Cognitive WSN access based on local WLAN traffic estimation

Transcription

1 Cognitive WSN access based on local WLAN traffic estimation MARCELLO LAGANÀ Master s Degree Project Stockholm, Sweden XR-EE-LCN 211:13

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3 KTH ROYAL INSTITUTE OF TECHNOLOGY School of Electrical Engineering Laboratory of Communication Networks COGNITIVE WSN ACCESS BASED ON LOCAL WLAN TRAFFIC ESTIMATION Marcello Laganà

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5 A B S T R A C T The recent growth in the deployment of wireless networks that access the unlicensed ISM bands has introduced the issue of co-existence between different kinds of wireless technologies. In this work we address the enhancement of the performance of the low-cost Wireless Sensor Network nodes competing for spectrum access with the WLAN devices, that are more powerful in terms of transmission and computational power. Those low-cost WSN nodes are usually battery powered devices with low computational capabilities and limited battery capacity, thus the traditional methods for accessing the wireless spectrum should be revised, including throughput and energy efficiency optimization while considering the hardware constraints of the nodes. In this work we provide tools for spectrum activity prediction that will be used by a WSN cognitive MAC to minimize the collision probability with WLAN transmissions, hence the number of retransmissions, increasing consequently the energy efficiency of communication. We propose two approaches for WLAN spectrum activity modeling considering different probability distributions for the idle WLAN spectrum periods. The first considers the universal activity of the WLAN, while the second takes into account the hardware limitations of the sensors in terms of the detection range. For both modeling options we provide low-complexity algorithms based on either numerical estimation methods or neural networks for the parameter estimation that are suitable for the employment in the WSN. We complete the work with an extensive performance analysis in terms of estimation accuracy and we show that, even with a spatial limited knowledge of the WLAN traffic, the sensors could derive a good approximation of the universal activity model. iii

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7 A C K N O W L E D G E M E N T S I would like to express my gratitude to all those who made this work possible. First of all, I would like to thank my supervisor in Italy, Prof. Chiara Petrioli, for giving me the great opportunity to study abroad and my supervisor in Sweden, Prof. Viktoria Fodor, for her invaluable advice and guidance during this whole master thesis project. I owe my deepest gratitude to Ioannis Glaropoulos for taking me under his wing and especially not only for supporting me throughout this work with constructive suggestions that helped me sort out different details, but also for the time spent carefully reading and commenting the drafting of this thesis. I also would like to thank all the members of the Laboratory of Communication Networks of KTH for their kindness and their hospitality in this laboratory, where I found a friendly and comfortable working environment, making this period a pleasant experience. Most importantly, none of this would have been possible without the encouragement of my family and their absolute confidence in me. v

8 C O N T E N T S 1 Introduction Scenario Main Contribution Related Work I Estimation 9 2 Model 11 3 Global View Estimation Generalized Pareto Mixture Distribution Maximum Likelihood Estimation Moment Evaluation Results Local View Estimation Compound MLE/MV Active Distribution Generalized Pareto Parameters Mixture Distribution Percentage of Observable Load Neural Network Point Selection Results Combined MLE-MV estimation Neural Network II Simulation 45 5 Simulation Simulator Structure Traffic Generation Estimation Library vi

9 5.2 Results Model Compliant Traffic Complex Traffic Model Conclusions 59 Acronyms 61 Bibliography 63 vii

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11 1 1 I N T R O D U C T I O N Wireless Sensor Network (WSN) is an emerging technology that has a wide range of current or potential applications including environment monitoring, smart spaces, medical systems and robotic exploration [3]. Such a network normally consists of a large number of distributed nodes that organize themselves into a multi-hop wireless network. Each node has one or more sensors, embedded processors and low-power radios, and is normally battery operated [2]. Typically, these nodes coordinate to perform a common task through communication in the unlicensed Industrial, Scientific and Medical (ISM) band. The spectrum in this band has gradually become a scarce resource, due to the growing proliferation of wireless devices like Wireless Local Area Netwoks (WLANs), WSNs, Bluetooth nodes, cordless phones and RFIDs [1]. Thus, random wireless medium access or even multiple access based on carrier sense are schemata that do not guarantee anymore effective co-existence, especially when accounting for the disparity in the hardware capability between heterogeneous technologies. For example, a sensor node cannot achieve the transmission power of a laptop or an Access Point (AP) and its communication could be easily disturbed since it will not be detected by those devices, leading to packet collisions. In this work we investigate solutions for energy-efficient co-existence of a WSN with more powerful, in terms of Tx power, technologies using a sort of implicit coordination. Specifically, instead of accessing the spectrum concurrently, we propose an approach that effectively exploits the idle spectrum space in the spatial and time domains between transmissions of more powerful users, represented by based WLAN devices. The proposed method relies on the existence of enough space to allow sensor transmissions; indeed it has been shown in [12, 14] that in most cases sufficient whitespace will be available; for example, considering a continuous conference call using the popular Voice-over-IP (VoIP) client Skype, the channel is idle the 89 percent of the time [14]. To access the idle spectrum spaces, this access method requires from the sensors a knowledge of the wireless environment status and the adaption of the Medium Access Control (MAC) to this environment. Cognitive radio technology is the key technology that enables this dynamic spectrum usage. The term, cognitive radio, is formally defined as follows [8]:

