Improving the Accuracy of Boolean Tomography by Exploiting Path Congestion Degrees

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1 Improving the Accuracy of Booean Tomography by Expoiting Path Congestion Degrees Zhiyong Zhang, Gaoei Fei, Fucai Yu, Guangmin Hu Schoo of Communication and Information Engineering, University of Eectronic Science and Technoogy of China fcyu, Abstract Booean tomography is based on expoiting performance eve correations of end-to-end measurements to identify the congested inks. Most work to date attempts to find the congested inks according to the observed pattern of congested paths and the prior ink congestion probabiities. In their work, the prior ink congestion probabiities are either assumed to be unreaisticay equa or estimated by a computationay compex agorithm. Furthermore, a congested paths are mapped down to the same bad state regardess of their congestion degrees, then separate causes of congestion may be identified as a common cause. In this paper, we propose a fast Bottom-Up Approach named to estimate the prior probabiities based on a sma number of measurement snapshots. is computationay simper than the existing approaches since it computes the congestion probabiity of each individua ink through an expicit function of the measurements. We then extract the subsets of congested paths that might traverse the same congested inks in current measurement snapshot according to their congestion degrees. The inks that cause the congestion of each subset of paths are identified with the aid of the earnt probabiities. Simuations in different network scenarios demonstrate that our approach is abe to improve the accuracy of the identification procedure. I. INTRODUCTION Booean tomography is a powerfu too for network troubeshooting. It identifies the poory performing inks using reguar unicast probes at the edge of the intervening networks. The appea of this approach is that it is abe to monitor the states of network inks without the coaboration of network interna devices and direct monitoring each singe ink. Endusers can use Booean network tomography to assess the performance of Internet Service Providers (ISPs) so as to choose reiabe ISPs. ISPs coud use it to troubeshoot probems in their own networks or monitor the performance of their peers. In principe, Booean tomography reduces the probem to a binary probem in which inks and paths can have one of two states: good and congested (aso referred to as bad ) [] [2]. If a inks on a path are good then the path is good, otherwise the path is congested. Booean tomography tries to identify the ink states from the observed path states. This probem is underdetermined, i.e., there are many different ink states matching the path states, since the number of inks aways far exceeds the number of paths. Thus additiona information is required to find a proper soution. Based on the assumptions that congested inks are rare and a inks have the same probabiity of being congested, N. Duffied proposes a fast and simpe agorithm SCFS, which designates the east inks that are consistent with the observed pattern of bad paths as congested [] [2]. Considering that a inks have the same congestion probabiity is unreaistic in practice, H. Nguyen et a. propose to find the prior ink congestion probabiities uniquey by matrix inversion based on mutipe measurement snapshots [3]. According to the estimated prior probabiities and the measurements in subsequent snapshot, the congested inks can be identified with higher accuracy by a Maximum A-Posteriori estimator. However, the ink probabiities estimation method proposed in [3] is not computationay efficient for arge-scae networks. Based on the work in [3], D. Ghita et a. study Booean tomography in the scenarios that inks the ink states are not mutuay independent [4] [5]. Booean tomography is motivated to be more practica by their work. There is another set of approaches that expoit secondorder moments of end-to-end fows. It is assumed that oss rates of congested inks have high variances and oss rates of most good inks have zero first- and second-order moments [6] [7]. After cacuating the ink variances from mutipe measurements, the authors approximate the oss