Congestion-adaptive Data Collection with Accuracy Guarantee in Cyber-Physical Systems

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1 Congestion-adaptive Data Collection with Accracy Garantee in Cyber-Physical Systems Nematollah Iri, Lei Y, Haiying Shen, Gregori Calfield Department of Electrical and Compter Engineering, Clemson University, USA Department of Compter Science, Georgia State University, USA {niri, gcalfi, Abstract Data collection by wireless sensor networks is a fndamental and critical fnction for cyber-physical systems (CPS) to estimate the state of the physical world. However, nstable network conditions impose great challenges in garanteeing data accracy, which is essential for reliable state estimation of the physical phenomena. For nderlying sensor networks, withot efficiently resolving congestion in data transmission, packet loss at congested nodes can considerably increase the estimation error. However, previos congestion control schemes relying on redcing transmitted data samples also increase the estimation error. Ths, we propose a Congestion-Adaptive Data Collection scheme (CADC) to efficiently resolve the network congestion while garanteeing the overall data estimation accracy. CADC mitigates congestion by adaptive lossy compression with garantee that a given overall data estimation error bond is satisfied. Besides, since a CPS application may have different priorities for different data items, we frther propose a weighted CADC scheme sch that the data with higher priority has less distortion. Extensive experimental reslts demonstrate the effectiveness and efficiency of or CADC schemes. I. INTRODUCTION Wireless Sensor Networks (WSNs) enable the sensing of physical phenomena in a large scale and become fndamental infrastrctres in cyber-physical systems (CPS). The sensor nodes in a WSN sample the physical phenomena sch as temperatre and light, and transmit the data to the base station (or controllers) for the state estimation of physical phenomena. Sch data collection, however, faces great challenges from nstable network conditions. A WSN sally consists of hndreds to thosands of sensor nodes, which generate a tremendos amont of sensed data and deliver it to the base s- tation sing mlti-hop wireless transmission. The large amont of data and nstable wireless links easily lead to network congestion, which incrs sbstantial packet loss. The packet loss can significantly increase the estimation error of CPS, thogh the controllers reqire garanteed estimation accracy of the physical phenomena for reliable control decisions. To avoid network congestion, many congestion control schemes for WSNs have been proposed. Most schemes [1] [5] are based on rate control, which redces the data generation rate at sorce nodes or compresses the spatio-temporal samples at the intermediate relay nodes. However, by redcing data samples to be transmitted to avoid congestion, these schemes also concrrently increase the estimation error. Therefore, a congestion control scheme shold work arond a sweet spot that avoids the congestion while still garantees the estimation accracy of applications. In this paper, we consider the problem of ensring reqired estimation accracy while redcing congestion. We assme nodes transmit data pwards to the sink throgh a roting tree [5]. We propose a Congestion-Adaptive Data Collection scheme (CADC), which determines the maximm tolerable distortion of data de to compression at each node to garantee a given estimation accracy at the sink. In order to redce data distortion by compression, each node novelly ses the k- means clstering algorithm for lossy data compression with its maximm tolerable distortion bond. When congestion occrs, by adaptively adjsting the maximm tolerable distortion allowed at sensor nodes, CADC makes best effort to garantee the given estimation accracy at the sink and redce the congestion. Besides, we also consider the different priorities of data measrements. A CPS application may have different priorities for data items in different vale ranges. For example, the safety monitoring system may be more interested in high temperatre readings, ths the temperatre measrements with higher vales are more important and shold have lower distortion hence compression degree. To this end, we propose weighted CADC scheme, which assigns weights to the measrements according to their priorities and aims to minimize the weighted estimation error. We condct extensive simlations to evalate or CADC schemes in comparison with previos schemes. Experimental reslts demonstrate the high effectiveness and efficiency of or schemes. The rest of paper is organized as follows. Section II smmarizes the related work. Section III illstrates or system model and objective. Section IV presents or congestionadaptive data collection schemes in detail. Section V presents the performance evalation of or schemes in comparison with previos methods. Section VI concldes this paper with remarks on or ftre work. II. RELATED WORK We present previos works for data collection to the sink in WSNs in three categories: data compression, roting and congestion control. Data compression. Many works have exploited data correlation to compress data in transmission to redce commnication cost. Cristesc et al. [6] tilized the Slepian-Wolf coding to compress correlated readings and addressed the problem of finding the optimal rate allocation for each node to minimize total data transmission cost. Silberstein et al. [7] proposed CONCH, which exploits the spatio-temporal data

