Improved Particle Filtering-Based Estimation of the Number of ...
IEEE SIGNAL PROCESSING LETTERS, VOL. 15, 2008
87
Improved Particle Filtering-Based Estimation of the Number of Competing Stations in IEEE 802.11 Networks Jang-Sub Kim, Erchin Serpedin, and Dong-Ryeol Shin
Abstract—This letter proposes a new method to estimate the number of competing stations in IEEE 802.11 networks. Due to the nonlinear/non-Gaussian nature of measurement model, a nonlinear filtering algorithm, called the Gaussian mixture sigma point particle filter (GMSPPF), is proposed herein to estimate the number of competing stations. Since GMSPPF represents a better alternative to the conventional extended Kalman filter (EKF), unscented Kalman filter (UKF), particle filter (PF), and unscented particle filter (UPF) for nonlinear/non-Gaussian (or Gaussian) tracking problems, we apply this filter for IEEE 802.11 WLANs. GMSPPF provides a more viable means for tracking in any conditions the number of competing stations in IEEE 802.11 WLANs relative to EKF, UKF, PF, and UPF. Further, GMSPPF presents both high accuracy as well as prompt reactivity to changes in the network occupancy status. For the more accurate method (GMSPPF), the combined access mode is shown to maximize the system throughput by switching between the basic access mode and the RTS/CTS access mode. Index Terms—Estimation, filtering, network.
I. INTRODUCTION N a bid to realize mobile Internet, IEEE 802.11 Wireless LAN (WLAN) has emerged as one of the most deployed wireless access technologies. In its medium access control (MAC) layer protocol, the distributed coordination function (DCF) employs two techniques for packet transmission: a two-way hand shaking technique called basic access mechanism and an optional four-way handshaking technique called request-to-send/clear-to-send (RTS/CTS) mechanism [1]. The IEEE 802.11 standard is designed to allow both basic access and RTS/CTS access modes to coexist. The standard suggests that the RTS/CTS access model should be chosen when the packet payload exceeds a given RTS threshold. However, it has been shown in [2] and [3] that the RTS threshold which maximizes the system throughput is not a constant value, and significantly depends on the number of competing stations . Clearly, this operation requires each station to be capable of estimating [3].
I
Manuscript received August 13, 2007; revised September 26, 2007. This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD: KRF-2006-214-D00111). The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Kainam Thomas Wong. J.-S. Kim and E. Serpedin are with the Department of Electrical Engineering, Texas A&M University, College Station TX 77843-3128 USA (e-mail: [email protected]; [email protected]). D.-R. Shin is with School of Information and Communication Engineering, Sungkyunkwan University, Suwon 440-746, Korea (e-mail: drshin@ ece.skku.ac.kr). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LSP.2007.911182
In order to estimate , [3] proposed EKF, which in practice suffers from some well-known drawbacks: EKF might be divergent since the measurement model exhibits a highly nonlinear dependence with respect to the state and it relies on a first-order Taylor series expansion of the nonlinear terms around the mean values. Since IEEE 802.11 networks require prompt reactivity to changes in the network occupancy status, the EKF-based estimator might frequently diverge in non-saturated conditions. A new filtering method, called UKF, was proposed to tackle the nonlinearity and to show its effectiveness in terms of divergence reduction and error propagation [4]. The performances of EKF and UKF are not optimal in the presence of non-Gaussian observations. Therefore, PF [5], also called sequential Monte Carlo method, was introduced and successfully applied to the problems arising in IEEE 802.11 WLANs [6], [7] due to nonGaussian observations. Since general PF employs no information from observations in proposing new samples, its use is often ineffective and leads to poor filtering performance. In order to overcome this weakness, UPF techniques were proposed for IEEE 802.11 WLANs [8]. Although UPF presents large estimation performance benefits over the standard PF, it still incurs a heavy computational burden since it has to run a UKF for each particle in the posterior state distribution. This letter proposes a new mechanism named GMSPPF [9] to solve the above-mentioned problems with better efficiency and accuracy than EKF, UKF, PF, and UPF-based estimators in IEEE 802.11 networks. GMSPPF combines an importance sampling (IS)-based measurement update step with a UKF-based Gaussian sum filter for the time-update and proposal density generation. Since GMSPPF employs new observations and uses the expectation-maximization (EM) algorithm to obtain the Gaussian mixture model (GMM), GMSPPF has better estimation performance when compared to standard PF and UPF. Compared to UPF, the computational cost can be reduced because a small number of particles is needed for GMSPPF. EKF and UKF present the limitation of not being applicable to general non-Gaussian distributions. However, PF, UPF, and the proposed GMSPPF-based estimator can be applied to both sequential nonlinear and nonGaussian conditions, while the best performance is achieved by GMSPPF. It can also be shown that the GMSPPF-based estimator presents better performance than EKF, UKF, PF, and UPF for any non-Gaussian network condition. II. PROBLEM FORMULATION AND OBJECTIVES In this section, we show that, starting from the model proposed in [3], it is immediate to derive a formula that explicitly relates the number of competing stations to a performance figure
