Estimating Internal Link Loss Rates Using Active Network Tomography

Transcription

1 Estimating Internal Link Loss Rates Using Active Network Tomography by Bowei Xi A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Statistics) in The University of Michigan 2004 Doctoral Committee: Associate Professor George Michailidis, Co-Chair Professor Vijayan N. Nair, Co-Chair Associate Professor Jeffrey A. Fessler Assistant Professor Kerby Shedden

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3 c Bowei Xi 2004 All Rights Reserved

4 To my parents ii

5 ACKNOWLEDGEMENTS I am greatly indebted to my advisors, Professor George Michailidis and Professor Vijayan N. Nair, for guiding me through my Ph.D program. I have learned a lot from them. This will help me throughout my career. I would also like to thank my committee members, Professors Kerby Shedden and Jeffrey A. Fessler. I am especially grateful to Professor Shedden for answering many questions on programming and computations, and for his valuable advice. Last but not least, I thank my parents for their unwavering support during these five years. I feel fortunate that I followed their advice to pursue a Ph.D degree five years ago. iii

6 TABLE OF CONTENTS DEDICATION ACKNOWLEDGEMENTS ii iii LIST OF FIGURES vi CHAPTER I. Introduction Motivation of Network Tomography Literature Review Recovery of Path-Level Information Estimating Network Internal Characteristics Topology Identification Overview II. Modeling Framework Framework and Notation Stochastic Model Probing Schemes Pure Bicast Experiments for Binary Trees Bicast Experiments for General Trees and Other Topologies k cast Schemes Identifiability Single-Source Topology Identifiability Condition Multi-Source Topology Identifiability Condition III. Statistical Inference Maximum Likelihood Estimation and Properties Bicast Experiment for Single-Source Topology k cast Experiment for Single-Source or Multi-Source Topology Regression Framework A Linear Model Formulation Least Squares Estimation Methods Distributed Algorithm Bicast Projections IV. Performance and Design Issues Efficiency Study of Bicast Experiment Optimal Allocation of Probes iv

7 4.1.2 Efficiency Comparisons with Multicast Schemes Efficiency Study on Multi-Source Topology Performance Evaluation of OLS, GLS and IRWLS Finite Sample Performance Asymptotic Efficiency Considerations NS-2 Simulation CBR Background Traffic TCP Background Traffic V. Summary and Future Work BIBLIOGRAPHY v

8 LIST OF FIGURES Figure I.1 A Small Network I.2 A 4 Node Network I.3 A Tree Topology I.4 Back-to-Back Unicast Probing Scheme I.5 A Sandwich Probe II.1 A General Tree Topology II.2 3-Layer Symmetric Binary Tree II.3 General Binary Tree II.4 A General Tree Topology For Which A Pure Bicast Experiment Suffices To Estimate The Link Loss Rates II.5 Cases 1 and 2 of the sufficiency of Proposition II II.6 Demonstration of (2.6)-(2.7) II.7 A Single Source Acyclic Graph Topology II.8 The Corresponding Multicast Tree for The Single Source Acyclic Graph Topology. 27 II.9 A Multi-source Network Topology II.10 Topology for counter example to Proposition II III.1 Convergence of the Log-Likelihood Function III.2 Convergence of Selected ˆα for A 3-Layer Tree III.3 Number of EM iterations for a 3-layer tree as a function of the success probability. 37 III.4 A Reduced Tree III.5 Reduced Tree for 2, 6, III.6 Subtree Covered by 2, 6, vi

