Interdependency Measure and Analysis for Cross-Layer Design in WLAN

Interdependency Measure and Analysis for Cross-Layer Design in WLAN

11

_{Rui Chen,}

2

_{Wennai Wang}

*

_,

3

_{Zhengkun Mi}

1 Key Lab of Broadband Wireless Communication and Sensor Network Technology,

(Nanjing University of Posts and Telecommunications), Ministry of Education, China;

School of Communications engineering, Nanjing Institute of Technology, China,

[email protected]

2 Key Lab of Broadband Wireless Communication and Sensor Network Technology, (Nanjing

University of Posts and Telecommunications), Ministry of Education, China,

[email protected]

3 Key Lab of Broadband Wireless Communication and Sensor Network Technology, (Nanjing

University of Posts and Telecommunications), Ministry of Education, China,

[email protected]

Abstract

Interdependency among system parameters may significantly affect the cross-layer design and optimization performance of wireless networks. However, it is very difficult to derive the interdependency among system parameters in a dynamic complex networking system due to the uncertainty of data observation and system modeling. In this research, we propose a new approach for dynamic interdependency measure among system parameters by using non-additive measure theory. In particular, the Choquet integral model is applied to distinguish the interaction among system parameters towards the objective function through a set of non-additive measures. Then the most significant effect subset of system parameters on the performance metrics of interest for system objective function is identified. The simulations show that it is only need to adjust the parameters in the optimized subset according to the interdependency measure can improved the network throughput, consequently, the radio resource is utilized more reasonably.

Keywords

:Interdependency measure; WLAN; Choquet integral; Cross-layer design

1. Introduction

Wireless Local Area Network (WLAN) has been received widely attention and research in recent years with its advantages of simplicity and low cost. The IEEE 802.11 MAC sublayer provides a fairly controlled access to the shared wireless medium through two different access mechanisms: the basic access mechanism, called the distributed coordination function (DCF), which is based on carrier sense multiple access with collision avoidance (CSMA/CA), and a centrally controlled access mechanism, called the point coordination function (PCF) [1]. DCF is contention-based and uses the so called Binary Exponential Backoff (BEB) scheme. Many schemes have been proposed to improve the DCF performance, and they can be classified into two broad categories: adaptation by scheduling and adaptation by varying channel parameters [2].

The first category of schemes adapts the channel by relying on a scheduling policy. In wireless environment, channel errors are burst and location dependent. Therefore, when several stations compete in a WLAN, each has its own channel condition (good channel state or bad channel state). Opportunistic scheduling consists in always selecting stations in a good channel state for transmission. To maintain some level of fairness, such schemes rely on swapping transmission opportunities among the stations to improve the network throughput. Many schemes have been proposed in this area (see for example [3], [4] and the references therein), they differ mainly in how stations “borrow” opportunities from other stations and how they “refund” them back.

1_{This work is financially supported by NSFC Project No. 60872018, PAPD of Jiangsu Higher}

Education Institutions, NSF of Jiangsu Province Project No.BK2009351, Program for Postgraduates Research Innovation in University of Jiangsu Province Project No.CX10B_185Z

In the second category, frame size and transmission rate are the two most important parameters that can be dynamically changed to improve the performance of the system. Under severe error rates, decreasing the transmission rate lowers the bit error rates, whereas decreasing the frame size reduces the frame error probability, thereby contributing toward improving the throughput. On the other hand, decreasing the transmission rate lowers the throughput of the network, whereas decreasing the frame size increases the overhead thereby reducing the throughput. Therefore there are tradeoffs between data rate, frame size, overhead and throughput. Many frame size adaptation schemes ([5], [6]) and rate adaptation schemes ([7]) have been proposed in the literature. These approaches are based on system modeling ([8], [9]). But it is hard to model the actual network systems due to the system parameters are time-varying and uncertain. For example, in a dynamic complex system, traditional probability measures might not be accurately derived due to the fact that system events could be non-repeatable in a huge event space. So, these probability-based system models are no longer valid due to the models’ inaccuracy. In addition, most of the literatures, only consider the impact of one parameter or several parameters on network performance respectively, and did not take into account the correlation between different system parameters.