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IEEE SIGNAL PROCESSING LETTERS, VOL. 15, 2008
that can be measured in runtime by each station. We consider a scenario composed of a fixed number of competing stations, each operating in saturation conditions. Channel conditions are ideal. Using the fundamental assumption that, regardless of the number of retransmissions suffered, the probability of collision is constant and independent at each transmission attempt, from [3, equation (3)], an explicit expression of versus the conditional collision probability takes the form
(1) stand for the maximum backoff stage and conwhere and tention windows sizes, respectively. Since the conditional collision probability can be independently measured by each station by simply monitoring the channel activity, it follows that each station can estimate the number of competing stations. If, stations in the system, then the condiat time , there are , where tional collision probability can be expressed as represents the inverse function of (1). Hence, in accordance with the results from [3, Section V-A]
III. IMPROVED PARTICLE FILTERING PF [5] is based on Monte Carlo simulations with sequential importance sampling (SIS). These methods allow for complete representation of the posterior distribution of the states using sequential importance sampling and resampling of the various probability densities. If the true posterior PDF embodies all the available statistical information of the channel estimates, it is optimal in the sense that all the available information has been used. Whereas the standard EKF suffers from the limitation that it cannot be applied to general non-Gaussian distributions, PF makes no assumptions on the form of the probability densities in , question, i.e., non-Gaussian. The posterior density and , constitutes where the complete solution to the sequential estimation problem. In computing the marginalized minimum mean square error (MMMSE) estimate (4), PF requires generation of random samfrom the posterior distribution . Then ples (4) can be approximated by a set of samples with associated weights, denoted by (6)
(2) is a binomial random
where is the Dirac delta function. From (5), the conditional mean state and the corresponding error covariance can be calculated as follows:
where denotes the size of the observation window. From the monitoring procedure for estimating the packet collision prob[3], we know that follows a binomial distribution ability with B trials and success probability
(7) At the end of each recursion, the particles are resampled to ensure they occur with the same probability as the weights. We next present a further refinement of PF and UPF called GMSPPF [9]. GMSPPF combines the IS-based measurement update step with a UKF-based Gaussian sum filter for the time-update and proposal density generation. In the time update stage, GMSPPF approximates the prior, proposal, and posterior density functions as GMM using banks of parallel UKFs. The predicted and updated Gaussian component means and covariances for each mix are calculated using the bank of UKFs. In the measurement update stage, GMSPPF uses a finite GMM representation of the posterior filtering density
, and where variable with zero mean and variance
(3) of The system state is trivially represented by the number stations in the network at discrete time . From [3, equation (8)], the network state model evolves as (4) in the system at time is given where the number of stations plus a random variable by the number of stations at time which accounts for stations that have been activated and/or stands for the noise terminated during the last time interval. . This state-space with zero mean and covariance matrix model can be easily cast as a hidden Markov model. Equations (2) and (4) provide a complete description of the state model for the system under consideration. Our goal is to find the estimator (5) with estimated error covariance given by
where samples up to time .
is the set of received
(8) where
is the number of GMM, are the mixing weights, and is a normal distribution determined from the th UKF with predicted mean vector and positive definite covariance matrix . This is recovered from the weighted posterior particle set of the IS-based measurement update stage, by means of an EM [10] step. The EM algorithm can be used to obtain Gaussian mixture approximations from these particles and weights. With this mechanism, the EM-based recovery from the posterior GMM further mitigates the “sample depletion” problem through its inherent “kernel smoothing” nature. The EM algorithm provides an iterative method to select which satisfies (9)
KIM et al.: IMPROVED PARTICLE FILTERING-BASED ESTIMATION
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TABLE I SIMULATION PARAMETERS
with the Gaussian mixture specified by the parameter set . Specifically, the EM algorithm is a two-step iterative algorithm which works , it finds the next value via as follows: given a • E-step: ; • M-step: .
