Suppose we have with where
We can start by finding the Fourier transform of :
Now factor out and find the :
We recognize that the signal in time domain is
Thus we know that the output has convolved with the input (product in frequency domain)
What is the expected value of ?
Since , then
Since the second term is a geometric series.
We see that as time increases, the mean of is changing, thus is not a stationary process.
Packets arrive at probability ; packets depart at probability . The buffer can hold up to packets. Let be the number of packets in the buffer at time .
Show that the system can be modelled by Markov Chain:
Since at any time we don’t care about the number of packets in the buffer at time (history) if we already have all the information . In particular,
The conditional stuff (after that symbol) in the probability is useless information as far as the buffer is concerned.
There are total of states in the Markov chain: state
For state 0: there are two possible states to go to:
For state : there are also only two possible states to go
For any state in between (state 1 to state ) there are three possible outcomes:
Therefore, the transition matrix is:
To find the stationary distribution, we use the fact that
Do the matrix multiplication and we obtain equations for variables: .
The equations are:
Then we substitute every equation in terms of , and for general , we find the pattern:
Then find by setting an initial condition.
The variance for two normal random variables added together is the sum of two variances. (Proof later)