ELEC 321
# Review Session

Updated 2017-12-06

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 donâ€™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:

- There is a probability that we will stay in state 0
- probability to go to state 1

For state : there are also only two possible states to go

- probability that we stay in state
- is the probability that we go to state

For any state in between (state 1 to state ) there are three possible outcomes:

- probability of of going to a lower state; since we need one packet to be transmitted and no packets arrive. The probabilities are and respectively. Since packet being received by the buffer and transmitted by the buffer is independent, the probability that goes from some number to is .
- probability of of going to a higher state (same argument from above applies)
- probability of of staying in the same state. This occurs when (no packets are received no packets are transmitted) (packet arrives packet transmitted).

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)