9 III.7 Left: The Subtree Covered by Unicast; Right: The Corresponding Reduced Tree. 54 III III III.10 2 Layer Tree IV.1 A 4 Layer Symmetric Binary Tree IV.2 IV.3 Efficiency comparison between a min experiment and a min + 1 experiment on a 3 Layer symmetric binary tree with α 2 = α 3 = P 1 and all others equal to P Efficiency comparison between a min experiment and a- min + 1 experiment on a 4 layer symmetric binary tree with equal loss rates IV.4 Variances of the MLEs for selected links in a 3-layer tree with equal loss rates (= P ). 89 IV.5 IV.6 IV.7 IV.8 IV.9 IV.10 IV.11 D-optimal allocations for the min + 1 bicast scheme for a 3-layer symmetric tree with α 2 = α 3 = P 2 and all others equal to P A-optimal allocations for the min + 1 bicast scheme for a 3-layer symmetric tree with α 2 = α 3 = P 2 and all others equal to P Efficiency comparison between the uniform allocation and the optimal allocation for a min + 1 experiment on a 3-layer symmetric tree with α 2 = α 3 = P 1 and all others equal to P Efficiency comparison between the uniform allocation and the optimal allocation for a min + 1 experiment on a 4-layer symmetric tree with equal loss rates A-optimal allocation for a 4-layer tree with equal loss rates (= P ) for all links. Left panel corresponds to the total allocation for the nodes that split at layer L 3 and right panel to those that split at layer L A-optimal allocation restricted to internal links only for a 4-layer tree. Equal values of loss rates (= P ). Left panel shows total allocation for nodes splitting at layer L 3 and right for nodes splitting at layer L Ratio of variances of MLEs under min + 1 and multicast schemes for a 4-layer tree with α 1 = α 8 = = α 15 =.95 and α 2 = = α 7 = P. Left panel: node 2; Right panel: node IV.12 Multi-Source Topology Used for Efficiency Study IV.13 Probe allocations for multi-source topology for the experiment C 1 with α(s 2, I 3 ) = α(i 2, I 3 ) = α(i 3, R 4 ) = α(i 2, R 3 ) = P 2 and the rest equal to P 1. Left panel: A-optimal allocation. Right panel: D-optimal allocation IV.14 Probe allocations for multi-source topology for the experiment C 2 with α(i 2, R 2 ) = α(i 2, R 3 ) = P 2 and the rest equal to P 1. Left panel: A-optimal allocation. Right panel: D-optimal allocation vii

10 IV.15 Ratio of variances of ˆα GLS/IRWLS (I 2, R 3 ) under experiment C 1 and C 2. α(i 2, R 2 ) = α(i 2, R 3 ) = P 2 and the rest equal to P 1. Left panel: Using A-optimal allocation. Right panel: Using D-optimal allocation IV.16 IV.17 Boxplots of Manhattan distances between the true link loss probability vector and its LS, GLS and IRWLS estimates for a 4-layer tree topology. The 4 panels correspond to 1000 probe packets with all the link success probabilities set equal to.7,.8,.95 and.9 respectively Boxplots of Manhattan distances between the true link loss probability vector and its LS, GLS and IRWLS estimates for a 4-layer tree topology. The 4 panels correspond to 10,000 probe packets with all the link success probabilities set equal to.7,.8,.95 and.9 respectively IV.18 Boxplots of selected elements of α. The success probabilities equal IV.19 Boxplots of selected elements of α. The success probabilities equal IV.20 IV.21 IV.22 IV.23 IV.24 IV.25 IV.26 IV.27 IV.28 IV.29 IV.30 Boxplots of MSE for OLS, GLS and IRWLS estimates of α with α i =.85 based on R cast measurements with a total sample size of 12,000 probes Boxplots of MSE for OLS, GLS and IRWLS estimates of α with α i =.85 based on R cast measurements with a total sample size of 12,000 probes and processing the data using the minimum bicast projection scheme Boxplots of MSE for OLS, GLS and IRWLS estimates of α with α i =.85 and total sample size 12, Boxplots of MSE for OLS, GLS and IRWLS estimates of α with α i =.55 and total sample size 4, Ratios of variances (in log-scale) for select elements of α and for different successful transmission probabilities Ratios of variances (in log-scale) for select elements of α and for different successful transmission probabilities Ratios of determinants of variance-covariance matrices (in log-scale) for different successful transmission probabilities Variance ratios of LS to GLS estimates for the FBP scheme for different links of a 4-layer tree topology, for different successful transmission probabilities Variance ratios for different links of a 4-layer tree topology, for different successful transmission probabilities. The symbol corresponds to the FBP scheme, while the to the MBP one Variance ratios (in log-scale) for different links of a 4-layer tree topology, for different successful transmission probabilities. The symbol corresponds to the R cast scheme, while the to the FBP one Variance ratios of GLS for a minimal collection of independent bicast schemes to those of the MBP scheme for different links of a 4-layer tree topology, for different successful transmission probabilities viii