2 correlation to sppress nnecessary vale transmissions in continos data collection to redce energy cost. Lo et al. [8] proposed to apply compressive sampling theory to sensor data gathering to redce global scale commnication cost. Gpta et al. [9] proposed to select a small sbset of sensor nodes that may be sfficient to reconstrct data for the entire sensor network within predefined error bond. Wang et al. [1] proposed an approximate data collection, in which the network is partitioned into clsters, and clster heads constrct the local estimation model with prespecified error bonds to approximate the readings of sensor nodes in the clsters. The sink then estimates the data based on the model parameters sent by clster heads. Roting. Many roting protocols have been proposed for data collection to redce energy cost or latency. Chang et al. [11] addressed the optimal roting problem with the objective to maximize the network lifetime. Park et al. [12] developed an online heristic for the problem of roting message to maximize the network lifetime. Lee et al. [13] proposed a collision-free schedling method for data collection roting to optimize energy consmption and reliability. Han et al. [14] addressed the problem of minimizing the expected total transmission power for reliable data dissemination in dtycycled WSNs. S et al. [15] aimed at achieving optimal rate allocation for data aggregation in WSNs with the goal to maximize resorce tilization and proposed a distribted algorithm for joint rate control and schedling. Wan et al. [16] stdied the problem of finding the minimm-latency schedle for data aggregation in wireless networks nder the interference constraint. Control congestion. The previos control congestion schemes can be classified into two classes: centralized rate control schemes and distribted rate control schemes. Eventto-Sink Reliable Transport (ESRT) [1] lets the base station adjst the reporting freqency of sensor nodes sch that the reqired information can be obtained with minimm energy considering one-hop commnication between nodes and the base station. Bian et al. [17] proposed a centralized rate allocation scheme that assigns sending rates to all sensors in the roting tree based on the wireless link characteristics. Zho et al. [2] proposed a sorce reporting rate control mechanism (PORT), which is aware of transmission cost of the sorces, and adjsts the sorce reporting rates with a garantee that the sink can still obtain enogh information. Paek et al. [3] proposed the rate controlled reliable transport protocol (RCRT), where the sink is responsible for congestion detection and rate allocation of sensor nodes based on AIMD (Additive Increase - Mltiplicative Decrease). CODA [4] is a distribted rate control scheme for congestion avoidance. In these rate control based schemes, since decreasing data rate redces the nmber of spatio-temporal samples, they cannot control the accracy of the state estimation. To address this problem, Ahmadi et al. [5] takes into accont the estimation error in the congestion control. Using least-error smmarization, their scheme eliminates congestion while incrs the least possible overall error in sensing the physical environment. In this paper, we consider the congestion isse in data collection and aim to design a congestion-adaptive data collection scheme for WSNs with estimation error bond. Or work is most related to [5]. However, the scheme in [5] is naware of the accracy reqirements of applications, and the data collection with sch congestion control scheme may fail to achieve the reqired data accracy. Instead, or scheme aims to ensre the pre-specified error bond when congestion occrs. III. SYSTEM MODEL AND OBJECTIVE A. System Model We assme a WSN for data collection, in which N sensor nodes are deployed to monitor a physical phenomenon of the environment and periodically send their sensor readings to a sink. De to commnication limitations of the sensor nodes, they transmit their sensing data in a mlti-hop fashion to the sink (denoted by r), which is responsible for collecting and processing the measrements. As shown in Figre 1, we assme a roting tree rooted at the sink as or network layer [18], [19], denoted by T r. The depth of a sensor node i is defined as the hop distance between node i and the sink, denoted by h i,r. Node i is the ancestor of node j if j is in the sbtree rooted at i (denoted by T i ). 1 2 n Fig. 1: Roting tree. TABLE I: Notations Parameter Description T i Sbtree of the roting tree rooted at node i r Sink node, i,i Notations for sensor nodes other than the sink x i Data generated at sensor node i ˆx i Vale of x i reconstrcted by i s ancestor based on the received compressed data e Sm of the errors between received vales of data at sensor node and their actal vales ɛ Maximm tolerable error at node (pper bond for e ) d Sm of the errors between received data at node and their vales after compression at node Maximm tolerable distortion of data de to compression at node (pper bond for d ) w i Priority coefficient of x i generated by node i To describe or scheme, we first assme that network tree topology is fixed. We will discss how or scheme adapts to network topology changes in Section IV-E. Data is forwarded along T r to the sink. Each node periodically sends its measred data and also forwards its received data from children to its parent. Given this system model, congestion occrs when a n- ode cannot transmit data messages at the rate they are received and generated cased by insfficient bottleneck resorce [5]. One of the main components in congestion control schemes is congestion detection. For this prpose, we can se a previosly proposed congestion detection scheme [5]. That is, a node compares its otpt bffer size with a threshold, and it is congested if its bffer size is higher than a threshold. B. Motivation and Objective Data qality is critical for the controllers to accrately estimate the state of the physical phenomenon. However, r