12 2 INTRODUCTION 1 A Cognitive Radio is a radio that can change its transmitter parameters based on interaction with the environment in which it operates. The implied awareness can be expressed as the capability of the WSN to predict the WLAN s activity and to adapt its medium access accordingly. A model of spectrum activity that allows us to define the probability distribution of the idle and active periods has been proposed in [12]. Knowing the actual idle time distribution means that the sensors can evaluate a priori the collision probability of their packets with the WLAN frames and consequently adapt the packet lengths and the transmission policy to decrease that probability (and, thus, the number of unsuccessful transmissions), while meeting the requirements of the applications but also extending their battery life by reducing the number of retransmissions. The knowledge comes with the cost of sensing the spectrum activity and collect enough samples to estimate accurately the parameters of the active and the idle distributions. Due to hardware limitations, the detection range of the sensors is reasonably smaller than the WLAN area and therefore they can see only a portion of the transmissions in the WLAN and hence its active periods. Thus, our solution for the spectrum activity modeling considers not only the universal status of the entire WLAN area as the model described in [12], but also it takes into account the individual point of view of the sensors. We call that model and our extension as Global View and Local View, respectively. Clearly, the WSN nodes could gain and use the global view only if there is sensing information exchange among them. They should exchange for example active period values with timestamps to build this global activity model. On the other hand, the use of the proposed local view is the only possibility if the channel occupancy model is built without sensors cooperation. Thus, in this work we investigate the models regarding both global and local view, and we propose efficient algorithms for the estimation of their parameters. We also compare and statistically evaluate the proposed solutions. Furthermore, we develop a simulation testbed to study different aspects of the proposed algorithms that remain hidden with the statistical evaluation, such as a common understanding of the channel status probability distribution, a desirable property for Cognitive WSN. The report is organized as follows. In the remaining part of this chapter, we describe in detail the considered networking scenario (Sec. 1.1), we introduce the main contribution provided (Sec. 1.2) and finally we discuss the related work on cognitive spectrum access (Sec. 1.3). In Chapter 2 we define the global and the local view model, while we provide algorithms for their parameter estimation in Chapter 3 and 4, respectively. In Chapter 5 we describe the simulation testbed and we provide the related results.

13 1.1 SCENARIO SCENARIO The considered networking scenario consists of a single WLAN access point area with a WSN deployed inside (Figure 1.1) [15]. The WLAN operates in that area of coverage with a radius of 1-15 meters, while communicating in the ISM band according to the IEEE standard with a transmission power in the order of 15-2 dbm. On the other hand, the transmission power of the WSN nodes, that operates within a band inside the WLAN channel, is assumed to be around two orders of magnitude lower than that of the coexisting WLAN. Due to this power difference, we also assume that the WLAN devices are unaware (blind) of the WSN communication [19], since their carrier sense fail to detect low-power WSN transmission and consequently they do not defer from accessing the medium. The opposite does not hold and it is possible to assume that the WLAN communication is not affected by the WSN operation in the ISM band. The number and the locations of the WLAN terminals are unknown, independently and identically distributed inside the AP coverage area, while the sensor locations are also uniformly distributed but with WSN topology known to all nodes. WSN WLAN AP - Zone Figure 1.1: The considered networking scenario. WSN nodes operate in ad-hoc manner and under a light load regime. This leads to the assumption of a negligible channel contention among them. Since they are energy constrained, we need to minimize the energy consumption for operation, limiting their awake time. Thus, their MAC is based on asynchronous duty-cycled schemes [4], enhanced by cognitive functionality, such as WLAN traffic sensing and prediction, as well as next-hop & frame length optimization [15]. The duty cycle approach drastically reduces the energy consumption, since the nodes periodically alternate between the awake and longer sleep states.

14 4 MAIN CONTRIBUTION 1.2 Then we could intervene on their communication to cut more power consumption, decreasing the number of unsuccessful transmissions and, hence, the consequent retransmissions. Thus, the frame transmission time (frame size) and the neighbor destination are optimized accordingly with the collision probability model that has been proposed in [15], which is built on the WLAN idle period distribution. Cognitive Cycle Measure process MAC duty-cycle Measure process MAC duty-cycle... Figure 1.2: The proposed cognitive cycle. To derive the WLAN traffic model, the WSN duty cycles are interleaved in a Cognitive Cycle (Figure 1.2) with a measurement period. During each measurement period the sensors perform continuous sensing of the channel, and collect statistics of the idle times wich are then used to estimate the idle time distribution. Once the WLAN activity has been statistically modeled, the sensors could use this information in their communication afterwards. Although the accuracy increases with the number of samples, the measurement process should not be long because it consumes energy and forbids the data communication. The sensors need to repeat the measure process to follow the changes of the WLAN traffic pattern on a larger time-scale. This issue will be addressed in future work. 1.2 MAIN CONTRIBUTION In this report we first extend the model of the busy and idle transmission periods in the WLAN, proposed in [12]. We enhance the original model with a local view component, by considering also the limited detection range of the sensors. Since we focus on sensors that are hardware-constrained devices, we also propose computationally efficient ways to estimate the parameters of the model and its extension. For both models we use mathematical methods like the Maximum Likelihood Estimation (MLE) and the Moment Evaluation (MV), but in

15 1.3 RELATED WORK 5 addition we employ a Neural Network based approach for the model extension for the local view. Furthermore, the parameter estimation processes are accomplished locally and without any collaboration among sensors, since the communication for exchanging information increases the energy overhead and, hence, it drains the battery. Our contribution includes a statistical evaluation of the algorithms proposed. We show that the estimation can achieve good approximation of the parameters for idle period distribution of the WLAN traffic, even when sensor can only see partially this distribution. The proposed algorithms will be included in the energy-efficient cross-layer communication framework, building a cognitive communication protocol that is able to optimize packet transmission time and distance, accordingly to the parameters estimated by only measuring the wireless spectrum activity. We also include an evaluation based on NS-based simulation and we provide the support software for the simulations, featuring the implementation of the proposed algorithms, the channel sensing functions, and also a modular traffic generator that covers different traffic models. This will allow us not only to proceed with the evaluation for our algorithms, but also it could be employed for testing the performance of future MAC implementations. 1.3 RELATED WORK Co-existence among wireless technologies has become an interesting problem because of the increasing density of the deployment of the various kind of wireless networks. A homogeneous case of coexisting networks is discussed in [21], where the authors consider a typical scenario of multiple overlapping WLANs. They propose a new protocol, with lower overhead than Request-To- Send/Clear-To-Send (RTS/CTS) schema, to mitigate the staggered collisions, that happen when a device either is interrupted by or it interrupts an hidden node transmission, by evaluating their probability locally. The traffic state of the medium is modelled at each node by combining the node sensing results with information received with AP broadcasts and the collision probabilities are evaluated using a hard-coded table. Different issues are faced when dealing with heterogeneous wireless networks. Because of the low transmission power, WSNs are particularly vulnerable to the interference introduced by other wireless technologies. Focusing on IEEE devices, the authors in [29] address the problem of reducing the inhibition loss only, that is the frame dropping caused by subsequent channel access failures. Since the nodes cannot fairly compete for spectrum access

