CHAPTER 4 Mathematic Model
4.1. Markov Chain
We observed the behavior of distributed scheduling in the mesh mode in previous chapter.
We found this behavior does not depend on all of past history. In other words, it is a“Time Homogeneous” and suitable for being modeled by stochastic process. The delay time in the period of MSH-DSCH transaction is what we are interested in this thesis, which the Markov Chain is easy using to observe it. Depends on Leonard Kleinrock’s description in his book
“QUEUEING SYSTEMS VOLUME I: THEORY”, Markov processes may be used to describe the motion of a particle in some space. We consider discrete-time Markov chains, which permit the particle to occupy discrete positions and permit transitions between these positions to take place only at discrete times.[11]
Assume Xn is denoted as a state in our consequent Markov Chain model that a node stays at a certain time to transmit MSH-DSCH. Time unit is an opportunity. A set of random
variable {Xn}forms a Markov chain if the probability that the next state is Xn+1 depends only upon the current state Xn and not upon any previous stations. Base on our analysis in previous chapter, the next state merely depends on the current competing result, neither on the last nor on all of past history. Thus we have a random sequence in which the
dependency extends backwards one unit in time. If this node’s Temp Xmt Time overlaps with its neighbors, it implies the competing is occurred with them. If it wins or there is no competition, it will set this Temp Xmt Time as its Next Xmt Time. If it loses, it will back one opportunity to run this behavior again until it wins. In order to simplify the notification, we assume integer 1,2,3 … represent each of certain state Xn , the physical concept of our proposed Markov Chain are depicted as Figure 4.1.
Figure 4.1: Each state corresponds to the Next Xmt Time-1
With this concept of Figure 4.1, we can model this behavior with a vertical chain as Figure 4.2. The states and transition definitions are defined as Table 4.1. From state 1 to state 2 implies the time duration of one Next Xmt Time. Suppose we have N nodes x totally, the probability which a node wins N-1 nodes can be expressed by formula (5).
Oppositely, the probability of a node loses them can be expressed by formula (6).
(5)
(6)
Table 4.1: The Notation definitions in the Markov Chain Notation Description
Integers in the state
The state probability that the transmission time backs to certain opportunity
Prob The transition probability to indicate the probability that the node wins.
X Exponent of Xmt Holdoff Time
N The number of nodes
For example, if our observed node loses, it transfers from state 1 to state 2, the transition probability is 1-ProbN-1. If it wins, it stays at state 1, the transition probability isProbN-1.
1
-ProbN
Win=
1
-ProbN
-1 Lose =
Figure 4.2: One vertical chain
In order to model it easily, we assume that as long as the node lose this competition, it does not back one opportunity. It has to back a length of Next Xmt Time. That’s why the transition probabilities during the inter-states are always 1 in Figure 4.2. Thus, the state transfers to the second vertical chain are shown as Figure 4.4. If this node loses again, it will back a length of Next Xmt Time again to the third vertical chain …etc.
Figure 4.3: Each state corresponds to the Next Xmt Time-2
Figure 4.4: Two vertical chains
At last, a Markov chain is organized as Figure 4.5.
Figure 4.5: The Markov Chain
4.2. Mathematical Evaluation
The Markov Chain we proposed presents the variations of state transitions. We hope to induce an equation to evaluate the average delay time. The dependency of delay time relates to a probability of win. Before evaluating the delay time, we have to induce this probability formula initially. Then the expected value, the average delay time we target, is the product
of probability and time.
4.2.1. Probability Theory and Assumptions
To begin with, we assume the number of nodes is N. Each observation of a node competes with neighbors is independent and represents one of two outcomes "competing" or
"non-competing". So by using (7), we can get the competing probabilityPcompeting. This is the binomial distribution we know. The P in the (7) is a probability that one node competes c with one another node. More details can be retrieved from chapter 3.4 and Figure 3.6.
competing
P is different from P in our assumptions. c P is the condition happened between c
one node and one node in a very short time. Nevertheless, Pcompeting is the condition while at least one of the following events is happened: between one node and one node, or
between one node and two nodes, or between one node and more another nodes. So formula (7) means one of the following situations is occurred: observed node competes with one neighbor, or observed node competes with two neighbors, or observed node competes with three neighbors …etc.
Table 4.2: Notations of equations Notation Description
Prob Probability of (competing∩ win)
competing
P Probability of competing, at least one of more events happens
P c Probability of competing between one node to one of another node.
N Number of nodes
(7)
Moreover, the win probability should be an inverse proportion of the number of competing nodes. So we get (8) from (7).