rates of the inks with sma variances by zeros and compute the oss rates of the remaining inks by soving a fu rank inear system of equations. Thus the state of each ink can be identified by comparing the estimated oss rate with a threshod. However, the approximation wi cause the oss rates of some inks to be under-estimated and those of the remaining inks to be overestimated, and it is impossibe to te to what extent these errors degrade the identification accuracy. In this paper, we focus on identifying the ossiest inks since high oss rate is one of the most significant characteristics of congested inks. We make two contributions. First, we propose a fast Bottom-Up Approach named to estimate the prior ink congestion probabiities using a sma number of measurements. Contrary to the existing work that computes the probabiities of a inks by soving a high dimensiona inear system [3], is more computationay efficient since it cacuates the probabiity of each individua ink based on an expicit estimator from bottom up. Second, given the estimated ink congestion probabiities, we combine them with the end-to-end data in subsequent snapshot to identify currenty congested inks. Existing approaches identicay cassify a congested paths as congested regardess of their actua congestion degrees and that is too coarse-grained for characterizing the /2/$3. 22 IEEE 725

2 state of congested paths. Actuay, congestion degrees of the paths can be used to improve the identification accuracy. The gist is the foowing: the paths traversing the same congested inks have simiar oss degrees, since a congested ink identicay degrade the performance of the paths traversing it [6] [7] and packet osses of a congested path mainy happen at its congested constituent inks [8]. Based on this property, we divide the set of a congested paths into severa subsets through a simpe comparison method according to their oss degrees, and identify the congested inks in each subset of the congested paths using the knowedge of the prior probabiities. The remaining sections of the paper are organized as foows. We describe the network and performance mode in Section II and derive the ink probabiity estimation approach in Section III. In Section IV, we show how to divide a the congested paths into severa subsets according to their congestion degrees and identify the congested inks in each subset of congested paths. Then, we present the simuations used to evauate our method in Section V and concude this paper in Section VI. II. THE NETWORK AND PERFORMANCE MODEL We start in Section II-A by presenting our terminoogy for networks. Section II-B formaizes the mapping of inks and paths into good and congested states. A. Network Mode Simiar to [] [2], we consider the networks consisting of the paths from one source to mutipe receivers. The network topoogy is known and remains unchanged throughout the measurement period. Let the directed tree T =(V,L) denote the ogica network topoogy [9], where V is the set of nodes and L is the set of inks. The source is the root node s V which sends probe packets to the receivers. The receivers are ocated at the eaf nodes R V. Apart from s, each node k V has a unique parent f(k) and k L denotes the ink from f(k) to k. Weusej k to denote k is descended from j. Then j k signifies j k or j = k. Let a(j, k) denote the nearest common ancestor of j and k. For each interna node k V \ (R {s}), it has a set of chid nodes D(k) ={i f(i) =k, i V }. D(k) 2 since a interna nodes with a singe chid are deeted when obtaining T [9]. We consider s has a singe chid d s. If it does not hod, we divide the origina tree into mutipe sub-trees so that within each sub-tree, s has ony one chid d s, then process each sub-tree separatey. For any r R, we use P r to denote the end-to-end path from s to r. Let P = {P r r R} denote the set of a endto-end paths. So, there are L inks and P = R end-to-end paths in the network. In addition, et T (k) =(V (k),l(k)) denote the subtree of T rooted at node k, and R(k) =R V (k) denotes the set of eaf nodes contained in T (k). We define T ( k )=(V (k) {f(k)},l(k) { k }) as the subtree T (k) pus a ink k. B. Performance Mode A random variabe α k is used to represent the transmission