Fig. 1. GMSPPF estimation in saturated and non-saturated network conditions. Clean means the real number of stations during the simulations. TABLE II STATE ESTIMATION RESULTS: MSE AND VARIANCE OF ENSEMBLE SAMPLES (X) FOR EKF, UKF, PF, UPF, AND GMSPPF
See [10] for more detailed explanations of the EM algorithm for GMM. Finally, the conditional mean state estimate and the corresponding error covariance can be calculated as follows:
(10) The computational complexities of EKF, UKF, PF, UPF, and GMSPPF are next evaluated and compared. The computational complexity is represented as Big O [i.e., O(N)] notation and only the major computational steps are considered. Both analytical and simulation results will be shown. From [3], EKF since the matrix times matrix mulis approximately is the most time-consuming step, while UKF tiplication due to the matrix times matrix is multiplications . Notice also that PF is due to the matrix times vector multiplication and the sampling step , and UPF is since each particle is updated by UKF. Finally, GMSPPF is due to the UKF step and due to the EM step , then GMSPPF in the case of maximum complexity. If . This indicates that PF and UPF are approximately is 100 times slower than EKF and UKF in an application with and , whereas GMSPPF exhibits almost the same per, , formance as PF and UPF in an application with ( ). From an MSE viewpoint, GMSPPF exhibits the best and performance. GMSPPF is about two to five times faster than UPF, , , and ) and ( , in applications with ( , and ), respectively, and its MSE performance is superior as well. Notice that this analysis considers only the major computational steps. Computer simulations will be shown next to assess realistically all the computational times. IV. SIMULATION RESULTS We now examine the performance of GMSPPF for estimating for an IEEE 802.11 network and compare it with the performance of EKF [3], UKF [4], PF [5], [6], and UPF [8]. We use
the event-driven simulator in [3], and the arrival and departure of competing terminals from the network follow a random Markov chain as in (1)–(3). The simulation parameters are summarized is a stationary process with a given constant in Table I. . One aspect about using EKF [3] is that it variance . The number of partirequires a proper initialization cles is 100. In saturated conditions, we create a simulation scenario where the number of competing stations in the network changes with time as [2, 4, 8, 4, 8, 10, and 15 stations] (simulation time 500 s). In non-saturated conditions, we simulate a network scenario of a random Markov chain (simulation time 200 s). The ability of the GMSPPF-based estimator to track the number of competing stations is shown in Fig. 1 in saturated and non-saturated network conditions. As shown in Fig. 1, the tracking performance of GMSPPF is very accurate. To further quantify the performance of the estimators, the MSEs of estimators are presented in Table II. The MSE and variance of the GMSPPF-based estimator yield the best performance among all the filters in both network conditions. Two different GMSPPFs were compared. The first, GMSPPF (1-1-2), uses 1-component GMM for the state posterior, and 1-component GMM for the process noise density, and 2-component GMM for the measurement noise density. The second, GMSPPF (3-1-2), uses a 3-, 1-, 2-component GMM for the state posterior, process noise density, and measurement noise density, respectively. In this case, a 2-component GMM is used to approximate the binomial distribution of measurement noise. The performance of UKF is better than that of EKF due to the unscented transformation that exhibits second-order accuracy, while EKF presents first-order accuracy. However, PF exhibits the worst performance in saturated conditions, probably due to the random sampling errors
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Fig. 2. Performance comparison of GMSPPF, UPF, PF, UKF, and EKF for the state estimation. [Left side (1)] N = 100 for GMSPPF and N = 100 for UPF and PF. [Right side (2)]N = 20 for GMSPPF, N = 50 for UPF and N = 100 for PF.