3 congestion control has the two-sided inflence on the data accracy: (1) congestion elimination redces data loss and improves the estimation accracy, (2) bt the ways to control congestion sch as redcing sorce rate [1] [4], [17] and aggregation [5] increase the estimation error. Ths, the design of congestion control scheme in data-collection networks needs to achieve the trade-off between the two sides while ensres that the estimation error reslting from collected data is within the tolerable range of CPS applications. With the above motivation, we propose the Congestion- Adaptive Data Collection scheme (CADC), which redces congestion by redcing the data transmission rate with lossy compression, while still garanteeing the data accracy reqired by CPS applications. In CADC, when congestion occrs at a node, to redce the congestion, its children nodes redce their data transmission rates by lossy compression on the data to be forwarded, which however cases data distortion. Formally, we denote the measrement of sensor node i as x i, and denote the vale of x i reconstrcted by i s ancestor based on the received compressed data as ˆx i, which may not eqal to x i de to compression. We define the estimation error and data distortion as follows: Definition 3.1: (ESTIMATION ERROR) Estimation error (error in short) at node represents the sm of errors between its received data vales from its sbtree T and their actal vales, i.e., e = (ˆx. (1) i T T denotes the nmber of sensors in T. Definition 3.2: (DATA DISTORTION) Data distortion at node k represents the sm of errors between its received data vales from its sbtree T k and their corresponding vales after compression received by its parent, i.e., d k = (ˆx i ˆx k i ) 2. (2) i T k The data accracy reqirement of a CPS application is characterized by the maximm tolerable estimation error at the sink node r, denoted by ɛ r. Objective: Or objective is to avoid congestion while ensring that the reslting estimation error at sink r, denoted by e r,is less than ɛ r, i.e., e r ɛ r. IV. CONGESTION-ADAPTIVE DATA COLLECTION A. CADC Scheme Overview To achieve the above-stated objective, CADC determines two parameters for each node, maximm tolerable error (ɛ ) and maximm tolerable distortion ( ). ɛ and are the pper bonds for estimation error e and data distortion d at node (defined by Definitions 3.1 and 3.2), respectively. That is, e ɛ, d. (3) Consider a node and its children 1,..., n in the roting tree (Figre 1). In CADC, node ses ɛ to determine k of each of its children ( k ). Node k compresses its data based on k to redce its data transmission rate for congestion control. The vale of k for each child ensres e ɛ. Finally, the estimation error at the sink is no more than the fixed maximm tolerable estimation error at the sink (e r ɛ r ), which means that CADC helps to satisfy the constraint of the desired data accracy of CPS applications. CADC dynamically and distribtedly determines proper vales of maximm tolerable error (ɛ ) and maximm tolerable distortion ( ) for every node based on the network stats and a given ɛ r. Dring data collection, CADC first determines the initial vales of ɛ and for each node, and then dynamically pdates them based on the crrent network congestion stats and redce the congestion accordingly. If a node is congested, it asks each child i to transmit data in a lower rate to avoid congestion throgh data compression. Bt if the data compression with sch a lower rate incrs data distortion larger than i, node attempts to increase i to accommodate sch compression withot violating the constraint of maximm tolerable error ɛ. Only if there is no way to achieve the desired i while satisfy ɛ, reqests its parent to pdate ɛ. Sch parameter pdate cold repeat along the path to the sink to make the error at the sink less than the given error bond ɛ r. This case means that the overall system is highly congested and e r ɛ r cannot be satisfied in any way. Then, the sink will inform the applications of the off-specification of data. In the following, we present the details of the three parts of CADC. Given ɛ r, how to determine the maximm tolerable error (ɛ ) and distortion ( ) for every node to realize or objective (Section IV-B)? How can a node compress its data based on its k while minimizing the data distortion (Section IV-C)? How to condct congestion control and pdate ɛ and to achieve or objective in dynamic network stats (Section IV-D)? How to adapt to dynamic network topology (Section IV-E)? B. Determination of Maximm Tolerable Error and Distortion As we can see, in CADC, a fndamental problem is: Given maximm tolerable error ɛ at any node and crrent network stats, how to determine maximm tolerable errors (ɛ k ) and maximm tolerable distortions ( k ) for s children, 1,..., n, sch that e ɛ. After the problem soltion is fond, given a ɛ r on the sink, the (ɛ, ) of each of its children can be determined. Then, the (ɛ k, k ) of each of s children are determined and so on. Finally, the (ɛ i, i ) of each node i are determined in the top-bottom manner to achieve or objective. In this section, we address this problem in two cases: Non-priority case, in which all of the sensor measrements are eqally important for an application (Section IV-B1). Priority case, in which the measrements have different priorities (Section IV-B2).