rate of k (i.e. the oss rate of k is α k ). The ink threshod t, predefined depending on practica appications, is used to separate inks into good and congested subsets. We say that k is congested if α k <t, despite the origin of the osses. Otherwise k is good. Let X k denote the state of k : X k = if k is good and X k =if k is congested. Let p k = P(X k = ) = P(α k t ) denotes the probabiity that ink k is good and p =[p p 2 p L ] T. Let Y r denote the state of P r : Y r =if P r is good and Y r = if P r is congested. P r is good if and ony if its a constituent inks are good, so Y r = k P r X k, where denotes the booean OR operation. Under the assumption that the ink states are mutuay independent, P r is good with probabiity P(Y r =)= k P r p k. To infer the vector p, we divide a period of measurement time into m intervas (snapshots). In each snapshot, we cacuate the end-to-end transmission rates of a paths and obtain their states by comparing the transmission rates with a path threshod [] - [5]. Let y = [y () y (2) y (m) ] denote the path states in m snapshots, where y (n) = [Y (n) Y (n) 2 Y (n) P ]T ( n m). After earning p from y, we can combine it with the end-to-end transmission rates in the subsequent snapshot to identify the congested inks. III. ESTIMATING THE PRIOR PROBABILITIES It has been proved that p can be estimated from a sufficient number of snapshots if p ies in the parameter set Ω p = {p p k <, k L } in [3]. The authors suggest to construct a fu coumn rank system of inear equations and inverse the coefficient matrix of the system to find vector p. As the size of network increases, the inear system becomes high-dimensiona, which eads to increase in the computationa compexity. To overcome this probem, we present a fast Bottom-Up Approach, named, to estimate the prior ink probabiities. A. The Bottom-Up Approach For any k V, we define Z k = k k X k. Let q k be the probabiity that Z k =, i.e. the path segment from s to k is good. By convention, q s =. Provided that the ink states are independent from each other, q k = q f(k) p k, and so p k = q k /q f(k). () For any k V \{s}, q k can be estimated based on an expicit function of the path states. We define U k = min r R(k) Y r and U (n) k = min r R(k) Y r (n). Then U k = means that at east one path traversing k is good. u k = P(U k =)can be estimated by û k = m m n= U (n) k. For k R, it is obvious that q k = u k. So, q k can be estimated by q k = û k, for k R, (2) which is a consistent estimator of q k by the aw of arge numbers /2/$3. 22 IEEE 726

3 For each k V \(R {s}), et Q(k) ={{i, j} D(k) i j} be the set of distinct chid pairs of k. We define U i,j = max{u i,u j } and U (n) i,j =max{u (n) i,u (n) j } for any {i, j} Q(k). Let u i,j = P(u i,j =), then the estimator of u i,j is û i,j = m m n= U (n) i,j. The foowing theorem gives an expicit estimator for q k. Theorem : Assume p Ω p, then for k V \ (R {s}): (i) q k = u i u j /u i,j, for {i, j} Q(k). (ii) Define q k (i, j) = û i û j /û i,j. Then any convex combination {i,j} Q(k) λ i,jq k (i, j) (i.e. with the λ i,j and summing to ) is a consistent estimator of q k. proof : (i) U i,j =impies Z k =, hence u i,j = P(U i,j =,Z k =) = q k P(U i,j = Z k =) = q k P(U i =,U j = Z k =) (3) = q k P(U i = Z k =)P(U j = Z k =) = q k ( u i )( u j ). q k q k Therefore, q k = u i u j /u i,j. (ii) By the aw of arge numbers, for {i, j} Q(k), û i u i, û j u j and û i,j u i,j amost surey as m. {i,j} Q(k) λ i,j ui u j u i,j is evidenty continuous as p Ω p, then {i,j} Q(k) λ i,jq k (i, j) is consistent. Theorem -(ii) constructs a convex famiy of consistent estimators for q k. It is appeaing to find the estimator of minima variance. However, finding the minima variance estimator comes at some computationa cost []. For computationa simpicity, we adopt the uniform estimator (i.e. with the λ i,j =/ Q(k) ) q k = û i û j. (4) Q(k) û i,j {i,j} Q(k) Combining (2) and (4), estimates q k through an expicit function of the observed path states from the bottom up unti k = d s. Once q k and q f(k) are obtained, the prior probabiity of k can be estimated by p k = q k / q f(k), which is aso consistent evidenty. B. Computationa Compexity Anaysis In this section, we demonstrate that