Fig. 3. Network normalized throughput using GMSPPF estimation in non-saturated conditions.
and the simplification of the proposal distribution with the prior distribution. However, the performance of PF is better than that of EKF and UKF in non-saturated conditions. The computation time is presented in Fig. 2, for simulations implemented on an Intel Pentium IV 3.2-GHz processor using MATLAB 7.1. The computation time for GMSPPF, UPF, and PF is much greater than that of UKF and EKF. More importantly, GMSPPF presents a considerably shorter computation time than PF and UPF, even though their MSEs are comparable and superior. The difference in computation time increases proportional to the number of particles. As indicated in Fig. 3, the combined mode maximizes the system normalized throughput based on the proposed selection mode. In the case of a large variation in the number of competing stations, the combined mode has the added benefit that the WLAN normalized throughput is maximized. When we also have a context-based access mode selection process between the basic access mode and the RTS-CTS access mode, we focus on the access mode selection method which uses the number of competing stations as context information. The access mode se, is determined to maxilection parameter, the threshold mize the system throughput. The optimal threshold is adjusted to match the existing WLAN conditions (e.g., windows size, , we assume maximum stage number, etc.). Then, if that the RTS-CTS access mode is on; otherwise, the basic access mode is on. In this case, we compare the accuracy of access mode selection using EKF- and GMSPPF-based estimators. Fig. 4 shows 1) the number of competing stations, 2) nor-
IEEE SIGNAL PROCESSING LETTERS, VOL. 15, 2008
Fig. 4. Access mode selection using perfect estimation, GMSPPF, and EKFbased estimation.
malized saturation throughput in the two modes, 3) access mode switching in the case of perfect and GMSPPF-based estimation of the number of competing stations and the position of switching’s failure, and 4) access mode switching in the case of perfect and EKF-based estimation. The plots marked with circles and plus signs in Fig. 4 denote the RTS-CTS access and basic access modes, respectively. As shown in Fig. 4, the performance of the GMSPPF-based estimator runs closely to that of perfect estimation in terms of accuracy of the access mode switching. From 100 trial runs (Monte Carlo simulations), it can be seen that GMSPPF yields the best performance with a selection accuracy of 92%, relative to the accuracy of 85%, 75%, 77%, and 76% exhibited by UPF, PF, UKF, and EKF, respectively. Therefore, by estimating the number of competing stations more accurately, the accuracy of access mode selection is more efficient and the WLAN throughput is maximized. REFERENCES [1] IEEE Standard for Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications (P802.11), 1997. [2] G. Bianchi, “Performance analysis of the IEEE 802.11 distributed coordination function,” IEEE J. Sel. Areas Telecomm., (Wireless Series), vol. 18, no. 3, pp. 535–547, Mar. 2000. [3] G. Bianchi and I. Tinnirello, “Kalman filter estimation of the number of competing terminals in an IEEE 802.11 network,” in Proc. IEEE Infocom 2003, Mar. 2003, vol. 2, pp. 844–852. [4] J. Kim, H. Shin, D. Shin, and W. Chung, “Estimation of the number of competing stations applied with central difference filter for an IEEE 802.11 network,” Lecture Notes in Computer Science (LNCS), vol. 4239, pp. 316–330, Aug. 2006. [5] P. M. Djuric et al., “Particle filtering,” IEEE Signal Process. Mag., vol. 20, no. 5, pp. 19–38, Sep. 2003. [6] T. Vercauteren, A. L. Toledo, and X. Wang, “Batch and sequential Bayesian estimators of the number of active terminals in an IEEE 802.11 network,” IEEE Trans. Signal Process., vol. 55, no. 2, pp. 437–450, Feb. 2006. [7] D. Zheng and J. Zhang, “Protocol design and throughput analysis of opportunistic multi-channel medium access control,” in Proc. Communications, Internet, and Information Technology (CIIT), Nov. 2003. [8] D. Zheng and J. Zhang, “A unscented particle filtering approach to estimating competing stations in IEEE 802.11 WLANs,” in Proc. IEEE GLOBECOM 2005, Dec. 2005. [9] R. van der Merwe and E. Wan, “Gaussian mixture sigma-point particle filters for sequential probabilistic inference in dynamic state-space models,” in Proc. Int. Conf. Acoustics, Speech, and Signal Processing (ICASSP), Apr. 2003. [10] F. Pernkopf and D. Bouchaffra, “Genetic-based EM algorithm for learning Gaussian mixture models,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 27, no. 8, pp. 1344–1348, Aug. 2005.