16 6 RELATED WORK 1.3 due to longer back-off periods with respect to IEEE 82.11b/g devices, the authors propose an algorithm that adaptively raises and reduces their energy detection threshold, so the WSN nodes attemp to transmit even in presence of interference. Furthermore, the solutions for WSN co-existence with other more powerful technologies needs to account for the constraints of the WSN nodes, i. e. low computational resources or energy efficiency. In [22], the authors developed a low complexity channel ranking algorithm to provide at the sensor network initialization phase a list of best channels used for transmission channel selection. Instead of using a rarely refreshed channel list, a simple interference estimation mechanism to detect IEEE channels that overlap with used WiFi channels has been proposed in [24]. Before each multihop transmission, all the nodes on the path between source and destination sense the spectrum and then they coordinate to select the least noisy common channel to provide a fast and reliable virtual circuit, used for download operation. The previous methods focus on the noise detection and avoidance, while in [6] the authors propose an algorithm that first classifies the interference and then it adapts the WSN nodes transmission protocol accordingly. The main considered sources of interference in a typical office or residential scenario are WLANs and microwave ovens, thus the proposed method exploit the differences in the operating patterns (random access vs. fixed duty cycle) and then it interleaves the WSN communication in the duty cycles of a microwave device or it occupies the back-off of WLAN devices by forcing the access to the spectrum before the DIFS 1 expires. The scenario in [6] assumes that WSN nodes have full knowledge of the availability of all the channels, while the POMDP framework described in [3] introduces the concept of partial knowledge due to the hardware demanding and energy inefficient sensing process. Thus, that framework adopts a decisiontheoretic approach for spectrum sensing and channel access, with an underlying Markov process built on the partial observations of the spectrum activities. In all those works, the authors provide methods for reacting to a discovered static or dynamic interference. However, by knowing the WLAN spectrum activity pattern or being able to forecast it, the WSN nodes may increase the communication quality in terms of throughput and energy saving. That knowledge may be acquired by analysing the WLAN traffic and representing the trends of active and idle periods as probability distributions. In [17], the authors model with different granularities (system-wide or at APlevel) many aspects for the workload of a whole campus WLAN. Specifically, they describe at different level the traffic generation pattern, from the session arrivals to the flow sizes, using different distribution and estimating the parameters 1 DCF Interframe Space (DIFS): defined by IEEE as idle the time that the WLAN stations need to sense and wait before any transmission.

17 1.3 RELATED WORK 7 of those distributions through MLE. Although their models are accurate and detailed, they do not go down to the spectrum occupancy level, not giving a model for idle periods that are the opportunities for WSN transmissions. Different probability distributions for describing the spectrum idle periods have been proposed in [28]. The authors employ MLE or Expectation Maximization (EM) algorithms to estimate the parameters that best fit the distribution with traffic coming from real traces of an environment with heterogeneous wireless devices, and they show that the hyper-exponential distribution is an excellent candidate. In [12], on the other hand, the authors give a mixture distribution that better models the WLAN idle period distribution, since it captures the two basic sources of inactivity, that are short (almost uniformly distributed) back-off times or longer (heavy tailed) idle periods. They also propose an MLE approach for estimating the parameters of the truncated distribution without the back-off time component. The distribution lacks the local view perspective, essential when dealing with hardware-constrained sensors that have a limited detecting range. Another WLAN traffic estimation model has been proposed in [19] where the authors introduce the concept of blindness of the WLAN with respect the WSN. In [19], the authors describe and validate a model based instead on a Pareto distribution fitted through MLE for describing the WLAN activity trend and then they propose the frame control protocol WISE that adapts the frame length for reducing the collision probability with the WLAN transmissions. In this work we will rely on the same blindness concept but we will use the mixture distribution proposed in [12], since it covers realistic wireless scenarios and combine the two different sources of idle periods. We also emphasize the necessity of a locality in the traffic view due to the sensor limitations, that is missing in both of the previous quoted works.

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19 Part I E S T I M AT I O N

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21 11 2 M O D E L WLAN spectrum activity requires continuous time modeling, since WLAN does not have a slot structure. The model should include both the channel s states with their transition behavior and the duration of time that the system resides in each of the states. A semi-markovian model has been proposed in [14], that is an extension of a Continuous-Time Markov Chain (CTMC) with arbitrary occupancy periods. f A (t) Data 1 1 SIFS ACK 1 CW p Active 1 f (CW) I (t) WS 1 - p f (WS) I (t) Idle Figure 2.1: The general model with all its states. Figure 2.1 shows all the states of the WLAN channels and their merging in a two-state semi-markovian chain. The states of Data, Short Interframe Space (SIFS) and Acknowledge packet (ACK) transmission are grouped together into a single Active state, since their transition is effectively deterministic because the probability of observing the sequence Data SIFS ACK is close to one (this also holds when WLAN uses the RTS/CTS mechanism). Although the small gap between data packets and their ACK corresponds to an idle channel, it is too short to be used for any WSN transmission or even to be detected. Similarly the states of the channel being in the Contention Window (CW) or White Space (WS) are merged into a single Idle state.

22 12 MODEL 2 To predict how long the system remains in either state, we need to find distributions f A (t) and f I (t) for the holding (or sojourn) times in the Active and Idle state, respectively [13]. As proposed in [13], a uniform distribution in the range [α on, β] sufficiently models the active spectrum periods, t < α on, 1 f A (t) = β α on α on t β, t > β (2.1) while the idle spectrum distribution in [13] incorporates the effects from both CW and WS states. According to [13], the contention period shows an almost uniformly distributed sojourn time, while a free state exhibits heavy-tailed behavior that is well approximated by a generalized Pareto distribution. f I (t) Idle f A (t) Active Figure 2.2: The reduced model for Global View. We call the resulting model in Figure 2.2 as the Global View since it models the status of the whole WLAN coverage area and is the model that is observed if sensors can see the whole (global) WLAN activity. We can express the mixture distribution of the Idle state as [1]: f I (t) { p f (CW) I (t) + (1 p) f (WS) I t α bk p f (WS) (2.2) I (t) t > α bk where the f (CW) I (t) is the uniform distribution in the back-off range in [, α bk ] and f (WS) I (t) is a generalized Pareto distribution for the white spaces where its Probability Density Function (PDF) and Cumulative Distribution Function (CDF) are: g [ξ,σ] (t) = 1 ( 1 + ξ t ) ( 1 ξ 1) (2.3a) σ σ ( G [ξ,σ] (t) = ξ σ) t 1 ξ (2.3b)