11 IV.31 Tracking the Actual Loss (Dashed Line) by Inferred Loss (Solid Line) for Link 0-1 of the 3 Layer Symmetric Binary Tree with CBR Background Traffic IV.32 Tracking the Actual Loss (Dashed Line) by Inferred Loss (Solid Line) for Link 1-2 of the 3 Layer Symmetric Binary Tree with CBR Background Traffic IV.33 Tracking the Actual Loss (Dashed Line) by Inferred Loss (Solid Line) for Link 2-4 of the 3 Layer Symmetric Binary Tree with CBR Background Traffic IV.34 Tracking the Actual Loss (Dashed Line) by Inferred Loss (Solid Line) for Link 3-7 of the 3 Layer Symmetric Binary Tree with CBR Background Traffic IV.35 Tracking the Actual Loss (Dashed Line) by Inferred Loss (Solid Line) for Link 0-1 of the 3 Layer Symmetric Binary Tree with TCP Background Traffic IV.36 Tracking the Actual Loss (Dashed Line) by Inferred Loss (Solid Line) for Link 1-2 of the 3 Layer Symmetric Binary Tree with TCP Background Traffic IV.37 Tracking the Actual Loss (Dashed Line) by Inferred Loss (Solid Line) for Link 2-4 of the 3 Layer Symmetric Binary Tree with TCP Background Traffic IV.38 Tracking the Actual Loss (Dashed Line) by Inferred Loss (Solid Line) for Link 3-7 of the 3 Layer Symmetric Binary Tree with TCP Background Traffic IV.39 3 Layer Asymmetric Topology Used in the NS-2 Simulation IV.40 Tracking the Actual Loss (Dashed Line) by Inferred Loss (Solid Line) for Link 0-1 of the 3 Layer Asymmetric Tree with TCP Background Traffic IV.41 Tracking the Actual Loss (Dashed Line) by Inferred Loss (Solid Line) for Link 1-2 of the 3 Layer Asymmetric Tree with TCP Background Traffic IV.42 Tracking the Actual Loss (Dashed Line) by Inferred Loss (Solid Line) for Link 1-3 of the 3 Layer Asymmetric Tree with TCP Background Traffic IV.43 Tracking the Actual Loss (Dashed Line) by Inferred Loss (Solid Line) for Link 3-4 of the 3 Layer Asymmetric Tree with TCP Background Traffic IV.44 Tracking the Actual Loss (Dashed Line) by Inferred Loss (Solid Line) for Link 3-5 of the 3 Layer Asymmetric Tree with TCP Background Traffic IV.45 Tracking the Actual Loss (Dashed Line) by Inferred Loss (Solid Line) for Link 3-6 of the 3 Layer Asymmetric Tree with TCP Background Traffic IV.46 Tracking the Actual Loss (Dashed Line) by Inferred Loss (Solid Line) for Link 0-1 of the 4 Layer Symmetric Binary Tree with TCP Background Traffic IV.47 Tracking the Actual Loss (Dashed Line) by Inferred Loss (Solid Line) for Link 1-2 of the 4 Layer Symmetric Binary Tree with TCP Background Traffic IV.48 Tracking the Actual Loss (Dashed Line) by Inferred Loss (Solid Line) for Link 1-3 of the 4 Layer Symmetric Binary Tree with TCP Background Traffic ix