4 1) Non-Priority Case: The estimation error e at node eqals the accmlated errors from each of its children 1,..., k,... n : e = (ˆx i x i) 2 i T = (ˆx i x i) (ˆx i x i) 2 i T 1 i T n The error contribtion of child k, denoted by c k, eqals i T k (ˆx i x i) 2. Based on the definition of c k,wese Cachy-Schwartz ineqality to get: c k = (ˆx i T k = ((ˆx i ˆx k i )+(ˆx k i x i )) 2 i T k = (ˆx i ˆx k i ) 2 + (ˆx k i T k i T k + 2 (ˆx i ˆx k i )(ˆx k i x i ) i T k (ˆx i ˆx k i ) 2 + (ˆx k i T k i T k + 2 (ˆx i ˆxk i ) 2. (ˆx k i T k i T k c k k : k is child of = d k + e k +2 d k e k (4) where e k is the estimation error at child k and d k is data distortion de to data compression of k. As the maximm tolerable error (ɛ k ) and maximm tolerable distortion ( k ) are the pper bonds of e k and d k, respectively, we define maximm tolerable error contribtion of k : c m k = k + ɛ k +2 k ɛ k. (5) To garantee e = ɛ, the determination of k and ɛ k needs to ensre ɛ ( k +ɛ k +2 k ɛ k ) ɛ k c m k k : k is child of (6) In this way, since d k k and e k ɛ k, we can achieve that e = k c k ɛ. As a reslt, Formla (6) gives the principle to initialize and pdate parameters (ɛ, ) for each node. We present the parameter initialization below, and present the parameter pdate in CADC s congestion control in Section IV-D. Initialization of ɛ k and k : With a priori knowledge of network congestion stats, we can properly initialize the maximm tolerable error and distortion (ɛ k, k ) for each node k. In the rooting tree for data collection, a sbtree with a larger size tend to sffer more congestions becase it needs to forward a larger amont of data to the sink. As a reslt, a larger sbtree may introdce higher estimation error into the data to the pper node de to CADC s lossy compression. Ths, the root of a larger sbtree needs a larger maximm tolerable error to allow more data compression within the sbtree to mitigate the congestions. Based on this rationale, node initializes the (ɛ k, k ) for each of its children k according to the size of each child s sbtree. Based on Formla (6), to garantee e ɛ,welet k + ɛ k +2 k ɛ k = α k ɛ ( <α k < 1) (7) where k=1,...,n α k =1sch that e k ( k +ɛ k +2 k ɛ k )= We choose α k by k=1,...,n T k α k = k=1,...,n T k α k ɛ = ɛ. (8) where T k is the size of sbtree T k, sch that any node with a larger sbtree size can have a higher maximm tolerable error. To find sbtree sizes T k in Eqation (9), we se the same procedre as in [18]. Particlarly, each sensor node sends its sbtree size in its packet header. Each parent node sms p sbtree sizes of its children and adds one to it to find its own sbtree size, with sbtree size of leaf nodes being 1. After α k (hence α k ɛ ) is determined, based on Eqation (9), node needs to determine k and ɛ k to satisfy Eqation (7). In order to maximize the estimation accracy, we let every node send raw data withot data compression initially, i.e., k =. Later on, CADC adjsts k to avoid congestion when it occrs. With k = initially, from Formla (7), we have ɛ k = α k ɛ. In CADC s congestion control (Section IV-D), when congestion occrs at node, if e ɛ still can be satisfied by data compression for congestion control, does not need to pdate and only k needs to pdate. Therefore, setting ɛ k to the possible maximm vale (ɛ k = α k ɛ ) can avoid freqent pdates later on. As a reslt, we find a soltion for the problem indicated at the beginning of this section. Using this soltion, given a ɛ r at the sink, CADC can determine the (ɛ, ) of each node in the system in the top-down matter to garantee e r ɛ r. 2) Priority Case: In this section, we consider the scenario in which the data has different priorities. For example, for a fire detection or cooling application, high temperatre vales, which may indicate abnormality, have higher priority than low temperatre vales. High-priority data shold sffer less distortion, so that the event can be more accrately modeled and qickly detected. We se priority coefficients to show the importance degree of different data items. We assme that the priority coefficient is a fnction of data vale, which is known to all sensor nodes. Approximate vales will have the same or close priority coefficients. Then, when a sensor receives a data vale, it determines its priority coefficient based on the priority fnction and the data vale. We need to determine maximm tolerable error and distortion with the goal that the higher-priority data has less estimation error in order to achieve more accrate state estimation for CPS control. If priority coefficients are eqal for all data, the problem is redced to the previos non-priority case. (9)

5 We define weighted estimation error and weighted data distortion below with the consideration of data priority. e w = i T w i (ˆx i x i) 2, (1) d w k = i T k w i (ˆx i ˆx k i ) 2, (11) where x i denotes the data measred by node i with priority w i, ˆx i denotes the vale of x i received by node, and k is a child of. Accordingly, we define weighted error contribtion of s child node k as c w k = i T k w i (ˆx i x i) 2. Similarly, we have c w k = w i ((ˆx i ˆx k i )+(ˆx k i x i )) 2 i T k = w i (ˆx i ˆx k i ) 2 + w i (ˆx k i T k i T k + 2 w i (ˆx i ˆx k i )(ˆx k i x i ) i T k w i ((ˆx i ˆx k i ) 2 + w i (ˆx k i T k i T k + 2. w i (ˆx i ˆxk i ) 2 w i (ˆx k i T k i T k = d w k + e w k +2 d w k e w k (12) Eqation (12) is derived with the assmption that the priority coefficient of data x i at parent and child k remains the same in CADC. This is reasonable becase the compression method in CADC (Section IV-C) constrains the distortion of data in compression, and the data will most likely have the same or close priority coefficient after compression, which is confirmed in or experiments in Section V. As we can see from Formla (12), it has the same form as the non-priority case. Ths, in the priority case, we can se the same principle for determining the (weighted) maximm tolerable error and