In this paper, we propose a parameter adaptation system which performs adaptation according to the network conditions by varying the system parameters core. Firstly, the system parameters and their correlations are measured by fuzzy measures which no need to establish the accurate mathematical model. Then an optimal parameter set is determined according to the measuring results. Finally, we only adjust those parameters in the optimal parameter set to improve the system performance. The major contributions made in this paper are: 1) the network throughput is obtained under different MAC frame size, data rate, minimum contention window size and retransmission times; 2) quantitative significance analysis is proposed for dynamically identifying which subset of system parameters has the most significant effect on the performance metrics of interest for system objective function and 3) the set of parameters is narrowed according to the measurement result and network performance is optimized by adjusting the parameters in the narrowed set.

2. Methodology

Without loss of generality, let X={x1, x2,…, xn} be the design vector with n system parameters and Ψ

be the system performance metrics for cross-layer design, then a cross-layer optimization problem can be formulated as

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(

m

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to

Subject

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Maximize

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(1)

Where h1 to hm are m design constraints which are closely related to performance requirements

imposed on system parameters. For interdependency measure and significance analysis, we would like to know which set of parameters X’ has the most significant effect on the performance metrics of interest for system optimization, usually X’∈X.

2.1 The Choquet model

We introduce a new nonlinear statistical model based on non-additive integrals [10] _for

interdependency model and significance analysis. The distinguishing feature of this model is that the interaction among system parameters toward the performance metrics of interest can be properly measured through the nonlinear integral such as Choquet integral. The nonlinear data fusion model is described as follows.

The data consists of l observations of x1, x2, …, xn _{and Ψ, and are formed as Table 1. In Table 1, each}

row is an observation of system parameters x1, x2,…, xn _{and Ψ. The observation of x}1, x2,…, xn can be

regarded as a function f : X→(−∞,+∞); hence the j-th observation of x1, x2,…, xn is denoted by fi, and

we write fji=fj (xi) where 1 ≤ i ≤ n and 1 ≤ j ≤ l.

Definition 1: Set the power set of X is P(X), including all subset of X. The interdependency among system parameters toward the performance metrics of interest is described by a set function μ defined

on the power set of X, i.e. μ: P(X)→ [0, ∞]. The set function μ is called a non-additive measure if it satisfies the following conditions:

1）

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2）the monotonicity:

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Table 1. Data formation

x1 x2 … xn Ψ f11 f21 … fm1 f12 f22 … fm2 … … … … f1n f2n … fmn Ψ1 Ψ2 … Ψm

We relax the above two traditional restrictions on non-additive measures, the co-domain of the set function μ is instead of R+ _{and the monotonicity is also not necessary} [11]_{. So, the new non-linear}

multi-regression model based on Choquet integral is now expressed as

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where c is a regression constant,

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non-additive measure, and

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(

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is a normally distributed random perturbation with expectation 0 and variance

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2_{. The Choquet integral}

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_(c) of the data observation f, i.e. a non-additive measure μ, is defined as:

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2.2 To solve the parameters of the Choquet model

The basic idea to solve the Choquet model is a two-step procedure. The first step is to reduce the nolinear multi-regression model to the traditional linear multiregression model by converting each n-dimensional vector attribute datum to a 2n_{-dimensional vector datum, which is defined over the power}

set of attributes; the second step is to solve the linear model by using the standard least-square method. As mentioned in [11], direct reduction on raw data from the Choquet model to linear multiregression model may cause “bad” solutions, where non-additive measures on some subsets are often not able to be determined. So we present a data normalization approach based on median alignment. The main idea is to align the observed data of each predictive attribute with others’ data along their medians, so that if the data samples are reasonably large, it is expected to have non-zero aggregated observation for each subset S∈ P( X’ ); and its non-additive measure can therefore be determined properly.