12 IV.49 Tracking the Actual Loss (Dashed Line) by Inferred Loss (Solid Line) for Link 2-4 of the 4 Layer Symmetric Binary Tree with TCP Background Traffic IV.50 Tracking the Actual Loss (Dashed Line) by Inferred Loss (Solid Line) for Link 3-7 of the 4 Layer Symmetric Binary Tree with TCP Background Traffic IV.51 Tracking the Actual Loss (Dashed Line) by Inferred Loss (Solid Line) for Link 5-10 of the 4 Layer Symmetric Binary Tree with TCP Background Traffic x

13 CHAPTER I Introduction 1.1 Motivation of Network Tomography Modern networks are multi-layered, distributed systems that are loosely controlled. The largest of all is the Internet, and its structure can be described as follows. On the edge, there are local area networks (LAN) which are connected to wide area networks (WAN). WANs are further connected to the backbone. Figure I.1 is an example of a small LAN. The large number and variety of service providers has allowed modern networks to expand rapidly and independently. But this has also made performance assessment and monitoring efforts difficult. Numerous user applications have been developed for modern networks. They can be broadly classified into two categories: time-sensitive applications and timeinsensitive ones. Time-sensitive applications, such as voice over IP (VOIP), require extremely high quality links, while time-insensitive applications, such as , allow for the retransmission of corrupted messages. Different levels of service quality are required by different user applications. Hence performance assessment and network monitoring are critical to support the vast variety of user applications and for the service providers to meet the service level agreements. Tools have been developed to discover network connectivity structure, available 1

14 2 Figure I.1: A Small Network bandwidth of links, and other performance characteristics. (Refer to CAIDA ([4]) for more information). Despite these efforts, large scale quantitative network performance assessment is still very difficult, and the expectation of full cooperation of routers is unrealistic in most situations. Service providers (SPs) are capable of directly collecting measurements at the routers within their own network. The associated price is a heavy overhead of extra computing, communication and hardware support. This is impractical for very fast and/or heavily loaded networks. Moreover network traffic usually generates huge amount of data, and analyzing such data is not a trivial task. It is helpful for a SP to gather information from a network connected to its own network but not under its control. But SPs who do not own the network cannot get access to the internal routes to collect information such as traffic rates, individual link delays, available bandwidth.

15 3 These difficulties and challenges call for efficient data collection and analysis techniques that can be used by everyone. This is the motivation for the research on active network tomography. 1.2 Literature Review There are three directions in network tomography: recovery of path-level information, topology identification, and estimation of network internal characteristics. The first direction formulates the problem from a service provider s perspective while the other two assume little or no help from the network internal devices (routers etc.). A more detailed discussion of these three directions follows Recovery of Path-Level Information Vardi (1996) was one of the first to formulate the problem and coined the term network tomography. The term tomography arises from the connection to the inverse problem in medical tomography. For example, in positive emission tomography, counts of photons are collected from tubes attached to an experiment subject s head and the intensity of brain activities is estimated from the counts. Similarly in the network tomography problem considered by Vardi (1996), the measurements are aggregated incoming and outgoing traffic counts measured at the nodes of the network. The objective is to estimate (recover) the traffic intensities between all possible source-destination pairs. Figure I.2 shows such an example and is taken from Vardi (1996). There are 4 nodes in the network. The links are either unidirectional or bidirectional. There are 7 directed links (a b, b a, a c, c b, b c, c d, d c), hence 7 measurements, and 12 possible source-destination pairs (ab, ac, ad, ba, bc, bd, ca, cb, cd, da, db, dc). Suppose the network traffic is stationary and the routing is fixed (if multiple

