Membership Function and Fuzzy Rule Base

Denote the trapezoid function by g( )⋅ which is expressed as

1

where parameters y₁ and y₄ represent two terminals of lower parallel side, and y2 and y₃ represent two terminals of upper parallel side.

Also denote the triangular function by f(．) which is expressed as

1

In WLAN system, we mainly focus on the HTTP non-real-time (NRT) users, so we will specifically discuss about the NRT HTTP users in the following contents. In [15], we can know the delay requirement for NRT users in WLAN is 1000ms. From

this saying, we will purchase best utility of AP in the limit of delay requirement in 1000ms for WLAN users. In other words, CR systems have to try their best to ease the effect on WLAN and let delay of HTTP users in WLAN can limit below 1000ms.

We assume the membership function is ( )μ ⋅ . For the input ρ, we denote ( ( 1))

L k

μ ρ − , ( (μ ρ_M k−1)), and μ ρ_H( (k−1)) the membership functions for the linguistic terms “L”, “M”, and “H”, respectively. They are given below and shown in Fig. 4.4(a):

μ ρ_L( (k−1)) = ( (g ρ k−1); 0, 0, 0.2, 0.3), (4.6)

μ ρ_M( (k−1)) ( (= g ρ k−1); 0.2, 0.3, 0.55, 0.7), (4.7)

μ ρ_H( (k−1)) ( (= g ρ k−1); 0.55, 0.7, 1, 1). (4.8)

The reason we choose those parameter in μ ρ( ) is according to the simulation result of WLAN collision probability and WLAN average packet delay in Fig. 4.5 and Fig. 4.6. We define the offered WLAN traffic intensity as the ratio of the total average arrival rate of all WLAN users over the system maximum transmission rate which is 11 Mbps in 802.11b. Fig. 4.6(a) shows the average WLAN packet delay with traffic intensity varies from 0.1 to 0.55 where Fig. 4.6(b) is with traffic intensity varies from 0.1 to 0.6. In Fig 4.5, that there is an apparent change at the traffic intensity around 0.55. We can regard it as the WLAN traffic is getting high and the collision probability is increasing rapidly. When the offered traffic intensity is greater than 0.7, the slope of collision probability will decrease and we can regard the WLAN traffic is very high now and almost saturated. So we will set 0.55 and 0.7 as the values for left edge of trapezoid representing the term “High (H)”. In Fig. 4.5 and Fig. 4.6(a), when the traffic intensity of WLAN is smaller than 0.3, the average delay and collision

probability of WLAN users is very small and not much change, so we will take traffic intensity equals 0.2 and 0.3 into our consideration for the right edge of trapezoid representing the term “Low (L)” and the range for representing the term “Medium (M)” is bound by 0.2, 0.3, 0.55, and 0.7. So we arrange these numbers and set them to be the bound of the Low, Medium, High area.

Besides, in Fig. 4.6(a) and Fig. 4.6(b), the average delay of WLAN users will be below the delay requirement of 1000 ms and it will much higher than 1000 ms when the traffic intensity is greater than 0.6. So we can rewrite the constrain mentioned above, that is, CR systems have to try their best to ease the effect on WLAN and let delay of HTTP users in WLAN can limit below 1000ms at the traffic intensity equals 0.55. delay in WLAN system will fiercely increase when the traffic intensity is about 0.5 ~ 0.6 because of the binary exponential backoff scheme. It can be seen from Fig. 4.6(a), when traffic intensity is about 0.55, the average packet delay of WLAN users is below 1000 ms, closing to the delay requirement. When the traffic intensity is higher, the delay of WLAN users will be over 1000ms. This, it can be estimated that, in the

situation of traffic intensity in WLAN networks is lower than 0.55, CR BS can use about 25 to 35 frames for active period in one BUF and the delay of WLAN users will not increase obviously. Further more, the reason we choose the parameter in μ_M( )N and ( )μ_L N is that there will be obviously degradation in the performance of WLAN networks when CR users use about 45 or 55 frames in one BUF. Because of these apparent changes that can be observed from Figs. 4.6(a) and 4.6(b), we choose those parameters in μ_S( )N , ( )μ_M N and μ_L( )N . the traffic is very intense. We separate them into three cases,

(i) When ρ( )<k λ ρ₁⋅ (k− , put 1) EQP k( − into group 1) φ₁.

(ii) When λ ρ₁⋅ (k− ≤1) ρ( )k ≤λ ρ₂⋅ (k− , put 1) EQP k( − into group 1) φ₂.

(iii) When λ ρ₂⋅ (k−1)< ( )ρ k , put EQP k( − into group 1) φ₃.

For the output ΔN, we uniformly separate the number between -100 % to 100%.

So the membership function will be the follows shown in Fig. 4.4(d):

μ_SD(+N k( )) (= g +N k( ); 100, 100, 70, 60)− − − − , (4.15)

From above, we can have the diagram for the membership functions as shown in Fig. 4.4.

μ ρ( )

Fig. 4.5: Collision probability in WLAN system

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Fig. 4.6: Average WLAN packet delay

After setting up the membership functions of inputs and output, we will build 27 fuzzy rules as the fuzzy rule base in Table 4.1 to do the defuzzification:

Table 4.1: The rule base of fuzzy AP adaptor

Rule ρ N EQP ΔN

From the fuzzy rule base, we can know the strategies we should take according the inputs numbers. Next we will do the defuzzification procedure.

4.2 Defuzzification

We adopt the max-min inferred method and use the center of gravity (COG) method for defuzzification method [16]. To briefly explain max-min inference method, we assume now we only have two fuzzy rules for two inputs, one output fuzzy logic theory problem. For term sets of input variables x and y , T x = ( )

operator to obtain the min membership function values for fuzzy rule 1 and rule 2,

Subsequently, applying the “max” operator through the COG method, a crisp value is exported to decide which action will be applied. To be more specifically, we assume the COG of membership function

1( ) μ_C z and

2( )

μ_C z respectively locates at z=z and ₁ z=z . Then the crisp output value ₂ z is obtained as follows: _o

Fig. 4.7 is the total procedure of max-min method for defuzzification in this example.

Fig. 4.7: Procedure of max-min method for defuzzification

Chapter 5