Given observation data, the optimal regression coefficients can be determined by using the least square method in order to make



2 minimal[12]. The algorithm can be described as follows:

1) The initial settings are:

Number of the chosen system parameters = n; Number of observations = m 2) Data normalization:

For each set of observation data of xi, we denote the normalized results of xi as

x

i':

)

(

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x

median

x

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i i

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(4) 3) Construct the

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otherwise

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4) Find the least square solution of the linear equations having above augmented matrix for unknown variables

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₂n_1. The regression residual error

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can be calculated by:

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3. The interdependency measure and significance analysis among system

parameters in WLANs

In this section, we apply the Choquet model for interdependency measure and significance analysis on MAC layer parameter set of IEEE 802.11 WLANs. The parameter set includes the number of users, the minimum contention window size, MAC-frame size, retransmission times and data rate, etc. In particular, the simulation scenario is a square area of 200m×200m, and several system parameters of MAC and physical layers used in the IEEE 802.11b standard are shown in Table 2. The network throughput is obtained by using different combinations of MAC frame size, data rate, minimum contention window size and retransmission times.

Table 2. MAC and PHY parameters of the IEEE802.11b WLAN

Parameters Values

SIFS DIFS Time slot Data rate MAC header length

ACK length Minimum frame length Maximum frame length

Preamble length CWmin CWmax

Max/min number of users

10 μs 50 μs 20 μs 1, 2, 5.5, 11Mbps 214 bits 112 bits 500 Bytes 1500 Bytes 144 μs(long)and 72 μs(short) 16,32,62,128,256 1023 50/10

The experimental data sets have been collected by running simulations with different combinations of minimum contention window size, frame size, data rate and retransmission times, under different channel states (Good or Bad) and number of users. Table 3 shows that the throughput data collected from simulations results. As shown in Table 3, the minimum contention window size is increased exponentially from 16 to 256, the frame size is chosen from 500 bytes to 1500 bytes with the step size of 250 bytes, the data rates are selected from 1, 2, 5.5, and 11 Mbps, and the retransmission times are selected from 2 to 10 with the step size of 2.

Then, we use the proposed approach to quantitatively capture the interdependency among IEEE 802.11 MAC protocol parameters and identify which set of system parameters has the most significant effect on the throughput performance. Table 4 shows the quantified significance of system parameters

and interdependency patterns among them on the throughput performance. There are four different network conditions considering both network status (lightly-loaded with 10 users or heavily-loaded with 50 users) and channel quality (good channel quality BER=10-5 _{and bad channel quality BER=10} -3_{). Each μ[i] in the table gives the quantified significance (positive or negative) form a set of parameters}

(minimum contention window size, frame size, data rate and retransmission times corresponding to cw,

fm, dr and r as shown in Table 4 toward the throughput performance.

Table 3. The collected experimental data sets

CWmin Frame Size

(Byte) Data rate (Mbps) Retransmission times Throughput (bps) 16 16 16 … 32 32 32 … 64 64 64 … 256 256 256 500 1000 1500 … 500 1000 1500 … 500 1000 1500 … 500 1000 1500 1 2 11 … 1 2 11 … 1 2 11 … 1 2 11 2 4 10 … 2 4 10 … 2 4 10 … 2 4 10 0.445E+06 0.903E+06 3.177E+06 … 0.438E+06 0.918E+06 3.086E+06 … 0.451E+06 0.878E+06 3.088E+06 … 0.432E+06 0.852E+06 2.704E+06

According to the non-additive measure theory, the relation of the measure result μ[i] and the performance metric is:

1) The sign of each μ[i] indicates whether it is positive or negative significant effect on the throughput performance, posed by change of the parameter subset i. In other words, the positive sign of

μ[i] shows that increasing the values of parameter subset i can enhance the throughput performance.

Likewise, the negative sign of μ[i] indicates the throughput performance will be lowered when increasing the values of parameters set i.

2) The absolute value of μ[i] shows how strong the corresponding effect can be imposed by increasing the values of parameter subset i. Recall that the significance measure of a parameter subset turns into a function of the related parameters in nonlinear systems. Thus, each observation will generates a value of significance measure, as shown in Table 4.

From the experimental results, we may have the following conclusions, which are actually consistent to the literatures.

 Lightly-loaded network

1) When the network is lightly loaded, whether the channel quality is good or bad, μ[14] = 3.0609 or 1.9730 has the largest absolute value with positive sign indicates that the interdependency between frame size, data rate and retransmission times have the most significant effect on the throughput performance, i.e., increasing frame size, data rate and retransmission times will be the most effective way to improve the system throughput performance.

