Only answer the question 3. The article \"Markov Chain Model for Performance Ana

Question

Only answer the question 3.

The article "Markov Chain Model for Performance Analysis of Transmitter Power Control in Wireless MAC Protocol" 21st International Conference on Advanced Networking and Applications, 2007) describes a Markov chain model for the state of the communication channel using a particular"slotted non-persistent (SNP) protocol. The channel's possible state are (1) idle, (2) successful transmission and (3) collision. For the particular values of the authors' proposed four parameter model, we get the following transition matrix 0.50 0.40 0.10 P 10.02 0.98 0 0.12 0 0.88 L. Vent mati f n irga in Determine the steady state probabilities for this channel. 3. What proportion of the time is the channel idle, in the long run 4. What is the average number of time steps between successive collisions? 2.

Explanation / Answer

3) Let the long term stationary probability distribution in the 3 states: idle, successfull transmission and collision here be X, Y and Z respectively.

Then, we get from the second column, that:

X = 0.4X + 0.98Y

0.6X = 0.98Y

Y = 0.6122X

Also, from the last column, we get:

Z = 0.1X + 0.88Z

0.12Z = 0.1X

Z = ( 0.1 / 0.12)X = 0.8333X

Also the sum of all 3 probabilities should be equal to 1. Therefore, we get here:

X + Y + Z = 1

X + 0.6122X + 0.8333X = 1

X = 0.4089

Therefore 0.4089 is the long term proportion of time that is the idle time.

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