is computationay simper than the High-dimensiona Matrix Inversion based method () presented in [3]. For simpicity, we consider a uniform tree with branching ratio d. In cacuation of the empirica probabiities û k and û i,j, needs to cacuate the combined path states U k = min r R(k) Y r =min i D(k) U i, and U i,j =max{u i,u j }. This takes respectivey m(d ) and md(d )/2 times of booean operation for m snapshots. Thus, O(m( L P )) times of booean operation are needed for a interna inks. Thereafter, the computation of is very computationay efficient, just needing expicit computations on the empirica probabiities. In constructing a fu coumn rank system of inear equations, takes L P ( P )/2 and m P ( P )/2 times of booean operation to cacuate the coefficient matrix and the empirica probabiities, respectivey [3]. A significant amount of time is needed for arge-scae networks [6]. Besides, the computationa compexity of using a standard technique from inear agebra to invert the matrix is O( P 2 L 2 L 3 /3) [], which is far more compex than the expicit computation of. Therefore, is more computationay efficient than, and the computationa compexity of increases faster than that of as the number of inks increases. Preiminary simuation resuts presented in Section V-B aso vaidate this deduction. IV. IDENTIFYING THE CONGESTED LINKS In this section, we propose an Improved Congested Link Identification agorithm (ICLI), which combines the estimated ink probabiities introduced in the previous section with the end-to-end transmission rates in current snapshot to identify currenty congested inks. A. Dividing the Set of Congested Paths Previous work [] - [5] identicay cassifies the paths whose oss rates exceed the path threshod as congested, despite their actua congestion degrees. This might cause fase negative errors if one path contains mutipe congested inks and fase positive errors if separate congestion happen at different inks derived from the same parent node. To overcome this probem, it is necessary to differentiate paths with different congestion degrees. A natura method is to spit the congestion state into mutipe sub-states. However, the reation between the ink states and the path states wi be more compex [2]. In order to expoit the knowedge of the congestion degrees and avoid spitting the congestion state into mutipe substates, we adopt a simpe method which ony invoves comparison operation to divide the set of a congested paths into severa subsets. Before that, we need to identify and remove a the good paths from the origina tree T at first, by comparing the path transmission rates β = {β r } r R in current snapshot with the path threshod [] - [5]. Thus T is separated into severa maxima congested subtrees [2]. Then we can process each subtree separatey since the paths in different maxima congested subtrees cannot contain common congested inks. For the maxima congested subtree rooted at node k, the state of ink k is aso undetermined, so we use T ( k ) instead of T (k) to denote this subtree. Let P (k) ={P r r R(k)} denote the set of the paths in T ( k ). We first extract the east congested paths in P (k), identify the inks that resut in the congestion of these paths, and remove these paths from T ( k ). Then, we are eft with some congested subtrees of T ( k ) formed by the remaining paths. These subtrees do not share common inks, so we can appy the same operation with T ( k ) to process each remaining subtree unti a the congested paths are removed. In order to extract the east congested paths in P (k), we choose the path P m P (k) with the maximum transmission rate as a representative. For any path P n P (k) \{P m }, /2/$3. 22 IEEE 727