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Improved Particle Filtering-Based Estimation of the Number of Competing Stations in IEEE 802.11 Networks Jang-Sub Kim, Erchin Serpedin, and Dong-Ryeol Shin
Abstract—This letter proposes a new method to estimate the number of competing stations in IEEE 802.11 networks. Due to the nonlinear/non-Gaussian nature of measurement model, a nonlinear filtering algorithm, called the Gaussian mixture sigma point particle filter (GMSPPF), is proposed herein to estimate the number of competing stations. Since GMSPPF represents a better alternative to the conventional extended Kalman filter (EKF), unscented Kalman filter (UKF), particle filter (PF), and unscented particle filter (UPF) for nonlinear/non-Gaussian (or Gaussian) tracking problems, we apply this filter for IEEE 802.11 WLANs. GMSPPF provides a more viable means for tracking in any conditions the number of competing stations in IEEE 802.11 WLANs relative to EKF, UKF, PF, and UPF. Further, GMSPPF presents both high accuracy as well as prompt reactivity to changes in the network occupancy status. For the more accurate method (GMSPPF), the combined access mode is shown to maximize the system throughput by switching between the basic access mode and the RTS/CTS access mode. Index Terms—Estimation, filtering, network.
I. INTRODUCTION N a bid to realize mobile Internet, IEEE 802.11 Wireless LAN (WLAN) has emerged as one of the most deployed wireless access technologies. In its medium access control (MAC) layer protocol, the distributed coordination function (DCF) employs two techniques for packet transmission: a two-way hand shaking technique called basic access mechanism and an optional four-way handshaking technique called request-to-send/clear-to-send (RTS/CTS) mechanism [1]. The IEEE 802.11 standard is designed to allow both basic access and RTS/CTS access modes to coexist. The standard suggests that the RTS/CTS access model should be chosen when the packet payload exceeds a given RTS threshold. However, it has been shown in [2] and [3] that the RTS threshold which maximizes the system throughput is not a constant value, and significantly depends on the number of competing stations . Clearly, this operation requires each station to be capable of estimating [3].
I
Manuscript received August 13, 2007; revised September 26, 2007. This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD: KRF-2006-214-D00111). The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Kainam Thomas Wong. J.-S. Kim and E. Serpedin are with the Department of Electrical Engineering, Texas A&M University, College Station TX 77843-3128 USA (e-mail: [email protected]; [email protected]). D.-R. Shin is with School of Information and Communication Engineering, Sungkyunkwan University, Suwon 440-746, Korea (e-mail: drshin@ ece.skku.ac.kr). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LSP.2007.911182
In order to estimate , [3] proposed EKF, which in practice suffers from some well-known drawbacks: EKF might be divergent since the measurement model exhibits a highly nonlinear dependence with respect to the state and it relies on a first-order Taylor series expansion of the nonlinear terms around the mean values. Since IEEE 802.11 networks require prompt reactivity to changes in the network occupancy status, the EKF-based estimator might frequently diverge in non-saturated conditions. A new filtering method, called UKF, was proposed to tackle the nonlinearity and to show its effectiveness in terms of divergence reduction and error propagation [4]. The performances of EKF and UKF are not optimal in the presence of non-Gaussian observations. Therefore, PF [5], also called sequential Monte Carlo method, was introduced and successfully applied to the problems arising in IEEE 802.11 WLANs [6], [7] due to nonGaussian observations. Since general PF employs no information from observations in proposing new samples, its use is often ineffective and leads to poor filtering performance. In order to overcome this weakness, UPF techniques were proposed for IEEE 802.11 WLANs [8]. Although UPF presents large estimation performance benefits over the standard PF, it still incurs a heavy computational burden since it has to run a UKF for each particle in the posterior state distribution. This letter proposes a new mechanism named GMSPPF [9] to solve the above-mentioned problems with better efficiency and accuracy than EKF, UKF, PF, and UPF-based estimators in IEEE 802.11 networks. GMSPPF combines an importance sampling (IS)-based measurement update step with a UKF-based Gaussian sum filter for the time-update and proposal density generation. Since GMSPPF employs new observations and uses the expectation-maximization (EM) algorithm to obtain the Gaussian mixture model (GMM), GMSPPF has better estimation performance when compared to standard PF and UPF. Compared to UPF, the computational cost can be reduced because a small number of particles is needed for GMSPPF. EKF and UKF present the limitation of not being applicable to general non-Gaussian distributions. However, PF, UPF, and the proposed GMSPPF-based estimator can be applied to both sequential nonlinear and nonGaussian conditions, while the best performance is achieved by GMSPPF. It can also be shown that the GMSPPF-based estimator presents better performance than EKF, UKF, PF, and UPF for any non-Gaussian network condition. II. PROBLEM FORMULATION AND OBJECTIVES In this section, we show that, starting from the model proposed in [3], it is immediate to derive a formula that explicitly relates the number of competing stations to a performance figure