23 2 MODEL 13 The literature suggests location parameter µ =, so we have: f (CW) I (t) = 1 α bk f (WS) I (t) g [ξ,µ,σ] (t) = 1 σ ( 1 + ξ t ) 1 ( ξ 1) σ (2.4a) (2.4b) Thus we get for the mixture distribution: p 1 + (1 p) 1 α f I (t) bk σ p 1 ( 1 + ξ t σ σ ) ( 1 ξ 1) ( 1 + ξ t ) 1 ( ξ 1) σ t α bk t > α bk (2.5) Throughout this work, we use a fixed value of α bk. As a matter of fact, let τ = 2µs be the slot time, so α bk = CW ˆ τ. Assuming a low-load wireless environment we can put CW ˆ = CW min = 32. Thus we have a α bk.7ms. Although (2.5) shows an excellent fit with the Empirical Cumulative Distribution Function (E-CDF) f e (t) built on WLAN traffic samples [14], due to hardware limitations we consider that sensors can only observe a partial amount of the WLAN traffic, since their detection area is reasonably smaller than the whole WLAN area and they can rely only on their own local sensing processes. R CCA (γ) Sensor CCA Zone Figure 2.3: The Clear Channel Assessment Zone of a sensor node. Therefore, the local sensing process gives a Local View that is limited to a disk shaped zone (Figure 2.3), usually denoted as the Clear Channel Assessment (CCA) area A CCA with radius R CCA. Inside A CCA deterministic detection of WLAN user activity is practically assumed, but the model can be extended for probabilistic sensing in the future.

24 14 MODEL 2 f A (t) Active out CCA 1 f I (t) Idle p CCA 1 p CCA 1 f A (t) Active in CCA fĩ(t) Idle/ Active out CCA fã(t) Active in CCA Figure 2.4: The extended model for Local View. Consequently, we assume that each sensor estimates in own parameters its own local spectrum activity model (Figure 2.4) resulting in an extended threestate semi-markovian model, to account for the two different cases regarding the active state: WLAN user being detected by the sensor or unseen (missed) activity. Assuming that WLAN sources are statistically well-behaved, i.e. users have the same activity pattern, the branches can be weighted based on the percentage of those active WLAN users that lie inside or outside the A CCA of the considered sensor, p CCA and (1 p CCA ), respectively. Thus, we can derive the observable active and idle WLAN spectrum distributions, fã(t), fĩ(t) denoted as: fã(t) = f A (t), fĩ(t) = L 1 { f Ĩ (s)}, (2.6a) (2.6b) where f (s) includes a sequence of Idle and Active outside CCA periods, and Ĩ can be expressed easily by its Laplace transform: p CCA f (s) = Ĩ f I(s) (2.7) 1 (1 p CCA )f A (s)f I (s), and f A (s) = L {f A(t)}, f I (s) = L {f I(t)}. Our goal is to estimate the parameters of the distribution of the idle periods, thus in the following Chapters 3 and 4 we show both estimators for global and local view. The global case is applicable whenever a device could see all the WLAN activity. For the local case, instead, we focus on a non-cooperative approach, since the exchanging of information could increase the complexity or even be impractical due to the large number of sensors. Thus, the estimator relies only on the data provided by sensor s own sensing process.

25 15 3 G L O B A L V I E W E S T I M AT I O N The model for the Global View is defined in the Chapter 2 and it consists in a mixture distribution (2.5) that can be easily estimated with two different steps. Specifically, we have assumed that α bk is known, since it is given by the WLAN system parameters (Sec. 2). Then the parameters to be estimated are (ˆξ, ˆσ, ˆp) and in Section 3.1 we estimate before the parameters of the generalized Pareto distribution; once we have determined the values (ˆξ, ˆσ) for the heavy tail distribution, we proceed to estimate the ˆp value for the mixture in Section GENERALIZED PARETO The support of the uniform distribution f (CW) I (t) is limited to [, α bk ], so if we discard all observations x i [, α bk ], there is only a left-truncated generalized Pareto distribution and we can estimate its parameters directly [12]. In this way we split the search of the three parameters (ξ, σ, p) of the global view in two separated steps, each involves a reduced space where to search for the solution. Using an alternative parametrization with η = σ/ ξ, we can rewrite the PDF and the CDF as [5]: g [ξ,η] (t) = gpareto PDF [ξ,,σ](t) = 1 ( 1 + t ) (1+ξ) ξ ξ η η G [ξ,η] (t) = gpareto CDF [ξ,,σ](t) = 1 (3.1a) ( 1 + t η) 1 ξ (3.1b) Provided that we are dealing with a left-truncated set of samples, the Maximum Likelihood Estimation (MLE) [2, 27] of the parameters (ξ, η) is given by: (ˆξ, ˆη) = arg max (ξ,η) N i=1 g [ξ,η] (x i ) 1 G [ξ,η] (α bk ) (3.2) It is more convenient to work with the natural logarithm of the likelihood function. We assume that we are only dealing with ξ [,.5] for both practical

