Only answer the question 4. The article \"Markov Chain Model for Performance Ana

Question

Only answer the question 4.

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

Question 4:

Let the average number of time steps between consecutive collisions be Z. Now let the average number of times steps to go from idle state to collision state be X and that from successful transmission state to collision state be Y. Then, we have here:

From the third row, we get:

Z = 0.12(1 + X) + 0.88*0

Z = 0.12 + 0.12X

From first row, we get:

X = 0.5(1 + X) + 0.4(1 + Y) + 0.1*0

0.5X = 0.9 + 0.4Y

Now from the second row, we get:

Y = 0.02(1 + X) + 0.98(1 + Y)

0.02Y = 1 + 0.02X

Y = 50 + X

Put this in the equation, 0.5X = 0.9 + 0.4Y, we get:

0.5X = 0.9 + 0.4(50 + X)

0.5X = 0.9 + 20 + 0.4X

0.1X = 20.9

X = 209

Therefore, we get now,

Z = 0.12 + 0.12X = 0.12 + 0.12*209 = 25.2

Therefore 25.2 is the average number of time steps between successive collisions here.

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