4 Fig.. An exampe of maxima congested subtree T ( k ) P m and P n separate at node j = a(m, n) as shown in Fig.. Let P j,n denote the path segment from node j to n. We adopt the criterion that P n shoud not be put into the set of east congested paths if P j,n contains congested inks. Referring to the method adopted in [2] [3] for choosing path threshod in a weak separabe performance mode, we take t h(j,n) as the transmission rate threshod of P j,n for identifying whether it contains congested inks or not, where h(j, n) is the ength of P j,n. Unfortunatey, there is no way to accuratey compute the transmission rate of P j,n ony according to the end-to-end data. Here, we use β i,i to denote the transmission rate of the path segment from node i to i. Then, β j,n = β n /β s,j = β n β j,m /β m, where β n and β m can be observed whereas β j,m cannot. So we approximate β j,n by β j,n β j,m β j,n = β n /β m. That is to say, if β n /β m <t h(j,n), P s,n is excuded from the set of east congested paths. Through a simpe comparison operation, any path P n P (k) \{P m } can be identified whether it shoud be put in the set of east congested paths. It takes P (k) times of comparison to identify a the paths in P (k), so this operation is computationay efficient. To seect t h(j,n) as the transmission rate threshod of P j,n and approximate β j,n by β s,n /β s,m may cause some errors. On one hand, if one ink i P j,n is congested, the chance for P n to be put in the set of congested paths is P(β j,n /β j,m t h(j,n) i congested) = P(β j,n β j,m t h(j,n) α i <t ) P(α i β j,m t h(j,n) α i <t ) = P(α i [β j,m t h(j,n),t ) i congested). Note that this bound wi decrease as β j,m increases and h(j, n) decreases. So, choosing the path with the maximum transmission rate as the representative can ensure a sma bound. For the genera oss rate mode in [2], the bound is β j,m t h(j,n), e.g., about 5% for β j,m =.99 and h(j, n) =5. On the other hand, if a the inks in P j,n are good, then β j,n t h(j,n) β j,m t h(j,n). P n wi be aways correcty identified. (5) B. Identifying the Congested Links in the Set of Least Congested Paths Let P (k) P k denote the set of east congested paths, T ( k )=(V ( k ),L ( k )) denote the subtree of T ( k ) formed by the paths in P (k), and R ( k ) denote the eaf nodes in T ( k ). Then, the congested inks in T ( k ) can be identified with the aid of the prior ink probabiities p. This probem is equivaent to find a vector X L( k ) that maximizes the conditiona probabiity given p and P (k) are congested: P p (X L( k ))=P p (X L( k ) = x Y R( k ) = ). From Bayes rue, P p (X L ( k ) = x Y R ( k ) = ) (6) P p (X L ( k ) = x)p p (Y R ( k ) = X L ( k ) = x) and P p (Y R ( k ) = X L ( k ) = x) { k if = j r x j =for r R ( k ) (7), otherwise P p (X L( k ) = x) = j L ( k ) j L ( k ) p x j j ( p j ) xj x j og p j p j, we are eft with the equivaent optimization probem: argmax P p (X L ( k ))=argmax x j og p j p j j L ( k ) = argmin p (9) j x j og p j j L ( k ) subject to k j r x j =for a r R ( k ). The optimization probem in (9) is indeed the weighted set cover probem where the universe is P (k) and the weight of subset P ( j )=P ( j ) P (k) is og(p j /( p j )). The heuristic agorithm CLINK in [3] can be used to sove this probem. Contrary to [3] and most other reated work, which is to find the congested inks in each maxima congested subtree at once, ICLI extracts the east congested subtrees and finds their congested constituent inks. That makes ICLI achieve higher accuracy. V. PERFORMANCE EVALUATION In this section, we evauate our approach by simuations in different scenarios. We first describe the simuation configurations in Section V-A and then investigate the behavior of in V-B. Section V-C presents the simuation resuts of ICLI. A. Simuation Setup The network topoogies are randomy constructed with the number of inks ( L ) ranging from to 3. Apart from the root and eaf nodes, the branching ratio of each node (the number of chidren) is picked at random between 2 and. To evauate the approach under different congestion eves, the proportion of congested inks, denoted by f, is varying (8) /2/$3. 22 IEEE 728