1070-9908/$25.00 © 2007 IEEE
88
IEEE SIGNAL PROCESSING LETTERS, VOL. 15, 2008
that can be measured in runtime by each station. We consider a scenario composed of a fixed number of competing stations, each operating in saturation conditions. Channel conditions are ideal. Using the fundamental assumption that, regardless of the number of retransmissions suffered, the probability of collision is constant and independent at each transmission attempt, from [3, equation (3)], an explicit expression of versus the conditional collision probability takes the form
(1) stand for the maximum backoff stage and conwhere and tention windows sizes, respectively. Since the conditional collision probability can be independently measured by each station by simply monitoring the channel activity, it follows that each station can estimate the number of competing stations. If, stations in the system, then the condiat time , there are , where tional collision probability can be expressed as represents the inverse function of (1). Hence, in accordance with the results from [3, Section V-A]
III. IMPROVED PARTICLE FILTERING PF [5] is based on Monte Carlo simulations with sequential importance sampling (SIS). These methods allow for complete representation of the posterior distribution of the states using sequential importance sampling and resampling of the various probability densities. If the true posterior PDF embodies all the available statistical information of the channel estimates, it is optimal in the sense that all the available information has been used. Whereas the standard EKF suffers from the limitation that it cannot be applied to general non-Gaussian distributions, PF makes no assumptions on the form of the probability densities in , question, i.e., non-Gaussian. The posterior density and , constitutes where the complete solution to the sequential estimation problem. In computing the marginalized minimum mean square error (MMMSE) estimate (4), PF requires generation of random samfrom the posterior distribution . Then ples (4) can be approximated by a set of samples with associated weights, denoted by (6)
(2) is a binomial random
where is the Dirac delta function. From (5), the conditional mean state and the corresponding error covariance can be calculated as follows:
where denotes the size of the observation window. From the monitoring procedure for estimating the packet collision prob[3], we know that follows a binomial distribution ability with B trials and success probability
(7) At the end of each recursion, the particles are resampled to ensure they occur with the same probability as the weights. We next present a further refinement of PF and UPF called GMSPPF [9]. GMSPPF combines the IS-based measurement update step with a UKF-based Gaussian sum filter for the time-update and proposal density generation. In the time update stage, GMSPPF approximates the prior, proposal, and posterior density functions as GMM using banks of parallel UKFs. The predicted and updated Gaussian component means and covariances for each mix are calculated using the bank of UKFs. In the measurement update stage, GMSPPF uses a finite GMM representation of the posterior filtering density
, and where variable with zero mean and variance
(3) of The system state is trivially represented by the number stations in the network at discrete time . From [3, equation (8)], the network state model evolves as (4) in the system at time is given where the number of stations plus a random variable by the number of stations at time which accounts for stations that have been activated and/or stands for the noise terminated during the last time interval. . This state-space with zero mean and covariance matrix model can be easily cast as a hidden Markov model. Equations (2) and (4) provide a complete description of the state model for the system under consideration. Our goal is to find the estimator (5) with estimated error covariance given by
where samples up to time .
is the set of received
(8) where
is the number of GMM, are the mixing weights, and is a normal distribution determined from the th UKF with predicted mean vector and positive definite covariance matrix . This is recovered from the weighted posterior particle set of the IS-based measurement update stage, by means of an EM [10] step. The EM algorithm can be used to obtain Gaussian mixture approximations from these particles and weights. With this mechanism, the EM-based recovery from the posterior GMM further mitigates the “sample depletion” problem through its inherent “kernel smoothing” nature. The EM algorithm provides an iterative method to select which satisfies (9)
KIM et al.: IMPROVED PARTICLE FILTERING-BASED ESTIMATION
89
TABLE I SIMULATION PARAMETERS
with the Gaussian mixture specified by the parameter set . Specifically, the EM algorithm is a two-step iterative algorithm which works , it finds the next value via as follows: given a • E-step: ; • M-step: .