26 16 GENERALIZED PARETO 3.1 and theoretical reasons, since for ξ >.5 the generalized Pareto distribution has infinite variance and for ξ < the distribution has bounded support. Finally, we get: L(x ξ, η) = 1 N log N i=1 ( 1 ξ η ( = log (ξ η) 1 + ξ ξ N 1 + x i η 1 + α bk η N i= ) 1+ξ ξ ) 1 ξ = ( log 1 + x ) i + 1ξ ( η log 1 + α ) bk η (3.3) Now the derivative of l(ξ, η) with respect of ξ is: ( ) L(x ξ, η) N log 1 + α bk N ξ + ( ) N η i= log 1 + x i η = (3.4) ξ N ξ 2 Equating this derivative to zero, we get the following result, used for calculating ξ directly from η: ˆξ = ξ(η) ξ(η, s) = 1 N N i=1 ( log 1 + x ) ( i log 1 + α ) bk η η (3.5) So we can reduce the search space to one dimension, getting the profilelikelihood given by: L p (x η) = log (ξ(η) η) 1 + ξ(η) 1 ξ(η) N + 1 ( ξ(η) log 1 + α ) bk = η = log(ξ(η) η) 1 N N i=1 N i=1 ( log 1 + x ) i η ( log 1 + x ) i 1 (3.6) η And the solution that gives us the best estimation for η will be: ˆη = argmax L p (x η), (3.7) η when we can find the maximum numerically with the Newton s Method: The values of (ˆξ, ˆσ) will then be ˆη i+1 = ˆη i L p(x ˆη) L p(x ˆη) (3.8) ˆξ = ξ(ˆη) ˆσ = ˆη ˆξ (3.9a) (3.9b)

27 3.1 GENERALIZED PARETO 17 The derivative of (3.6) with respect η is: L p (x η) η = ξ (η) ξ(η) 1 η 1 N = ξ (η) ξ(η) 1 N N i=1 N x i i=1 η (1 2 + x i η ) = 1 η + x i (3.1) Where ξ (η) is the derivative of ξ(η) with respect of η, that is ξ(η) η = 1 N x i α bk + (3.11) N η 2 + η x i η 2 + η α bk i=1 The second derivative of (3.6) with respect η is: 2 L p (x η) η 2 = ξ (η) ξ(η) + ξ (η) 2 ξ(η) N N i=1 1 (η + x i ) 2 (3.12) Where ξ (η) is the second derivative of ξ(η) with respect of η, that is 2 ξ(η) = 1 N x i (2 η + x i ) η 2 N (η 2 + η x i ) α bk (2 η + α bk) (3.13) 2 (η 2 + η α bk ) 2 i=1 For faster convergence and higher accuracy, we can also approximate the starting point η from the Mean value evaluation, considering that for truncated distribution: b a E(X) = xf(x)dx F(b) F(a) = 1 N x i (3.14) N In our case: That gives us: 1 N N i=1 a = α bk b = f(x) = 1 ξη F(b) = 1 F(a) = 1 x i = η ξ α bk ξ 1 η (ξ) = i=1 ( 1 + x ) 1 ξ 1 η (3.15a) (3.15b) (3.15c) (3.15d) ( 1 + α ) 1 ξ bk (3.15e) η α bk + (ξ 1) 1 N ξ N i=1 x i (3.16)

28 18 MIXTURE DISTRIBUTION 3.2 The (3.16) is function of ξ which is bounded in [ :.5], so we can easily choose or randomize a ξ value in this interval and use η (ξ ) as starting point. 3.2 MIXTURE DISTRIBUTION Once we have estimated the (ˆξ, ˆσ) parameters for the generalized Pareto distribution, the search for the value of p of the mixture distribution is a task that can be accomplished in two different ways Maximum Likelihood Estimation Recalling the equation (2.5), then the MLE for p will be: log L(x p) = = = N log f I (x i ) = i=1 N [ ] log p f (CW) I (x i ) + (1 p) f (WS) I (x i ) = i=1 N i=1 [ p log + 1 p α bk σ ( 1 + ξ x i σ ) 1+ξ ] ξ (3.17) Using again the Newton s Method for finding the maximum point of the log-likelihood function, we get ˆp i+1 = ˆp i log L (x p) log L (x p) (3.18) Where the first derivative with respect of p is: log L(x p) p = N i=1 1 α bk 1 σ p α bk 1 p σ ( 1 + ξ x i σ ( 1 + ξ x i σ ) 1+ξ ξ ) 1+ξ ξ (3.19) And the second derivative with respect of p is: 2 log L(x p) 2 p = ( N ( i=1 1 α bk 1 σ p α bk 1 p σ ( 1 + ξ x i σ ( 1 + ξ x i σ ) 1+ξ ) 2 ξ ) 1+ξ ) 2 (3.2) ξ We could start with a first approximation ˆp from the samples mean evalu-

29 3.2 MIXTURE DISTRIBUTION 19 ation. E[f I ] = p E[f (CW) I ] + (1 p) E[f (WS) I ] E[f I] E[f (WS) ˆp = E[f (CW) I I ] ] E[f (WS) I ] (3.21) Where the means, considering also the truncation of the generalized Pareto distribution, are: N i= E[f I ] = x i N ] = α bk 2 E[f (CS) I E[f (WS) I ] = = αbk x g [ξ,σ] (x) G [ξ,σ] (α bk ) G [ξ,σ] () [ (x + σ)(1 + ξ x σ ) 1 ξ 1 (1 + ξ α bk σ ) 1 ξ ] αbk (3.22) Moment Evaluation In this case we build the estimation for ξ and σ values, and then we exploit the number of samples that have fallen in some specific ranges. Let ˆN T be the expected total number of samples that came from the generalized Pareto distribution: 1 ˆN T = N (αbk, ) (3.23) q (αbk, ) Where q (t1,t 2 ) is the probability to get a generalized Pareto sample in the range [t 1, t 2 ], that is: q (t1,t 2 ) = G [ξ,,σ] (t 2 ) G [ξ,,σ] (t 1 ) (3.24) Now, let ˆN (α,α bk ) be the expected number of generalized Pareto samples that fall in the range of [, α bk ], when we are dealing with a mixture distribution: ˆN (α,α bk ) = ˆN T N (αbk, ) 1 q (αbk, ) 1 q (,αbk ) (3.25) The value N (αbk, ) is the actual number of samples gathered in the interval [α bk, ). So now we can estimate the p value as the ratio of the difference between the actual number of samples in the range [, α bk ] and its expected value of

30 2 MIXTURE DISTRIBUTION 3.2 Generalized Pareto s, over the expected total number of Generalized Pareto samples plus the estimated number of Uniform Distribution samples, that is: ˆp = N (α,α bk ) ˆN (α,α bk ) ˆN T + N (α,α bk ) ˆN (α,α bk ) (3.26)