distortion, and choose the same initial vales. C. Data Compression Scheme To redce the congestion, the nodes compress the received data based on their maximm tolerable distortion ( ) before transmitting the data to their parents. Sensor readings may be redndant becase nodes in the same neighborhood can have approximate readings in WSNs. Unlike the previos compression methods that do not focs on minimizing data distortion in compression, or data compression scheme aims to select most representative data samples that minimize the data distortion. Accordingly, we se the k-means clstering algorithm (k-means in short) [2] for data compression. Given a set of data points V in real d-dimensional space R d and an integer k, k-means clstering is to partition the points into k clsters; each with a center (i.e., clster head) not necessarily belonging to the set of points, with the goal of minimizing the mean sqared distances of each point to its nearest clster head. Formally, it is to minimize C(V )= x V (x c(x)) 2, (13) where C(V ) is the cost of clstering and c(x) is the center of the clster that data x belongs to. C(V ) actally reflects the data distortion. Ths, in CADC, to compress the data, a node condcts the k-means clstering on its received and generated data and sends the vales of clster heads and corresponding clster sizes to its parent. CADC represents data in the form of tples < (v 1,n 1 ),...,(v i,n i ),...,(v m,n m ) >, where v i is the sample vale, and n i is the nmber of sensor readings (each from a sensor node) with vale v i. n =1if the data represents a single sensor reading. For example, if a node receives a set of sensor readings {(3, 1), (4, 1), (6, 1), (8, 1), (1, 1), (12, 1)} from 6 nodes, with 2-means clstering, this dataset is partitioned to two clsters {3, 4, 6} and {8, 1, 12}, with centers eqal to 4.33 and 1, respectively. Then, the compressed dataset is represented by {(4.33, 3), (1, 3)}. In order to apply the k-means clstering method to the priority case, we modify the cost fnction C(V ) for k-means clstering to C(V )= w(x)(x c(x)) 2, (14) x V where w(x) is the priority coefficient of data x. Ths, data with higher priority will have less distortion. In the congestion control (Section IV-D), CADC ses the k-means clstering algorithm for data compression throgh two methods nder the constraint that the data distortion after compression (i.e., cost of clstering) C(V ) is less than a given bond. In the first method, a node needs to redce the available data into k samples with a given vale of k. For this prpose, we can directly se an existing k-means clstering algorithm sch as Lloyd s algorithm [2]. In the second method, a node needs to find minimm k for data compression. For this prpose, we can simply enmerate all possible vales of k from 1 to the total nmber of data points. For each vale of k, we se Lloyd s algorithm to find k clsters and the cost of clstering. Once the cost of clstering becomes no more than the given bond, the algorithm retrns crrent k and clster heads. D. Congestion Control In this section, we introdce the procedre of congestion control in CADC, inclding adaptive adjstments of the maximm tolerable error and distortion (ɛ, ), and the corresponding congestion control. Consider an arbitrary node and its children 1,..., n with maximm tolerable error and distortion, ɛ i and i (i = 1,..., n). When node is congested, to avoid the congestion, needs to redce its inpt data arrival rate r in to less than its otpt transmission rate r o by asking its children to redce their transmission rates throgh data compression. When r in <r, o node s bffer size will decrease and the congestion can be resolved finally. Node first comptes the ratio of r in to r, o r in r, and then sends a congestion o notification with this ratio to each of its children. Since r in is the sm of data transmission rates of all s children, decreasing each child s crrent transmission rate by a factor of at least rin r o can redce r in to less than r o. To do this, each

6 child decreases its crrent data compression ratio by r in /r o times. The data compression ratio is the ratio of the nmber of compressed samples to the nmber of available data tples to be transferred. The compression ratio is 1 if the node sends data withot compression. We se γ i and γ i to denote the previos and new data compression ratio of node i : γ i = γ i / rin r. For the o dataset of available data tples to be forwarded at node i < (v 1,n 1 ),...,(v i,n i ),...,(v m,n m ) >, we se N i = m j=1 n j to denote the total nmber of data readings in the dataset. Node i compresses data with compress ratio γ i by sing the k-means clstering with k = N i γ i (Section IV-C). Sch data compression with γ i will lead to data distortion (denoted by d γi ) compted by Formlas (13) and (14) in the non-priority and priority cases, respectively. Recall that to ensre e r ɛ r, node i has maximm tolerable distortion, i, defined over all the T i data readings from its sbtree T i. Ths, node i needs to compress data with distortion not exceeding N i i T i, where i T i is the average maximm distortion allowed for each data reading from each node in the sbtree T i. N i i T i means the distortion allowed for the dataset with N i readings. Then, if d γi N i i T i, i jst sends the compressed samples to the parent. If d γi which means that the compression ratio reqired to redce the congestion cannot satisfy the distortion constraint hence e r ɛ r, the parameters (ɛ, ) for congestion control then mst be pdated. Next, we explain how to pdate parameters to ensre e r ɛ r while redce congestion in this case. >n i i T i, When d γi > N i i T i, node i tries to compress data as mch as possible with data distortion not exceeding data distortion constraint N i i T i by finding the minimm nmber of clster heads for k-means clstering nder the constraint (Section IV-C). This data compression makes data distortion smaller than d γi, so