Fig. 1. GMSPPF estimation in saturated and non-saturated network conditions. Clean means the real number of stations during the simulations. TABLE II STATE ESTIMATION RESULTS: MSE AND VARIANCE OF ENSEMBLE SAMPLES (X) FOR EKF, UKF, PF, UPF, AND GMSPPF
See [10] for more detailed explanations of the EM algorithm for GMM. Finally, the conditional mean state estimate and the corresponding error covariance can be calculated as follows:
(10) The computational complexities of EKF, UKF, PF, UPF, and GMSPPF are next evaluated and compared. The computational complexity is represented as Big O [i.e., O(N)] notation and only the major computational steps are considered. Both analytical and simulation results will be shown. From [3], EKF since the matrix times matrix mulis approximately is the most time-consuming step, while UKF tiplication due to the matrix times matrix is multiplications . Notice also that PF is due to the matrix times vector multiplication and the sampling step , and UPF is since each particle is updated by UKF. Finally, GMSPPF is due to the UKF step and due to the EM step , then GMSPPF in the case of maximum complexity. If . This indicates that PF and UPF are approximately is 100 times slower than EKF and UKF in an application with and , whereas GMSPPF exhibits almost the same per, , formance as PF and UPF in an application with ( ). From an MSE viewpoint, GMSPPF exhibits the best and performance. GMSPPF is about two to five times faster than UPF, , , and ) and ( , in applications with ( , and ), respectively, and its MSE performance is superior as well. Notice that this analysis considers only the major computational steps. Computer simulations will be shown next to assess realistically all the computational times. IV. SIMULATION RESULTS We now examine the performance of GMSPPF for estimating for an IEEE 802.11 network and compare it with the performance of EKF [3], UKF [4], PF [5], [6], and UPF [8]. We use
the event-driven simulator in [3], and the arrival and departure of competing terminals from the network follow a random Markov chain as in (1)–(3). The simulation parameters are summarized is a stationary process with a given constant in Table I. . One aspect about using EKF [3] is that it variance . The number of partirequires a proper initialization cles is 100. In saturated conditions, we create a simulation scenario where the number of competing stations in the network changes with time as [2, 4, 8, 4, 8, 10, and 15 stations] (simulation time 500 s). In non-saturated conditions, we simulate a network scenario of a random Markov chain (simulation time 200 s). The ability of the GMSPPF-based estimator to track the number of competing stations is shown in Fig. 1 in saturated and non-saturated network conditions. As shown in Fig. 1, the tracking performance of GMSPPF is very accurate. To further quantify the performance of the estimators, the MSEs of estimators are presented in Table II. The MSE and variance of the GMSPPF-based estimator yield the best performance among all the filters in both network conditions. Two different GMSPPFs were compared. The first, GMSPPF (1-1-2), uses 1-component GMM for the state posterior, and 1-component GMM for the process noise density, and 2-component GMM for the measurement noise density. The second, GMSPPF (3-1-2), uses a 3-, 1-, 2-component GMM for the state posterior, process noise density, and measurement noise density, respectively. In this case, a 2-component GMM is used to approximate the binomial distribution of measurement noise. The performance of UKF is better than that of EKF due to the unscented transformation that exhibits second-order accuracy, while EKF presents first-order accuracy. However, PF exhibits the worst performance in saturated conditions, probably due to the random sampling errors
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Fig. 2. Performance comparison of GMSPPF, UPF, PF, UKF, and EKF for the state estimation. [Left side (1)] N = 100 for GMSPPF and N = 100 for UPF and PF. [Right side (2)]N = 20 for GMSPPF, N = 50 for UPF and N = 100 for PF.