5 TABLE I RUNNING TIME OF AND (IN MILLISECONDS) L the Logarithm of Running Time Cumuative Distribution Absoute Error (a) CDF of the absoute errors when the number of inks is..9 Cumuative Distribution Absoute Error (b) CDF of the absoute errors when the number of inks is Number of Links ( L ) Cumuative Distribution.8.7 Cumuative Distribution.8.7 Fig. 2. Running time of and when f =5%and L is varying from to 4. from 5% to 35%. Given f, the prior ink probabiities p are randomy generated such that the average number of congested inks equas to L f. The number of measurement snapshots used for earning p, i.e. m, is set to be. Within each snapshot, ink k is randomy designated to be congested with probabiity p k. We use the genera oss rate mode in [2] to assign transmission rates to inks, where the transmission rates of congested inks and good inks are uniformy picked from [,.99] and [.99, ] respectivey. The ink threshod is t =.99 and the path threshod for identifying path states is t p = t h, where h is the number of inks traversed by the path. We repeat our simuations times for each configuration, where each repetition has a new topoogy and prior ink probabiity assignments. We first appy to estimate the prior ink probabiities p using the end-to-end data in m snapshots and then proceed to identify the congested inks in a new snapshot. B. Evauation of prior probabiity estimation agorithms Here, we compare and with respect to their running time and accuracy of estimating the prior ink probabiities. It is hard to impement such that the simuation competes within a reasonabe amount of time and memory space, given a arge-scae network, so we assess their performance with the number of inks varying from to 4. The running time of the two agorithms are independent on f, so the resuts for different proportions of congested inks are simiar. We ony give the comparisons for f =5%, as shown in Tabe I and Fig. 2. We observe that the running time of is much onger than that of under the same conditions... Absoute Error (c) CDF of the absoute errors when the number of inks is 3... Absoute Error (d) CDF of the absoute errors when the number of inks is 4. Fig. 3. Accuracy of and when f = 35% and L is varying from to 4. As the size of network increases, the running time of amost increases ineary, whereas that of increases more significanty. This is consistent with our anaysis resut in section III-B. Fig. 3 shows the Cumuative Distribution Functions (CDFs) of the absoute errors of the estimated prior probabiities. We just give the resuts for one of the ten trees with f = 35% since the resuts for other trees and f are simiar. The errors in come from two sources: (i) the samping errors in estimating the empirica probabiities, which aso exists in ; (ii) the errors accumuated from estimation of the downstream inks, which is indeed the cost that incurred by s computationa simpicity. Even under these errors, we observe that achieves accuracy comparabe to in genera. Actuay, the sma differences between the prior probabiities estimated from and do not make the fina identification resuts different. C. Evauation of congested ink identification agorithms To evauate the performance of the congested ink identification agorithms, we use the foowing metrics: the detection rate (DR), which is the proportion of inks that are identified as congested correcty, and the fase positive rate (FPR), which is the proportion of inks that are good but we identify them as congested. If F denotes the set of the actua congested inks, and X denotes the set of inks identified as congested, then DR= F X / F, FPR= X \ F / X /2/$3. 22 IEEE 729

6 Fig. 4. DR and FPR of the CLINK and ICLI agorithms for -ink networks when f is varying from 5% to 35%. We compare ICLI with the CLINK agorithm proposed in [3]. Simiar to ICLI, CLINK combines the earnt prior probabiities with the current measurements to identify the congested inks. In order to evauate the performance of ICLI and CLINK in arge-scae networks, we ony use the prior probabiities earnt from as input. In fact, the prior probabiities earnt from and can both be used as input to ICLI and CLINK. However, the resuts of the same identification agorithm are simiar in most scenarios even though different prior probabiity estimation methods are used. That is because the orders of the weight of each ink, i.e. the vaue og(p j /( p j )), in the congested subtrees are rarey different. Fig. 4 shows the DR and FPR of the two congested ink identification agorithms when the network is under different congestion eves. We note that both ICLI and CLINK perform we when f changes. This bois down to the fact that they use prior probabiities to overcome the bias against shared inks. Meanwhie, ICLI aso differentiates the congested paths with different congestion eves and therefore it