the compression ratio is still larger than γ i reqired to avoid congestion. Sch data compression can mitigate congestion bt cannot eliminate it. In order to avoid the sbseqent congestions, i.e., achieve γ i, i reqests its parent to increase its maximm tolerable distortion ( i ) sch that d γi N i i T i. In this case, T i d i γi N i. To avoid freqent sch reqests and parameter pdates, i can be set to the historically largest vale. To do that, each node maintains two parameters: maximm necessary distortion ( i ) and maximm necessary error (ɛ i ). i keeps track of the maximm distortion reqired to remove congestion within a fixed time window. If no congestion occrs in a time window, i =. ɛ i is derived based on Formla (4) based on i.givend γi, node i comptes its i and ɛ i : i (t γi ) = max { T t γi w t t i (t),d i γi N i } (15) γi ɛ i = ik ( ik + ɛ ik +2 ik ɛ ik ) (16) where t γi is the crrent time, w is the time window, and ik + ɛ ik +2 ik ɛ ik is the pper bond of ik s error contribtion c ik according to Formla (4). Node i asks its parent to pdate its i and ɛ i to i and ɛ i, respectively, sch that the desired data compression ratio γ i can be achieved to avoid congestion. At node, the parameters (ɛ i, i ) always need to satisfy Formla (6) in order to ensre e ɛ, that is, ( k +ɛ k +2 k ɛ k ) ɛ c m k ɛ k k : k is child of However, the increase of ( i, ɛ i )to( i, ɛ i ) may violate Formla (6). Note that thogh s other children are assigned (ɛ k, k ) hence maximm tolerable error contribtion (c m k ), they may generate no or a little error if they experience no or little congestion, i.e., ( k, ɛ k ) are or small vales. Ths, node can redce the c m k of ncongested children and increase the c m k of congested children to satisfy Formla (6). Accordingly, node first attempts to change (ɛ k, k )to( i, ɛ i ) for each of its children. It then calclates its ɛ based on pdated k and ɛ k by Eqation (16), and then compare ɛ and ɛ to decide the next step as follows: (1) If ɛ ɛ, it means that pdating each child k s parameters with ɛ k = ɛ k and k = k can garantee Formla (6), becase ɛ ɛ = ( k + ɛ k +2 k ɛ k ) k : k is child of = ( k + ɛ k +2 (17) k ɛ k ). k : k is child of Therefore, then pdates each child k s parameters with ɛ k = ɛ k and k = k Conseqently, node i has i = i, allowing the data compression with ratio γ i at i. (2) If ɛ >ɛ, it is obvios that the previos pdating soltion cannot garantee Formla (6). Ths, node attempts to pdate each child k s parameters with ɛ k = ɛ k and k = k, by reqesting its parent node to assign ɛ as s maximm tolerable error (ɛ ) so that Formla (6) can be satisfied. Node pdates maximm tolerable distortion and error of its children in the same way of pdating parameters of s children by considering two cases ɛ ɛ and ɛ >ɛ.ifɛ >ɛ, will frther reqest pdate from its parent node. This process can repeat along the path towards the sink ntil reaching either a node that sccessflly reassigns these parameters for all its children, or the sink. If the reqest reaches the sink, the sink informs its application of the lower data accracy than the specified vale. After congested node s children decrease their compression ratios to redce their transmission rates, the inpt data arrival rate to starts to decrease ntil it is not congested anymore. Once the congestion is eliminated, node notifies its children that it is not congested anymore and they can increase their compression ratios. However, in order to avoid oscillation, children do not abrptly increase their compression ratio to 1 (which means no compression). Instead, a node gradally increases its compression ratio by γ i (t +1)=γ i (t)+ρ times, where ρ is a constant vale. E. Adaptivity to Dynamic Network Topology The setting of maximm tolerable errors and distortions (ɛ, ) in CADC depends on the topology of the roting

7 tree. However, since the roting tree can dynamically change becase of common failres of nodes and links in WSNs, CADC needs to adaptively adjst the parameters of (ɛ, ). To handle the failres of nodes or links, the roting tree is rebilt, in which some nodes leave a sbtree and join in another sbtree along with the sbtrees rooted at them. Sppose that node leaves original parent and chooses another node as its new parent becase the failre of or the link between and. CADC lets the setting of (ɛ k, k ) remain the same for all nodes in s sbtree T. In order to have the same maximm tolerable error at node in the new sbtree T, based on Eqation (6), needs to increase its maximm tolerable error to ɛ + ɛ + +2 ɛ.. Ths, reqests pdate from its parent, following the same pdating procedre in Section IV-D. V. PERFORMANCE EVALUATION In this section, we evalate the performance of CADC in the non-priority and priority cases throgh simlations in comparison with previos schemes. In particlar, we measred the estimation error incrred at the sink, data delivery ratio and the network overhead nder different network conditions. Data delivery ratio is measred by the percentage of nodes whose sensor readings are received by the sink (i.e., represented by compressed samples received by the sink). Network overhead is measred by the total nmber of packet (i.e., data tple) transmissions of all nodes in a rond of data collection, in which each node generates one sensor reading. We compared CADC with the following data collection schemes with congestion control, which do not have maximm error bond at the sink. (1) Spatio-Temporal data collection (ST) [5]. It ses adaptive smmarization as a compression scheme to mitigate congestion while aims to minimize the estimation error. Assme node k has m data vales. The first level smmarization ses every two consective vales to obtain m 2 samples. Contining this process yields k-th smmarization, which comptes the average of every 2 k consective vales to obtain m 2 samples. k (2) Spatio-Temporal data collection with sorted adaptive smmarization (ST-SortAdpSm). It is a variant of