31 3.3 RESULTS RESULTS We tested the mixture distribution parameters estimation by using 1 randomized 3-parameters vector (ξ, σ, p). The parameters are generated following the rules in Table 3.1 and their trends are represented in Figure 3.1. Parameter Distribution Min Max Mean StdDev ξ Truncated Gaussian σ Truncated Gaussian 1e p Uniform Table 3.1: Parameters Randomization: Global View Randomization of ξ PDF Randomization of σ PDF Randomization of p PDF Value of ξ Value of σ Value of p Figure 3.1: Parameters Randomization: trends. Then from each vector we generate samples to: 1. Build the MLE for (ξ, σ) (3.1); 2. (a) Estimate p with MLE (3.2.1); (b) Estimate p through the MV (3.2.2). In Figures 3.2a and 3.2b we can see the Mean Percentage Error (MPE) for (ˆξ, ˆσ) calculated as: MPE + : MPE : 1 N pos 1 N neg N pos i=1 N neg i=1 a i f i a i if a i > f i (3.27) a i f i a i if a i < f i (3.28)

32 22 RESULTS 3.3 where N pos and N neg are the number of over and under estimations, respectively. Error for ξ evaluation Error for σ evaluation Mean Percetage Error % Fail ξ: MPE + ξ: MPE - Missed Estimation Mean Percetage Error % Fail σ: MPE + σ: MPE - Missed Estimation Number of samples Number of samples Figure 3.2: Global View: Estimation of (ξ, σ) with MLE. We have tested the generalized Pareto parameters estimation with 4-samples sets with a size of increasing order of magnitude. We have also counted the number of Missed Estimation, that is the times that Newton s Method with a precision of 1 1 returns always negative values for ˆη after choosing 6 different starting points of η (ξ ) with ξ uniformed distributed in [.5,.55]. The iteration through different starting points gives us a lower failure rate compared to a single run with ξ =.25 in (3.16) (Figure 3.3). It follows that the choice of the starting point could have a role in the process, but the single-run rate for larger vector s dataset shows a constant behaviour while the iterative rate shows a decreasing trend following the vector size. That means that keep repeating a failed test with new starting points will finally discard all errors related to that bad choice, leaving only those that come from the noise contained in the samples. The high error in Figure 3.2 that comes with the (ξ, σ)-estimation with 1 or 1 samples is justified by the lack of information contained in so few samples. As a matter of fact, plotted in Figure 3.4 there is the difference between CDFs of: Global Idle Distribution given by (2.5); Empirical Distribution built on the samples dataset; Estimated Global Idle Distribution. As it can be seen, the curve of the dataset CDF determines the accuracy of the estimation: the MLE will always try to fit as best as possible to that line, so

33 3.3 RESULTS 23 1 Single Run Different Starting Points (multiple runs) 8 6 % Fail Number of samples Figure 3.3: Global View: missed estimations of (ξ, σ) with single run or iteration on the starting point CDF(t).8.75 CDF(t) Real Estimated Empirical Real Estimated Empirical Time (t) Time (t) (a) Estimation with 1 samples (b) Estimation with 1 samples Figure 3.4: Global View: Differences between the CDFs of Real, Estimated and the samples dataset.

34 24 RESULTS 3.3 the estimation error is not caused by the estimation algorithm, but is due to the noise in the data acquired. Error for p evaluation with MLE Mean Absolute Error % Fail p: MAE + p: MAE Number of Samples Missed Estimation Figure 3.5: Global View: Estimation of p with MLE. Now we evaluate the accuracy of p parameter estimation. Figures 3.5 and 3.7 show the Mean Absolute Error (MAE) since now we are dealing with already normalized values in the interval [, 1]. Therefore, the MAE is: N MAE + 1 pos a i f i if a i > f i (3.29) N pos i=1 N MAE 1 neg a i f i if a i < f i (3.3) N neg i=1 The results depicted in Figure 3.5 show that MLE goes too often out of the range [, 1] and, even when it converges, the error is always high, around ±.3 of the real value. That is because the log-likelihood function for p-mle does not have a maximum in that range but is always increasing (Figure 3.6), that ensue from both generalized Pareto distribution and uniform distribution have a similar homogeneous increasing trend. Since they almost coincide, it is impossible for the estimator to tell, given the samples, the percentage p of influence of the uniform distribution. The MV method, instead, gives an high accuracy (Figure 3.7): it retrieves estimated values that differs only by ±.1 from the actual ones, despite it relies on the previous step as the MLE method. Hence, the errors ξ and σ are canceled when evaluating (3.23) and (3.25).

35 3.3 RESULTS log-likelihood(p) log-likelihood p value Figure 3.6: Global View: Log-Likelihood function. Error for p evaluation Mean Absolute Error % Fail p: MAE + p: MAE Number of Samples Missed Estimation Figure 3.7: Global View: Estimation of p with MV.

36 26 RESULTS 3.3 But, as we can see, the missed estimation value is always the same as for the generalized Pareto parameter s estimation because the failure of the Newton s method on η is propagated first to (ξ, σ) and then, without that parameters, it is impossible to estimate p with MV. Parameter estimation for the global view based on MLE for (ξ, σ) and MV for p gives accurate (Figures 3.2 and 3.7) and almost failure-free (Figure 3.3) results for datasets with 1 samples. Considering a WLAN traffic with an average inactive time of 1ms, it is about 1s of measurement process, which is acceptable. The complexity of the estimation process is bounded by the MLE function that is O(k n) which is acceptable too. The value O(k n) comes from the iteration of k times for the Newton s method and its point selection, and evaluating each time functions that are linear with the dataset size n.