(8)
In opposition, the losing probability can be derived as (9).
(9)
4.2.2. Delay Time
By the observation of Markov Chain (Figure 4.5), if the node wins at state 1, the transition probability of win can be expressed by using (10). If the node wins at the state2 that is the x
...
end of first vertical chain, the probability of win can be expressed by using (11). If the node wins at the state 2⋅ that is the end of second vertical chain, the probability of win can be 2x expressed by using (12)…etc.
(10)
(11)
(12)
This probability distribution gives the trial number of the first success, so it is a geometric distribution. Substitute (8) and (9) into (10), (11) and (12), we can derive the probability (13), (14) and (15).
1
-Prob
N2 -N 1
-N
) Prob
Prob
-(1 ⋅
3 -N 2
-N 1
-N
) (1 - Prob ) Prob
Prob
-(1 ⋅ ⋅
(13)
(14)
(15)
So far, each probability on the corresponding vertical chain has been derived. The expected value can be calculated by the summation of these probabilities and multiplied by time, Then formula (16) can be obtained. The unit of time in this formula is opportunity.
)
....
Finally, we generalize our equation, as (17). In conclusion, the input parameters are N and x. It means the delay time is affected by the number of nodes and holdoff exponent.
(17)
4.2.3. The Success Probability of MSH-DSCH Transmission
Except for delay time, we hope to know what the mean value of wining probability is that a node transmits MSH-DSCH. We know Markov Chain is more suitable to get the average
∑∏ ∑ ∑
probability of each state. If
π
(k) is the probability of certain state at certain time k in our proposed Markov Chain,π
(k-1) is its probability of certain state at the time before k. P is the transition probability from the state of probabilityπ
(k-1) to the state ofprobability
π
(k)(Figure 4.6). Formula (18) and (19) are applied to evaluate the wining probability that a node transmit MSH-DSCH. These two formulas imply a recursivefunction and converge at
ξ
denoted as a convergence value.π = { π
1, π
2, π
3...}
is avector and each element
π
1, π
2, π
3,...
within the vector is denoted as theprobabilities of corresponding state 1, 2, 3….in Figure 4.5.
P
is a two dimension matrixwhich the size equals to
2
x⋅ (N - 1) × 2
x⋅ (N - 1)
. For example x=2, N=5, the matrixP
will be shown as equation (20).
P P
Figure 4.6: The transition probability of certain state
(18)
(19)
(20)
In order to simplify the expressions, P1 and 1-P1 within the matrix
P
stand for the formula (5) and (6). With this same rule, we simplify other expressions as P2, 1-P2, P3, 1-P3… Then P(success) can be calculated by (21).(21)
ξ π
π
(k)-
(k-1)<
win
Prob
WinP(success) = ∑ ⋅ π
CHAPTER 5 Simulation Results
In this chapter, we validate our model. The transmission behavior simulation base on the Figure 3.7and Figure 3.8 was implemented by the C code. The mathematic evaluation base on our proposed schemes throughout the CHAPTER 4 was computed by the MATLAB 7.0.
There are two major items that we will evaluate, which are the delay time and the success probability of MSH-DSCH transmission.
5.1. Delay Time
The formula (17) is our proposed scheme to evaluate the delay time of one certain node transmitting its scheduling information MSH-DSCH. The MATLAB 7.0 is applied to calculate this complex operation in our numeric validations. Following parameters are applied:
Exponent = 2
Node ID: random number between 1~4095
Probability: Pc= 0.5
And the result is shown as Figure 5.1. The “sim” denotes a curve by simulation;
“math” denotes a curve by mathematics. With this figure, it shows our mathematical model approaches the simulation result. By the way, the error rate is analyzed by the statistic method, as Figure 5.2, presents the difference in distance between the method by behavior simulation and by our proposed mathematic formula. The error is under 10% while the nodes of number between 2 to 20. Except for the exponent x=2, we are also interested in x=3 and x=4. These simulation results are shown from Figure 5.3 to Figure 5.6. The errors are under 10% throughout the above simulations. The stable accuracy is performed all over these simulation results; even the different exponents are applied.