ST with sorting available data at each node before performing adaptive smmarization. It is easy to see that nder the same data distortion constraint, sorted adaptive smmarization leads to fewer samples (i.e. higher compression) and conseqently a lower transmission rate. (3) ESRT [1]. It is a rate based congestion control scheme, which mitigates the congestion by adjsting the reporting rate of sensor nodes. (4) Pre congestion elimination (PreElimination). Itisa congestion control scheme, which jst ses lossy compression to mitigate congestion. In particlar, we let it se the adaptive smmarization method to compress data to the extent that can eliminate congestion. A. Experimental Setp In or simlation, a random roting tree for the WSN was generated. The average nmber of children for any node was set to 3. The sensor reading of each node was randomly generated, following the Gassian distribtion with mean μ = 5 and variance σ 2 = 5 [21], [22]. In the priority case, we determine the priority coefficient of vale x by the interval x belongs to. The entire range of data vale is divided into small ranges (,μ/5), [μ/5,μ/4), [μ/4,μ/3),..., [μ/2,μ/1), [μ, 2μ), [2μ, 3μ),..., [4μ, 5μ), [5μ, + ), each is associated with the corresponding priority coefficient in {2, 3, 4, 5, 6, 7, 8, 9, 1}. We implemented the above for schemes, which operate on the same roting tree in order to perform comparable experiments. We assme there is no non-congestion-indced loss for all links to emlate a reliable wireless medim. The measrement reslts for each scheme are the average vales over 3 rns. B. Validity of CADC In this test, nless otherwise specified, the network size was set to 2 and the maximm tolerable error at the sink was set to 2. We varied the maximm tolerable error at the sink from 5 to 12 with 1 increase in each step. We also evalated CADC nder different network sizes in {1, 2, 4, 6, 8, 1}. We sed both k-means clstering and adaptive smmarization as or compression scheme in CADC, referred to as CADC-Kmeans and CADC-AdpSm, and compared their performance. We tested CADC in both the non-priority and priority cases. In the method names, we se the sffix -P for the priority case and se -N for the nonpriority case. 1) Estimation Error Incrred At the Sink: We first verify the validity of CADC for achieving or primary objective to keep estimation error at the sink below the given assigned maximm tolerable error. Figre 2(a) shows the error incrred at the sink (e r ) verss the maximm tolerable estimation error at the sink (ɛ r ) for different CADC methods. We see that in both the non-priority and priority cases, the error incrred at the sink is lower than the maximm tolerable error. Also, as the maximm tolerable error increases, the incrred error increases. We frther observe that in each case, the k-means compression generates higher errors than the adaptive smmarization. The reason lies in the fact that k- means achieves higher compression ratio and hence lower transmission rate than the adaptive smmarization. Since the samples received at the sink with k-means are less than those with the adaptive smmarization, k-means gives higher estimation error. However, this error is still nder the maximm tolerable error, and the lower transmission rate and hence energy-efficiency is an evidence of speriority of k-means over adaptive smmarization for data compression. 2) Data Delivery Ratio and Network Overhead: De to network congestion, packets carrying data tples may be dropped and the sensor readings they represent are lost in the transmission. Figres 2(b) and 2(c) show these two metrics verss the maximm tolerable error at the sink (ɛ r ). We see that as ɛ r increases, the delivery ratio increases and the nmber of packet transmissions decreases. With a higher maximm tolerable error at the sink, data is allowed to be compressed with a higher compression ratio, which mitigates the congestion and ths redces the nmber of missing sensor readings cased by congestion. A higher compression ratio also redces the total nmber of packets transmitted in the network.

8 Estimation error at the sink CADC-KMeans-N CADC- KMeans-P CADC-AdpSm-N CADC-AdpSm-P Maximm tolerable error at the sink (a) Error at the sink Delivery ratio CADC- KMeans-N.65 CADC-KMeans-P.6 CADC-AdpSm-N.55 CADC-AdpSm-P Maximm tolerable error at the sink (b) Delivery ratio The total nmber of packet transmissions Fig. 2: Performance vs. the maximm tolerable error CADC KMeans N CADC KMeans P CADC AdpSm N CADC AdpSm P Maximm tolerable error at the sink (c) Network overhead Distortion Priority coefficient Fig. 3: Distortion vs. priority coefficient. Delivery ratio CADC KMeans N CADC KMeans P CADC AdpSm N CADC AdpSm P Nmber of nodes (a) Delivery ratio The total nmber of packet transmissions CADC-KMeans-N CADC- KMeans-P CADC-AdpSm-N CADC-AdpSm-P Nmber of nodes (b) Network overhead Estimation error at the sink 45 ST N 4 ESRT N 35 ST SortAdpSm N 3 25 PreElimination N 2 CADC N Nmber of nodes (a) In the non-priority case Estimation error at the sink ST P ESRT P ST SortAdpSm P PreElimination P CADC P Nmber of nodes (b) In the priority case Fig. 4: Performance vs. the nmber of nodes. Fig. 5: Performance comparison: error at the sink. From the two figres, we also see that k-means has a higher data delivery ratio and a lower nmber of total packet transmissions than adaptive smmarization in both non-priority and priority cases. This is becase k-means can achieve higher compression ratio than adaptive smmarization nder the same distortion bond. This experimental reslt spports the speriority of k-means over adaptive smmarization. Figre 4(a) shows that the delivery ratio decreases with the network size. Figre 4(b) shows that the total nmber of packet transmissions increases with the network size. This is becase that a larger network has more data to collect, which leads to more transmissions bt also generates a higher probability to case congestion, leading to delivery ratio decrease. The two figres also show the speriority of CADC with k-means on redcing transmission rate and achieving higher delivery ratio compared to CADC with adaptive smmarization. 