37 27 4 L O C A L V I E W E S T I M AT I O N In this Chapter we address the problem of estimating the parameters for the local view model (Figure 2.4). Despite the similarity with the global view case, we do not have a closed form expression in the probability domain for (2.7). Thus, we could either try to use MLE in the probability domain, but also evaluating numerically the Laplace transform on each Newton s method step, that double the complexity. Or we could try to fit the functions directly in the Laplace domain, but this does not guarantee that the values found also maximize the likelihood in the probability domain. This is not the only difference with the global view case. In the formulation also f A (t) appears and, thus, we need to estimate α on and β as well. Thus, in the following sections we propose two different approaches for estimating the vector of parameters (ξ, σ, p, p CCA ) plus (α on, β) without dealing with the Laplace transform: analysis based on mixed MLE/MV estimation (Sec. 4.1) and employing neural networks (Sec. 4.2). In Section 4.3 we will prove that the first attempt would be unsuccessful compared to the neural network approach. 4.1 COMPOUND MLE/MV Direct Maximum Likelihood Estimation of the four parameters (ˆξ, ˆσ, ˆp, ˆp cca ) is impractical because we do not have the closed form in the probability domain. Instead of trying to estimate all parameters at the same time, we can exploit some property that comes with a version of mean-value analysis. As a matter of fact, given the (2.1) and (2.5) the minimum cycle length outside CCA (Fig. 2.4) is: But we can also assume that: min f A (t) > + min f I (t) > = α on + δ (4.1) t t α bk < α on, (4.2) because we can choose an Active traffic distribution that has the minimum greater than α bk. Indeed, constant rate UDP traffic of 512B has been measured in

38 28 COMPOUND MLE/MV 4.1 [11], resulting in length of.61ms for data packet and.2ms for the consecutive ACK, that is greater than the α bk value assumed in Sec. 2. So, in the time range [, α on ] we are sure that we are only dealing with a f I (t) (2.5) and if we filter also the samples with t > α bk, we finally get a truncated version of the f (WS) I (t) (2.4b): f (WS T ) I (t) = ( ξ α on σ ( ξ t ) 1 ξ 1 σ σ ) 1 ( ξ ξ α bk σ ) 1 ξ t (α bk, α on ] (4.3) We can exploit the (4.3) for building an algorithm that first finds (ˆξ, ˆσ) with samples in (α bk, α on ], then with samples in [, α on ] we can find p and finally with all the samples we can find p CCA : 1. Gather enough Idle/Active samples; 2. Estimate fã for retrieving α on and β; 3. Get (ˆξ, ˆσ) with MLE on interval samples (α bk, α on ]; 4. Evaluate ˆp in the interval sample [, α bk ] with MV; 5. Using (ˆξ, ˆσ, ˆp), calculate p CCA with MV estimation Active Distribution Since we are assuming that the f A (t) (2.1) is a Uniform Distribution in the range [α on, β] (Chapter 2), let us have a vector of samples x = (x (1), x (2),..., x (N) ), such that x (1) x (2)... x (N). (4.4) From (2.1) we have: α on x (1) β x (n) (4.5) So the likelihood function will be N L(x α on, β) = f A (x i α on, β) = (β α on ) N (4.6) But we also have: i=1 L(x α on, β) = α on > x (1) (4.7a) L(x α on, β) = β < x (N) (4.7b)

39 4.1 COMPOUND MLE/MV 29 Putting together (4.6) and (4.7), we get: { (β α on ) N α on x (1) and β x (N) L(x α on, β) = otherwise (4.8) The log-likelihood function will be: log L(x α on, β) = log ( (β α on ) N) = N log(β α on ) (4.9) And the partial derivative of (4.9) with respect α on yields log L(x α on, β) α on = N β α on > (4.1) So for α on x (1) and β x (N) the function log L(x α on, β) is a strictly increasing function in the variable α on. Hence, from (4.9) and (4.1) follows that the maximum likelihood estimator for α on is given by ˆα on = x (1) (4.11) Taking instead the partial derivative (4.9) with respect β gives log L(x α on, β) β = N β α on < (4.12) And we can conclude that for α on x (1) and β x (N) the log-likelihood function is a strictly decreasing function in the variable β. Hence, from (4.9) and (4.12) it follows that the MLE for β is given by ˆβ = x (N) (4.13) The value of α on is the left boundary of the distribution just estimated according to (4.11) Generalized Pareto Parameters The maximum likelihood estimator for the truncated generalized Pareto distribution f (WS T ) I (t) in the interval (α bk, α on ] will be: (ˆξ, ˆσ) = argmax ξ,σ N i=1 g [ξ,σ] (x i ) G [ξ,σ] (α on ) G [ξ,σ] (α bk ) (4.14)

40 3 COMPOUND MLE/MV 4.1 And the full log-likelihood function will be: L(x ξ, σ) = 1 N ( ) N log 1 σ 1 + ξ x i 1 ξ 1 σ i=1 1 ( ) 1 + ξ α on 1 ξ 1 + ( ) 1 + ξ α bk 1 = ξ σ σ [ ( = log 1 + ξ α ) 1 bk ξ ( 1 + ξ α ) ] 1 on ξ σ σ log σ 1 + ξ N ( log 1 + ξ x ) i ξ N σ i=1 (4.15) Now we can solve numerically the non linear system: That is k {on,bk} ( 1 + ξ α k L(x ξ, σ) ξ L(x ξ, σ) σ σ ) 1 ξ ( 1 + ξ α bk ) 1 ξ ( 1 + ξ α on ( σ σ 1 N ( log 1 + ξ x ) i + (1 + ξ) ξ N σ i= ξ N ( log 1 + ξ x ) i = ξ N σ i=1 = = ( ( ) log 1 + ξ αk σ ) 1 ξ ξ 2 N i=1 x i σ ( 1 + ξ x i σ ) α k ξ σ ( 1 + ξ α k ) (4.16) σ ) ) ( 1 + ξ α bk ) 1 ξ α σ bk σ ( ξ α bk σ ( 1 + ξ α bk σ ) ( 1 + ξ α on ) 1 ξ α σ on σ ( ) ξ α on ) 1 ξ ( 1 + ξ α on σ σ ) 1 1 ξ σ ξ ξ N N i=1 ξ x i σ 2 ( 1 + ξ x i σ ) = (4.17) And apply the Second Partial Derivative test, checking if the determinant of 2 L(x ξ, σ) 2 L(x ξ, σ) M(ξ, σ) = ξ 2 ξ σ 2 L(x ξ, σ) 2 L(x ξ, σ) (4.18) σ ξ σ 2 is positive while 2 L(x ξ,σ) ξ 2 <. The Newton s Method for non linear equation is: x k+1 = x k J 1 k f( x k ), (4.19)

