- #1
Kakashi
- 27
- 1
- Homework Statement
- A packet communication system consists of a buffer that stores packets generated by a source and a communication line that retrieves packets from the buffer and transmits them to a receiver. The system operates in pairs of time slots.During the first slot, the source generates packets according to a Poisson distribution with parameter λ. The buffer has a finite capacity of
b packets. If packets arrive when the buffer is full, the excess packets are discarded.
During the second slot, the system transmits packets from the buffer. It transmits either all packets currently stored or c packets, whichever is smaller, where 0<c<b.
(a) Assuming that the buffer is empty at the beginning of the first slot, find the probability mass function (PMF) of the number of packets stored
(i) at the end of the first slot, and
(ii) at the end of the second slot.
(b) What is the probability that some packets are discarded during the first slot?
- Relevant Equations
- Poisson Distribution
a) If the buffer starts empty the number of packets stored can be any integer $$ 0 \leq k_{1} \leq b $$. At the end of the second slot the system transmits either $$ k_{1} $$ or c depending which is less. If $$ k_{1}\geq c $$ the number of packets remaining in the buffer is $$ k_{1}-c $$ and if $$ k \leq c $$ the number of packets remaining is 0.
$$ P(X=k_{1} \epsilon \{0,1,2,..,b-1\})=\sum_{k=0}^{b-1} e^{-\lambda} \frac{\lambda^{k}}{k!} $$
$$ P(X=k_{1}=b)=1-\sum_{k=0}^{b-1} e^{-\lambda} \frac{\lambda^{k}}{k!} $$
The probability the buffer is empty after transmission
$$P(X=k_{2}=0)=\sum_{k=0}^{c} e^{-\lambda} \frac{\lambda^{k}}{k!} $$
The probability the buffer is not empty after transmission
$$P(X=k_{2}>0)=1-\sum_{k=0}^{c} e^{-\lambda} \frac{\lambda^{k}}{k!} $$
How can packets be discarded when in the first slot the buffer was empty?