3) Performance in the Priority Case: We then validate that in the priority case, CADC indeed incrs lower distortion to high priority data. We measred the average overall distortion incrred to data with different priorities as shown in Figre 3. We see that the experimental reslts confirm that data with higher priorities does have less distortion. Recall that when data is transmitted hop by hop along the roting tree from the sensing node to the sink, each forwarding hop may compress the data. After compression in a hop, some data vales are changed, and if a vale belongs to a different range, its priority coefficient may be changed. In or experiments, most of data has the same priority coefficients at different hops in the forwarding path. This is becase the k-means method clsters the most approximate data points, which have same or close priority coefficients. This validates or assmption in Section IV-B2 that the data compression does not change the priority of data in different hops. C. Performance Comparison We compared CADC with the ST, ST-SortAdpSm, ESRT, and PreElimination schemes. We varied the nmber of nodes by {1, 2, 4, 6, 8, 1} and set the maximm tolerable error (ɛ r ) at the sink to 2. Figres 5(a) and 5(b) show the error incrred at the sink of different schemes in the non-priority and priority cases, respectively. We see that this metric reslt increases as the nmber of nodes increases in all schemes. CADC scceeds in constraining the error incrred at the sink below the maximm tolerable error except at the largest network size, which generates high congestion. In this case, the sink alerts the application to increase ɛ r. The incrred error of CADC is considerably lower than those of other schemes. In other schemes, the incrred error keeps increasing with the nmber of nodes, and it exceeds the maximm tolerable error when the nmber of nodes is larger than 5. These schemes prodce ncontrollable errors, becase they mitigate the congestion withot being aware of any error bond reqirement at the sink. These experimental reslts indicate the advantage of CADC in constraining the estimation error below ɛ r at the sink. Figres 6(a) and 6(b) show the total nmber of packet transmissions verss the nmber of nodes in the non-priority and priority cases, respectively. We see that the total nmber of packet transmissions increases as the nmber of nodes increases. The nmber of transmissions of CADC is lower than that of ESRT becase of the data compression in CADC. It is higher than that of ST becase the maximm tolerable error in CADC limits the compression degree, while ST does not have estimation error constraint, which reslts in a high compression ratio. PreElimination has the lowest transmission rate becase it eliminates congestion abrptly, while ST mitigates congestion gradally. Figres 7(a) and 7(b) show the data delivery ratio verss

9 The total nmber of packet transmissions CADC-KMeans-N CADC-AdpSm-N ST-N ESRT-N ST-SortAdpSm-N PreElimnation-N Nmber of nodes (a) In the non-priority case The total nmber of packet transmissions CADC- KMeans-P CADC-AdpSm-P ST-P ESRT-P ST-SortAdpSm-P PreElimnation-P Nmber of nodes (b) In the priority case Delivery ratio 1.4 CADC KMeans N CADC AdpSm N 1.2 ST N ESRT N ST SortAdpSm N PreElimination N Nmber of nodes (a) In the non-priority case Delivery ratio 1.4 CADC KMeans P CADC AdpSm P 1.2 ST P ESRT P ST SortAdpSm P PreElimnation P Nmber of nodes (b) In the priority case Fig. 6: Performance comparison: network overhead. Fig. 7: Performance comparison: delivery ratio. the nmber of nodes in the non-priority and priority cases, respectively. We see that the delivery ratio in all schemes generally decreases with network size. This is de to the increase in total nmber of packet transmissions. In both cases, PreElimination, ST and ST-SortAdpSm prodce higher delivery ratios than CADC-KMeans and CADC-AdpSm, which have higher delivery ratios than ESRT. These observations can be explained by the experimental reslts of network overhead (total transmission rate) in Figres 6(b) and 6(a). The total transmission rate has inverse relationship with the delivery ratio. That is, a higher total transmission rate (hence higher network overhead) leads to a higher probability of congestion occrrence and hence lower delivery ratio. VI. CONCLUSION AND FUTURE WORK In CPS, it is critical to garantee estimation accracy of the physical environmental phenomena. Althogh many congestion control schemes have been proposed to redce congestion in order to increase estimation accracy, they also concrrently increase the estimation error de to data sample redction. Also, none of them can garantee the data accracy at the sink. To garantee the estimation accracy while control congestion, we presented a Congestion-Adaptive Data Collection scheme (CADC). Based on a given maximm tolerable error bond at the sink, CADC redces transmission rate of data while keeps the estimation error below the given bond. It novelly ses the k-means clstering algorithm to redce transmission rate in order to redce data distortion. CADC also distingishes data with different importance degrees so that more important data has less distortion, which benefits the accrate environmental phenomena monitoring. Extensive experimental reslts show the sperior performance of or schemes in comparison with previos schemes. In or ftre work, we will implement CADC and investigate its performance in the real testbed. We will also investigate extending CADC with other types of accracy measrement, since the CPS applications may have error reqirement for the reslts of specific state estimation fnctions not limiting to sqare error over all the data. 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