41 4.1 COMPOUND MLE/MV 31 where J is the Jacobian matrix. Instead of using the computational-expensive inversion of the Jacobian matrix, we can write the correction step s k as: x k+1 = x k + s k (4.2) J k s k = f( x k ) (4.21) The first step of the method is to evaluate f( x k ), that in our case is: L(x ξ k, σ k ) = L(x ξ k, σ k ) ξ L(x ξ k, σ k ) σ The second step is to evaluate the Jacobian matrix 2 L(x ξ k, σ k ) 2 L(x ξ k, σ k ) J(x ξ k, σ k ) = 2 ξ ξ σ 2 L(x ξ k, σ k ) 2 L(x ξ k, σ k ) ξ σ 2 σ Finally we can solve the system: (4.22) (4.23) J(x ξ k, σ k ) (ξ k, σ k ) = L(x ξ k, σ k ) (4.24) That is: ( Lξξ L ξσ L σξ L σσ ) ( ξ σ ) ( Lξ = L σ Or: { L ξξ ξ + L ξσ σ + L ξ = L σξ ξ + L σσ σ + L σ = Solving (4.26) for ξ and σ, we get ξ σ = = L ξσ σ L ξ L ξξ L ξ L ξξ L σ L σσ L2 σξ L ξξ ) (4.25) (4.26) (4.27) And finally: ξ k+1 = ξ k + ξ (4.28) σ k+1 = σ k + σ (4.29) We can stop the process if both ξ < ɛ and σ < ɛ.

42 32 COMPOUND MLE/MV Mixture Distribution For the estimation of p we can exploit the same considerations as those explained in Sec Because of the weak results on Maximum Likelihood Estimation of p for the global view (Sec. 3.3), we choose not to use this method but to apply only the Moment Method. Recalling equation (4.3), we have pure samples from the generalized Pareto distribution only in the interval [α bk, α on ]. Thus, the equations (3.23) and (3.25) need to be changed accordingly. So, in (3.23) instead of evaluating the expected total number N T of samples that came from the Pareto in the whole interval [α on, ), it will be: ˆN T = N (αbk,α on ) where q (t1,t 2 ) is defined in (3.24), that is: 1 q (αbk,α on ), (4.3) q (t1,t 2 ) = G [ξ,,σ] (t 2 ) G [ξ,,σ] (t 1 ) (4.31) And for (3.25), the expected number ˆN (α,α bk ) of generalized Pareto samples that falls in the range of [, α bk ] will be: ˆN (α,α bk ) = ˆN T N (αbk,α on ) 1 q (αbk,α on ) 1 q (,αbk ) (4.32) The p estimation follows the same equation in 3.26 and once again it is: ˆp = N (α,α bk ) ˆN (,αbk ) ˆN T + N (,αbk ) ˆN (,αbk ) (4.33) Percentage of Observable Load Since the closed form for the complete idle PDF can be expressed only in the Laplace domain and the MLE works in the Probability domain, we should use numerical approximation of the inverse transformation formula on each step. Thus, the complexity of the computation becomes unaffordable for the hardware-constrained devices. So we apply only the less accurate moment evaluation method. The Expected Value of the complete (local) idle spectrum period is: E[fĨ] = (E[f I ] + k E[f I + f A ]) (1 p CCA ) k p CCA = k= = E[f I ] + E[f I + f A ] 1 p CCA p CCA (4.34)

43 4.2 NEURAL NETWORK 33 Thus, the p CCA will be: E[f I + f A ] ˆp CCA = E[fĨ] E[f I ] + E[f I + f A ] = E[f I] + E[f A ] E[fĨ] + E[f A ] (4.35) Where the sample mean, pareto mean and active mean are respectively: E[fĨ] = 1 N N i=1 x i E[f I ] = p α bk + 2 E[f A ] = α on + β 2 (1 p) σ 1 ξ (4.36a) (4.36b) (4.36c) Putting together (4.35) and (4.36), we finally get: ˆp CCA = p α bk 2 1 N (1 p) σ + 1 ξ N i=1 + α on + β 2 x i + α on + β 2 (4.37) 4.2 NEURAL NETWORK As we will explain in the Section the analytic way of estimating the four parameters vector could be impractical due to the complex dimensional space and its exploration through different steps, where the support datasets are difficult to generate and errors can be easily propagated. Therefore, we propose also another approach with Neural Networks [16, 18]. The output layer is chosen to have the four-ordered vector of (ˆξ, ˆσ, ˆp, ˆp CCA ) as output, since we could estimate f A (t) and its parameter with the method proposed in Section The input layer on the other hand needs to be structured but also be small, so we get a simpler network that can be managed by memory constrained devices. At the same time the small input vector needs to contain as much information as possible about the idle times. Thus, the number of inputs should be decoupled from the number of acquired samples, since the more samples the sensor gathers the more is the information gained. A way to guarantee the decoupling is to build an empirical distribution from the dataset and choose as neural network s input a subset of points from its PDF or its CDF.

44 34 NEURAL NETWORK 4.2 The training process is memory and CPU intensive and should be clearly done off-line. Accordingly, we should produce a neural network to be saved on each sensor (using only 1 Kb), that is capable of generalizing every single case, not only the combination of the four parameters vector but also to account other hidden variables (the local view contains, indeed, the Active Distribution so it is related to α on and β, too). Those values could be part of the input vector since they are either known or estimated, but for increasing the accuracy we can also include other sources of information like the mean (µ) and the variance (σ 2 ) of gathered samples coming from fĩ(t). In Figure 4.1 we provide an example of architecture for a generic Neural Network that we used for the results, with all the four parameters (α on, β, µ, σ 2 ) in addition to k points of the Empirical Distribution. Input layer Hidden layer Output layer α on β µ σ 2 x ˆξ ˆσ ˆp ˆp CCA x i... x k Figure 4.1: Neural Network Point Selection We mentioned above that a vector of samples should not be directly used as input of neural network because it is not structured and the information is better coded in the PDF domain. Both problems can be easily resolved by selecting a fixed series of points in the time domain from which retrieve the CDF or a