16 4 a b c d Figure I.2: A 4 Node Network paths are possible between a source-destination pair, only one of them is used and it is known which one). The problem then becomes a linear inverse problem: (1.1) Y = A X, where Y = [Y 1,..., Y k ] is the vector of the aggregated traffic counts (the measurements), X = [X 1,..., X c ] the vector of the source-destination counts and A the routing matrix, which is determined by the routing tables held at every node. Every element of A is either 1 (the link is on the path of a certain source-destination pair) or 0 (the link is not on the path). The inverse problem arises from the fact that we measure Y but are interested in the traffic matrix X or its distribution. This source-destination information is of considerable interest to SPs and network engineers. This is a highly ill-posed inverse problem n, the number of observations of nodes is much smaller than p, the number of origin-destination pairs. So we need additional assumptions or must use

17 5 some type of regularization before we can estimate all the parameters. Vardi (1996) assumed that the source-destination pair counts X follow Poisson distributions with different parameters. Since the means and variances are the same for the Poisson distribution, this yields additional information (in the form of variances) to estimate the parameters. The EM-algorithm for obtaining the MLEs is computationally very intensive, so moment-based methods were also proposed. Cao et al. (2000) considered a more general formulation where variance mean c for c = 1, 2. Tebaldi and West (1998) obtained Bayesian methods. There are also several other papers in the literature on estimating the traffic matrix, including some that uses regularization methods (see for example [33]). Cao et al. (2000) is one of the few papers that uses real data collected through the Simple Network Management Protocol (SNMP) at Lucent Technologies. The measurements collected over 5-minute intervals are noisy and clearly time-dependent. Thus the stationary assumption assumed in most papers is no longer valid. To incorporate the temporal dependence structure, Cao et al. (2000) assumed local stationarity and used a moving window method to estimate the parameters. They also considered a integrated moving average model. This is one of the first to try and address temporal issues. Modeling spatial-temporal aspects of network data is an important area of current research Estimating Network Internal Characteristics Caceres, et al. (1999) is the first major work to estimate internal network characteristics using data from end-to-end measurements. This is referred to as active network tomography. The issue here is that one is not able to passively collect data at the routers. So Caceres et al. (1999) generated measurements by actively probing the network. Nodes are selected on the periphery of the network as a source node and

18 6 destination nodes. The source node actively sends probe packets to the destinations and the counts of received/lost probe packets are measured. The individual link loss rates are recovered from the end-to-end measurements. This is a different type of inverse problem. The probing scheme proposed in Caceres, et al. (1999) is called multicast probing scheme. In Caceres, et al. (1999) a probe is always sent to all the receivers on the network from the source node. The mechanism is the following. The probe is duplicated at the first branching node. Then individual copies are sent along the children links and further duplicated at the down stream branching nodes, until either reaching the receivers or being dropped on the way. Figure I.3 is a topology with 4 receivers, nodes 4, 5, 6, and 7. There are 16 possible outcomes from the multicast probing scheme. Each outcome is a 4 1 vector, recording which nodes receive the probe t and which do not. The measurements are the counts of the outcomes Figure I.3: A Tree Topology

19 7 Caceres et al. (1999) developed a clever way to compute the approximate maximum likelihood estimators (MLE) of the loss rates. Their algorithm works well for binary trees (the definition of binary trees will be introduced in Chapter II). It has been implemented in a project at AT&T Labs ([17]). The authors also tried to use similar ideas to estimate link delay distributions (Lo Presti et al. (2002)), but the method does not extend easily to the delay problem. The method in Lo Presti et al. (2002) for estimating discrete delay distributions is ad-hoc and very inefficient. Liang and Yu (2003) use a pseudo-likelihood approach with multicast experiments. Lawrence, Michailidis, and Nair (2004) have studied the use of EM-algorithms for computing the MLEs. Tsang et al. (2003) proposed a back-to-back unicast probing scheme to imitate the multicast protocol. The back-to-back unicast scheme is a useful technique in such cases. Unicast probing scheme has been studied some time ago ([29]). A unicast probe is a packet that is sent from the source node to one receiver node at a time. It suffers from identifiability problems: active probing experiments based on unicast schemes alone cannot identify all the internal link parameters of interest, such as loss rates and delay distributions. On the other hand, the multicast protocol is not supported by all the networks. Tsang et al. (2003) proposed the use of back-to-back unicast schemes (two closely time-spaced unicast probes) as an alternative. Figure I.4 explains the idea. At time t a unicast packet is sent to receiver i and at time t+δt a second unicast packet is sent to receiver j. If the time difference δt is set to be small enough, we would expect the two unicast probes to experience similar network environment on the common path that they share on the network. Tsang et al. (2003) did not quantify the correlation