2) For the case of lightly loaded network with good channel, μ[6]=3.0575 is very close to μ[14] and

μ[8]= 0.0120 is close to zero, i.e. μ{dr, fm} is close to μ{dr, r, fm} and μ{r} is close to zero, indicates

that the interdependency between fm and dr leads to great effect on the throughput performance, and the retransmission times has negligible significant effect on the throughput performance. This means the parameter set can further narrowed by neglecting parameter retransmission times r.

3) For the case of lightly loaded network with bad channel, μ[6]=1.9643 is very close to μ[14] and

μ[8]= -0.0216 is close to zero, i.e. μ{dr, fm} is close to μ{dr, r, fm} and μ{r} is close to zero, indicates

that the interdependency between frame size, retransmission times and data rate leads to most significant effect on the throughput performance, and the retransmission times has negligible significant effect on the throughput performance. So the result is similar to 2).

 Heavily-loaded network

1) When the network is lightly loaded, whether the channel quality is good or bad, μ[7] = 2.4404 or 1.3275 has the largest absolute value with positive sign indicates that the interdependency between frame size, minimum contention window size and data rate have the most significant effect on the throughput performance.

2) For the case of heavily-loaded network with bad channel, μ{fm, cw} = 0.0929 >0 even though both μ{fm}= -0.1278 and μ{cw}= -0.0012 have negative effects. It is true that increasing minimal contention window size may reduce the throughput performance directly, but on the other hand, it reduces the packet loss rate caused by collision as well. Thus, if the frame size is increased as the same time, it is possible to increase the throughput performance instead.

Table 4. The impacts on throughput performance under different parameters

μ Parameter

subset

Number of users=10 Number of users=50

BER=10-5 BER=10-3 BER=10-5 BER=10-3

μ[1] μ[2] μ[3] μ[4] μ[5] μ[6] μ[7] μ[8] μ[9] μ[10] μ[11] μ[12] μ[13] μ[14] μ[15] {cw} {fm} {cw, fm} {dr} {dr, cw} {dr, fm} {dr, fm, cw} {r} {r, cw} {fm, r} {r, fm, cw} {dr, r} {dr, r, cw} {dr, r, fm} {dr, r ,fm, cw} -0.0229 0.3437 0.3721 0.7644 0.5862 3.0575 3.0334 0.0120 0.0209 0.2998 0.3538 0.7358 0.5715 3.0609 3.0217 -0.0272 0.2358 0.2647 0.4961 0.3756 1.9643 1.8538 -0.0216 0.0178 0.2231 0.2345 0.4811 0.3359 1.9730 1.8546 0.0331 -0.3390 0.0193 0.2424 0.6372 1.6536 2.4404 -0.4497 -0.0285 -0.1841 0.1100 -0.1272 0.6385 1.2545 2.3174 -0.0012 -0.1278 0.0929 0.1547 0.3326 1.0212 1.3275 -0.1973 -0.0085 -0.0461 0.1736 -0.2542 0.2952 1.1846 1.2678

In summary, tradeoffs between the complexity and system performance, increasing the frame length and data rate is the most efficient method to improve the system throughput under lightly-loaded networks, and the most efficient method to improve throughput is increasing the minimum contention window size, frame length and data rate under heavily-loaded networks.

4. Simulation results and analysis

After the subset of parameters which has the most significant effect on the throughput performance is derived from the above measure results, the following step is verifying whether the simulation results will repeat the conclusions derived from the theoretical study. The results are as follows:

(1) Scenario with light-loaded

According to the results of Table 4, three largest μ values were selected, which are the μ[14], μ[6] and μ[15] respectively. Table 5 shows the simulation results derived from the scenario with good and bad channel quality. In this table, the first column is the subset of system parameters considered in the throughput optimization; the others are the average throughput. From Table 5, we can conclude that optimizing frame size, retransmission times and data rate simultaneously will yield the highest throughput performance. This conclusion supports the theoretical results derived from using the proposed approach for interdependency measure and significance analysis in Table 4, where μ[14] is the highest value of significance, meaning that the joint-optimization of frame size, retransmission times, and data rate has the most significant effect on the throughput enhancement. Furthermore, only adjusting the frame size and data rates can also get perfect results. In some actual network performance optimization case study, the frame size and data rates can be adjusted to effectively improve throughput performance in order to reduce cost and complexity. Fig. 1 shows the average throughput under different frame size and data rates.

