Bonafide certificate

3.4 NODE MODEL

From the above model the steady state probability of channel being idle is obtained. In order to obtain the throughput performance of node, the states of a node needs to be modeled. The node is modeled as a 3 state Markov chain again as in Wang and Aceves (2004). Our intention is to use this model which was used for CSMA/CA based networks to the IEEE 802.11 standard based networks.

Fig 7 Node moled for IEEE 802.11 MAC

The model has three states.

The wait state is a state when the node defers for other nodes or backs off, succeed is the state when the node can complete a successful transmission and fail is the state when the node initiates an unsuccessful handshake. We define the length of succeed and fail states as

T_succeed = T_long.

= l_rts + l_cts + l_data + l_ack + 3τ.

T_fail = T_short2.

= l_rts + l_cts + 2τ.

The duration of the node in wait state is τ, in this model. This is a simple approach and the work evaluates how good can it be modeled using a single state with a fixed time.

While there are many transition probabilities involved, it is sufficient for our purpose if the transition probability P_ws, from wait state to succeed state is determined. express P_ws can be expressed as follows,

P_ws = P₁. P₂. P₃.P_4. ------ (23)
P₁ = Probability that the node transmits = p’ .

P₂ = Probability that the destination of this node’s packet does not transmit in the same slot = (1-p’)^.

P₃ = Probability that none of the other neighs of the node transmits = (1-p’).

P₄ = P that none of the neighs of y transmit for vulnerable period = (1-p’)^{(2lrts + 2τ)} .

P_ws = 2.p’.(1-p’)².(1-p’)^(2lrts+2τ). ------ (24)
The transition probability from wait state to succeed state, P_ws is found out considering the transmission from this node to every other node. Since in a string network, a node sends to any of its two neighbors, we take twice the transition probability for each of its neighbors.
Similarly the probability that a node x continues to stay in the same state, P_ww is just the probability that no node including itself transmits in the same slot.

P_ww = (1-p’)³. ------ (25)

Let , , denote the steady state probability of state succeed, wait, and fail respectively. The transition probabilities from any state to wait state, P_sw, P_fw is 1 since before attempting to send a next packet, each node has to wait for DIFS time. From the markov chain we have,

P_ww ++= .

P_ww +1 - = .

=

= . ------ (26)

The steady state probability of state succeed, , can be calculated as :

= P_ws = = p_s. ------ (27)

This steady state probability is the previously unknown quantity p_s. With this knowledge of p_s, we can obtain p’ from equation for p’[].

From the node markov model we can also obtain the throughput of a node as follows,
Th = .

= .

= ------ (28)
We can use similar methods to derive the throughput for grid topologies. We give the transition probabilities for the channel model as follows.
P_ii = (1 - p´)⁴. ------ (29)
P_il = 4p_s(1- p´)³. ------ (30)
P_is1 = 1 – [(1 - p´)⁴ - 4p_s(1- p´)].

p´ ² [ 3p’² – 8p´ + 6]. ------ (31)

P_is2 = 1 - P_ii - P_il – P_is1.

4(1 - p´)³(p´ - p_s). ------ (32)

.

=.

p´ = p. _.

= p . [  + 4* p_s *(1 - p´)³T_long + p´ ²(3 p´² – 8 p´ +6)T_short1 + 4(1- p´)³(p´- p_s) T_short2 ].

------ (33)

And using the Markov model for node as in fig, the successful transmission probabilities and node throughput is derived for grid network.

P_ws = P₁. P₂. P₃.P_4.

P_ws = 4 p´(1 - p´)⁴(1- p´)^3(2.l_rts ^+τ). ----- (34)

P_ww = (1 - p´)⁵. ------- (35)

=

= . ------ (36)
= . P_ws. = = p_s. ------- (37)

p_s = .

Th =

= ----- (38).

The expressions obtained for string and grid topologies are evaluated in chapter 4.