20 8 t+ δ t t 0 s i j Figure I.4: Back-to-Back Unicast Probing Scheme on the common path; instead it was assumed that the loss experience and delays on the common path are identically the same. Under such an assumption the backto-back unicast probing scheme in Tsang et al. (2003) is equivalent to the bicast (multicast to a pair of receivers) protocol. However, the assumption of identical performance will not hold in all cases and it is difficult to verify this assumption Topology Identification Both of the network tomography problems discussed above implicitly assume that the routing matrix is given and fixed during the study period. It is reasonable to assume the routing tables remain the same if the probing experiment (or data monitoring period) lasts for only a short time. Then there are tools, for example traceroute, that could discover the connectivity structure with some help from the routers. Increasingly, due to security concerns, network routers do not respond to requests from tools such as traceroute, leading to difficulty in topology identification.

21 9 Thus, using active probing experiments to determine the topology has become an important area of research. The methods used in the literature ([9], [20], [7], [22]) include clustering techniques, maximum likelihood, and Bayesian approaches. There are also some clever ideas for probing experiments such as the sandwich probing scheme ([7]). Figure I.5 illustrates this idea. Two small unicast probe packets, packet 1 and 3, are sent to one receiver and the large unicast probe packet is sent to another receiver. But the large probe is sandwiched between the two small probes. Let d denote the time difference between probes 1 and 3 at the source node and d denote the observed time difference between the two probes at the receiver. The longer the common path in the pair of receivers, the longer the delay. Thus, the delay differences can be used to estimate the topology. Since only the measurements of local delay differences are made, the clock synchronization among the nodes on the network is not an issue. There are, however, still formidable issues involved in estimating the topology (see [7]) Overview This thesis focuses on estimating link loss rates a key network internal characteristic using end-to-end measurements. In our logical topology, a node could represent one router, one computer, or a sub-network; a link could be a physical link or a succession of links that connect two sub-networks. The link loss rates of interest could be the loss rates of certain physical links or the average loss rates between two sub-networks. All the links are unidirectional. The traffic always moves from the source nodes to the receiver nodes. Chapter II discusses more details of the modeling framework. In Chapter III two types of estimators are examined, the maximum likelihood estimator and the

22 10 d s i j d* Figure I.5: A Sandwich Probe least squares based estimators, for the statistical inference purpose. Some numerical results of the performance of the estimators are shown in Chapter IV. We also address the issue of designing a good probing experiment in Chapter IV. Chapter V concludes the thesis and states several problems that we would like to investigate in the future.