Table 5. The average throughput under different parameters with lightly-loaded

Good channel Bad channel

μ{dr, r, fm} μ{ dr, fm } μ{ dr, r ,fm, cw } 2.185Mbps 2.18Mbps 2.134Mbps 1.362Mbps 1.361Mbps 1.335Mbps 0 5 10 15 500 1000 1500 0 1 2 3 4 5 Data Rate(Mbps) Frame Size(Bytes) T hr o ugh tpu t( M bp s ) 0 5 10 15 500 1000 1500 0 1 2 3 4 Data Rate(Mbps) Frame Size(Bytes) T hro ugh tpu t(M bp s )

Figure 1.The average throughput under different fm and dr. Left: lightly-loaded network (10 users)

with BER=10-5_{; Right: lightly-loaded network (10 users) with BER=10}-3₎

(2) Scenario with heavily-loaded

Table 6 shows the simulation results derived from the scenario with good channel quality and bad channel quality. This conclusion supports the theoretical results derived from using the proposed approach for interdependency measure and significance analysis in Table 6, where μ[7] is the largest value of significance, meaning that the joint-optimization of frame size, minimum contention window size and data rate has the most significant effect on the throughput enhancement. Fig. 2 shows the average throughput under different frame, size minimum contention window size and data rates.

Table 6. The average throughput under different parameters with heavy-loaded

Good channel (BER=10-5) Bad channel (BER=10-3) μ[7] μ[15] μ[6] μ{dr, fm, cw} μ{dr, fm, cw, r} μ{dr, fm } 1.493Mbps 1.417Mbps 1.182Mbps 0.854Mbps 0.842Mbps 0.505Mbps 0 100 200 300 500 1000 1500 0 0.2 0.4 0.6 0.8 CWmin Data Rate=1 Mbps Frame Size(Bytes) T h rou g ht pu t( M b p s) 0 100 200 300 500 1000 1500 0 0.5 1 1.5 CWmin Data Rate=2 Mbps Frame Size(Bytes) T h rou g ht pu t( M b p s) 0 100 200 300 500 1000 15000 0.5 1 1.5 2 2.5 3 CWmin Data Rate=5.5 Mbps Frame Size(Bytes) Thr ou ght pu t( M bps ) 0 100 200 300 500 1000 15000 1 2 3 4 5 CWmin Data Rate=11 Mbps Frame Size(Bytes) Thr ou ght pu t( M bps ) 0 100 200 300 500 1000 1500 0 0.1 0.2 0.3 0.4 CWmin Data Rate=1 Mbps Frame Size(Bytes) T hro ught pu t( M b p s) 0 100 200 300 500 1000 1500 0 0.2 0.4 0.6 0.8 CWmin Data Rate=2 Mbps Frame Size(Bytes) T hro ught pu t( M b p s) 0 100 200 300 500 1000 1500 0 0.5 1 1.5 2 CWmin Data Rate=5.5 Mbps Frame Size(Bytes) T hr oug ht put (M bps ) 0 100 200 300 500 1000 1500 0.5 1 1.5 2 2.5 CWmin Data Rate=11 Mbps Frame Size(Bytes) T hr oug ht put (M bps )

Figure 2. The average throughput under different fm, cw and dr. Left: heavily-loaded network (50

5. Conclusion

Interdependency among system parameters for cross-layer design in WLAN is very critical to the performance metrics. In this paper, we propose a new approach based on the Choquet integral to measure the interdependency among system parameters, such as MAC frame size, data rate, the minimum contention window size and retransmission times. Then we carry out the significance analysis and identify which subset of the system parameters has the most significant effect on the performance metrics for cross-layer optimization. The proposed approach only relies on the managerial data and can be easily incorporated in the current cross-layer design framework, and the simulations show that it is only need to adjust the parameters in the optimized subset according to the interdependency measure can improved the network throughput, consequently, the radio resource is utilized more reasonably.

6. References

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[10] Zhenyuan Wang, “A new model of nonlinear multiregressions by projection pursuit based on generalized Choquet integrals”, In Proceeding(s) of IEEE International Conference on Fuzzy Systems, pp.1240-1244, 2002

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