2~20 Nodes, x=2
0 20 40 60 80 100 120
2 4 6 8 10 12 14 16 18 20
Number of nodes
opportunities(time slot)
sim math
Figure 5.1: The delay time of opportunities between simulation and mathematic model-1
2~20 Nodes Error Rate
0 10 20 30 40 50 60 70 80 90 100
2 4 6 8 10 12 14 16 18 20
Number of nodes
Error rate (100%)
x=2
Figure 5.2: The error rate between simulation and mathematic model-1
2~20 Nodes, x=3
0 50 100 150 200 250
2 4 6 8 10 12 14 16 18 20
Number of nodes
opportunities(time slot)
sim math
Figure 5.3: The delay time of opportunities between simulation and mathematic model-2
2~20 Nodes Error Rate
0 10 20 30 40 50 60 70 80 90 100
2 4 6 8 10 12 14 16 18 20
Number of nodes
Error rate (100%)
x=3
Figure 5.4: The delay time of opportunities between simulation and mathematic model-2
2~20 Nodes, x=4
Number of nodes
opportunities(time slot)
sim math
Figure 5.5: The delay time of opportunities between simulation and mathematic model-3
2~20 Nodes Error Rate
0
Number of nodes
Error rate (100%)
x=4
Figure 5.6: The delay time of opportunities between simulation and mathematic model-3
Besides, as shown in the Table 5.1 and Figure 5.7, there is an error rate comparison
between the [9] and proposed evaluation. In order to compare with original analysis, Table 5.1 follows the original table format in the [9], that’s why the exponent x is list here without regular order.
Table 5.1: Comparison between original and proposed evaluation Numbers of Nodes X Original (100%) Proposed (100%)
2 2 2.47 0.75
3 3 3.12 1.97
4 3 2.81 1.39
5 1 0.85 0.36
6 0 0.36 0.38
7 2 2.28 2.2
8 2 3.85 0.9
9 1 0.86 0.65
10 0 0.42 0.1
Compare between Original and Proposed Evaluation
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
2 3 4 5 6 7 8 9 10
Number of nodes
Error rate (100%)
original proposed
Figure 5.7: Comparison between original and proposed evaluation
The Figure 5.7 shows that the values predicted by our model have a smaller degree of error than the values generated by [9]’s model do.
5.2. The Success Probability of MSH-DSCH Transmission
The formula (18) is a recursive function, so the initial value should be assumed for the recursive calculation. The initial value of π is assumed as follows,
...) 0, 0, 0,
(0)
= (1, π
If we are interested in the exponent x=2, the probability of success is evaluated as Figure
5.8. The inverse ratio depicted in this figure shows that as the number of nodes increases, the probability of success decreases.
Success Probability of MSH-DSCH transmission
0 0.1 0.2 0.3 0.4 0.5 0.6
2 3 4 5 6 7 8 9 10
Number of Nodes
Probability
x=2
Figure 5.8: The mean of success probability that a node transmit MSH-DSCH
The Figure 5.9 shows the probabilities of success which three different exponents are compared. This figure shows that as the number of exponent increases, the probability decreases. So there is a concept that the small exponent can speed-up the MSH-DSCH transmission. This is useful in the future as a mechanism of QoS or call admission control.
For example, a node which has the small exponent may have the more probabilities to transmit its scheduling information MSH-DSCH. The time for transmitting the MSH-DSCH
may image as a call setup time at beginning of a link connection. Thus the higher
probability for transmitting scheduling information MSH-DSCH implies the higher chance or priority it will be to initialize a connection.
Success Probability of MSH-DSCH transmission
0 0.1 0.2 0.3 0.4 0.5 0.6
2 3 4 5
Number of Nodes
Probability x=2
x=3 x=4
Figure 5.9: The mean of success probability when x is non-identical
CHAPTER 6
Conclusions and Future Works
In this thesis, we have proposed a Markov Chain model which can be used to simulate MSH-DSCH transmission behavior in 802.16 mesh mode. This model considers the competing probability and back behavior of transmitting MSH-DSCH. It also helps us to realize the competing behavior more clearly. In the future, there will be more possibilities to design the WiMax mesh mode based on this model.
Based on this model, we derived a formula to evaluate an average delay time of MSH-DSCH transmission. Furthermore, this delay time may impact the starting time of a link connection. Thus the higher probability for transmitting scheduling information
MSH-DSCH implies the higher chance or priority it will be to initialize a connection. More important, the processing time of the following three-way handshaking is also influenced by MSH-DSCH transmission delay. By this model, we separate out the factors that affect the delay time. These factors are possibly useful for future researches.
Our scheme also evaluates the success probability of MSH-DSCH transmission. That is useful for QoS negotiation and adaptation. A conclusion is obtained that the success probability is inversely proportionate to the number of nodes. We may get a threshold to guarantee the connection is more stable by applying this probability in the future.
Finally, we have a simulation. It appears that results calculated from our mathematic model closely resemble the results from simulation. In other words, the theoretical model fits the experimental data well.
References
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