equal to 0.01 seconds and the means message inter-arrival time is equal to 4 slots.
(a) Prove that the number of slots required to transmit a message is also geometrically distributed. Determine the mean service time in units of slots and also calculate the service rate in units of [messages/slot ].
Partial Solution:
Firstly you need to know how many number of bits (denoted as C) can be transmitted during a slot. If we assume t to be a time slot (i.e. t = 0.01) then we obtain C = floor(Rt). Thus the number of slots required to transmit a message is
S = ceiling(L/C) where L denotes the number of bits per message and follow geometric distribution with a mean of 500 bits. For convenience you can denote P(L=k) = r(1-r)^(k-1) where r = 1/500. Then
P(S = n) = P(ceiling(L/C) = n) = P(n-1