Fig. 3. Network normalized throughput using GMSPPF estimation in non-saturated conditions.
and the simplification of the proposal distribution with the prior distribution. However, the performance of PF is better than that of EKF and UKF in non-saturated conditions. The computation time is presented in Fig. 2, for simulations implemented on an Intel Pentium IV 3.2-GHz processor using MATLAB 7.1. The computation time for GMSPPF, UPF, and PF is much greater than that of UKF and EKF. More importantly, GMSPPF presents a considerably shorter computation time than PF and UPF, even though their MSEs are comparable and superior. The difference in computation time increases proportional to the number of particles. As indicated in Fig. 3, the combined mode maximizes the system normalized throughput based on the proposed selection mode. In the case of a large variation in the number of competing stations, the combined mode has the added benefit that the WLAN normalized throughput is maximized. When we also have a context-based access mode selection process between the basic access mode and the RTS-CTS access mode, we focus on the access mode selection method which uses the number of competing stations as context information. The access mode se, is determined to maxilection parameter, the threshold mize the system throughput. The optimal threshold is adjusted to match the existing WLAN conditions (e.g., windows size, , we assume maximum stage number, etc.). Then, if that the RTS-CTS access mode is on; otherwise, the basic access mode is on. In this case, we compare the accuracy of access mode selection using EKF- and GMSPPF-based estimators. Fig. 4 shows 1) the number of competing stations, 2) nor-
IEEE SIGNAL PROCESSING LETTERS, VOL. 15, 2008
Fig. 4. Access mode selection using perfect estimation, GMSPPF, and EKFbased estimation.
malized saturation throughput in the two modes, 3) access mode switching in the case of perfect and GMSPPF-based estimation of the number of competing stations and the position of switching’s failure, and 4) access mode switching in the case of perfect and EKF-based estimation. The plots marked with circles and plus signs in Fig. 4 denote the RTS-CTS access and basic access modes, respectively. As shown in Fig. 4, the performance of the GMSPPF-based estimator runs closely to that of perfect estimation in terms of accuracy of the access mode switching. From 100 trial runs (Monte Carlo simulations), it can be seen that GMSPPF yields the best performance with a selection accuracy of 92%, relative to the accuracy of 85%, 75%, 77%, and 76% exhibited by UPF, PF, UKF, and EKF, respectively. Therefore, by estimating the number of competing stations more accurately, the accuracy of access mode selection is more efficient and the WLAN throughput is maximized. REFERENCES [1] IEEE Standard for Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications (P802.11), 1997. [2] G. Bianchi, “Performance analysis of the IEEE 802.11 distributed coordination function,” IEEE J. Sel. Areas Telecomm., (Wireless Series), vol. 18, no. 3, pp. 535–547, Mar. 2000. [3] G. Bianchi and I. Tinnirello, “Kalman filter estimation of the number of competing terminals in an IEEE 802.11 network,” in Proc. IEEE Infocom 2003, Mar. 2003, vol. 2, pp. 844–852. [4] J. Kim, H. Shin, D. Shin, and W. Chung, “Estimation of the number of competing stations applied with central difference filter for an IEEE 802.11 network,” Lecture Notes in Computer Science (LNCS), vol. 4239, pp. 316–330, Aug. 2006. [5] P. M. Djuric et al., “Particle filtering,” IEEE Signal Process. Mag., vol. 20, no. 5, pp. 19–38, Sep. 2003. [6] T. Vercauteren, A. L. Toledo, and X. Wang, “Batch and sequential Bayesian estimators of the number of active terminals in an IEEE 802.11 network,” IEEE Trans. Signal Process., vol. 55, no. 2, pp. 437–450, Feb. 2006. [7] D. Zheng and J. Zhang, “Protocol design and throughput analysis of opportunistic multi-channel medium access control,” in Proc. Communications, Internet, and Information Technology (CIIT), Nov. 2003. [8] D. Zheng and J. Zhang, “A unscented particle filtering approach to estimating competing stations in IEEE 802.11 WLANs,” in Proc. IEEE GLOBECOM 2005, Dec. 2005. [9] R. van der Merwe and E. Wan, “Gaussian mixture sigma-point particle filters for sequential probabilistic inference in dynamic state-space models,” in Proc. Int. Conf. Acoustics, Speech, and Signal Processing (ICASSP), Apr. 2003. [10] F. Pernkopf and D. Bouchaffra, “Genetic-based EM algorithm for learning Gaussian mixture models,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 27, no. 8, pp. 1344–1348, Aug. 2005.