has higher DR and ower FPR than CLINK. Furthermore, the confidence intervas of ICLI are smaer than those of CLINK. We beieve that to divide each maxima congested subtree into severa sma subtrees is one reason for the stabiity of ICLI. We aso observe that the DR of CLINK increases a bit as f grows from 2% to 35%. However, this comes at the cost of a high FPR ranging from 5% to 2%. Such a high FPR is unacceptabe when there are a ot of congested inks in the networks since the cost of fase positive is reativey higher than that of fase negatives. Fig. 5 and Fig. 6 show the scaing behavior of the CLINK and ICLI agorithms for arge-scae networks when f =5% and 35% respectivey. We observe that the reative performance remains quaitativey the same even for arger networks. ICLI performs better than CLINK with higher DR. When f =5%, ICLI and CLINK achieve comparabe FPRs which are ess than 5% for most of the networks. When f = 35%, the FPR of ICLI is much ower than that of CLINK, which is Fig. 5. DR and FPR of the CLINK and ICLI agorithms for networks of different number of inks when f =5%. Fig. 6. DR and FPR of the CLINK and ICLI agorithms for networks of different number of inks when f = 35%. consistent with Fig. 4. VI. CONCLUSION In this paper, we propose a fast and accurate approach to identify the congested inks by using the prior ink congestion probabiities and end-to-end data in current snapshot. Since our approach expoits the congestion degrees to differentiate the congested paths that have separate causes of congestion, it has the potentia to overcome some issues in the existing approaches, such as omitting some congested inks ocating in the same path and the bias against shared inks. To compute the prior probabiities quicky, we propose a consistent estimator which is an expicit function of the measurements. We vaidate the effectiveness of our approach through simuations and observe that it achieves better accuracy than the existing approaches especiay when the networks are under heavy congestion eves. ACKNOWLEDGEMENT This work was supported in part by the Nationa Natura Science Foundation of China (No and No /2/$3. 22 IEEE 73

7 679) and the Fundamenta Research Funds for the Centra Universities of UESTC (No E22525). REFERENCES [] N. Duffied. Simpe network performance tomography. In Proceedings of the 3rd ACM SIGCOMM conference on Internet measurement, pages ACM, 23. [2] N. Duffied. Network tomography of binary network performance characteristics. Information Theory, IEEE Transactions on, 52(2): , 26. [3] H.X. Nguyen and P. Thiran. The booean soution to the congested ip ink ocation probem: Theory and practice. In INFOCOM th IEEE Internationa Conference on Computer Communications. IEEE, pages IEEE, 27. [4] D. Ghita, K. Argyraki, and P. Thiran. Network tomography on correated inks. In Proceedings of the th annua conference on Internet measurement, pages ACM, 2. [5] D. Ghita, C. Karakus, K. Argyraki, and P. Thiran. Shifting Network Tomography Toward A Practica Goa. In Proceedings of the ACM Internationa Conference on emerging Networking EXperiments and Technoogies (CoNext), 2. [6] H.X. Nguyen and P. Thiran. Network oss inference with second order statistics of end-to-end fows. In Proceedings of the 7th ACM SIGCOMM conference on Internet measurement, pages ACM, 27. [7] D. Ghita, H. Nguyen, M. Kurant, K. Argyraki, and P. Thiran. Netscope: practica network oss tomography. In INFOCOM, 2 Proceedings IEEE, pages 9. IEEE, 2. [8] V.N. Padmanabhan, L. Qiu, and H.J. Wang. Server-based inference of internet ink ossiness. In INFOCOM 23. Twenty-Second Annua Joint Conference of the IEEE Computer and Communications. IEEE Societies, voume, pages IEEE, 23. [9] R. Caceres, N.G. Duffied, J. Horowitz, and D.F. Towsey. Muticastbased inference of network-interna oss characteristics. Information Theory, IEEE Transactions on, 45(7): , 999. [] N.G. Duffied and F. Lo Presti. Muticast inference of packet deay variance at interior network inks. In INFOCOM 2. Nineteenth Annua Joint Conference of the IEEE Computer and Communications Societies. Proceedings. IEEE, voume 3, pages IEEE, 2. [] G.H. Goub and C.F. Van Loan. Matrix computations, voume 3. Johns Hopkins Univ Pr, /2/$3. 22 IEEE 73