23 CHAPTER II Modeling Framework 2.1 Framework and Notation In the thesis, we will first restrict attention to single-rooted tree topologies (see Figure II.1). Sections 2.4.2, and will consider the generalization to multirooted trees. The network over which active probing is performed will be numbered according to the following canonical scheme: the sender is denoted by 0 and the remaining nodes 1, 2,..., while the links are assigned the number of the connected node below it. We consider tree topologies defined as follows: let T = (V, E) denote a tree with root 0 V, a set of nodes V and a set of edges/links E. A link between nodes i and j is an ordered pair (i, j) V V. The root node 0 represents the source (sender) of the transmitted packets. Let d(i) be a direct descendant (child) of node i, and D(i) = {j V : j = d(i)} denote the set of all direct descendants (children) of node i. (In Figure II.1, D(1) = {2, 3, 4, 5}.) The set of receiver nodes, denoted by R V, consists of all nodes without children, i.e., R = {i V : D(i) = }. (In Figure II.1, R = {2, 3, 6, 8, 9, 10, 11, 12, 13, 14, 15}.) The set of internal nodes I is comprised of the nodes that are neither the root nor the receivers (i.e. I = {s V {R {0}}). We assume throughout that each internal node has at least two children. Otherwise, the internal link characteristics (e.g. delays/losses) associated 11

24 12 with the node and its child cannot be estimated separately. For each node i V {0} there is a unique node j such that d(j) = i. We refer to this as the parent node of i and denote it as f(i). Defining f n (i) recursively by f n (i) = f(f n 1 (i)), we say that i is a descendant of j if j = f n (i) for some integer n > 0. (In Figure II.1, f(6) = 4, f 2 (6) = 1, and f 3 (6) = 0.) Let L j, j = 1, 2,... denote the j th layer of a tree, defined as the set of all nodes whose shortest path from the root node 0 has j links; i.e., L j = {i V : 0 = f j (i)}. (In Figure II.1, L 3 = {6, 7, 8, 9, 10, 11}.) Finally, we denote by P(i, j) a path between nodes i and j, which is comprised of a set of connected links (see Figure II.2). A special case of the tree topology are binary trees. For a binary tree, each internal node has exactly two children, i.e., D(i) = 2 for all i V (R {0}). Further, for a symmetric binary tree we have L j = 2 j 1, j = 1, 2,... Figure II.3 shows an example of a binary tree and Figure II.2 a 3-layer symmetric binary tree. 0 α 1 1 α 2 α α 5 4 α α α6 α 8 7 α α 12 α 13 α14 α 15 α 11 α Figure II.1: A General Tree Topology

25 13 0 α 1 α 1 α p(1,7) α α α α Figure II.2: 3-Layer Symmetric Binary Tree. 0 α 1 α 2 1 α3 2 3 α 4 α α 6 α Stochastic Model 6 7 Figure II.3: General Binary Tree. Suppose a probe packet t (t is the index of the packet) is sent from node 0 to node i. Define Z i (t) = 1 if the probe packet reached node i and zero otherwise. If we had

26 14 access to the internal nodes, we could observe the outcomes Z i (t) for all the nodes. However, we have access only to the receiver nodes r R, so we can observe only Z r (t), r R. For a unicast scheme, the root node transmits packets t = 1, 2,... to one receiver at a time, so we observe only Z r (t) for a single receiver r for each probe packet. For the multicast scheme, on the other hand, the sender transmits each packet simultaneously to all receiver nodes. So the observed outcome for the tth probe packet consists of Z r (t) for all r R. We will analyze the data under the following conditional independence model that has also been commonly used in the literature (Caceres et al., 1999). First let α i (t) = P (Z i (t) = 1 Z f(i) (t) = 1), i.e., the probability that the probe packet reaches node i given that it reached its parent node f(i). We assume throughout that α i (t) α i. Second, the children s loss behaviors are conditionally independent given that of the parent, i.e., P (Z j (t) = 1 j D(i) Z i (t) = 1) = Π j D(i) α j. These assumptions have been used as reasonable starting points in the literature to study the properties of various estimators and measurement schemes (Caceres et al, 1999; Castro et al, 2003; Coates et al., 2002). Extensions to situations with spatialtemporal dependence will be considered in future work. We do, however, consider a limited assessment of the assumptions using the ns-simulator ([18]) in Section 4.4. Under the above assumptions, P (Z i (t) = 1) = Π s P(0,i) α s. Further, P (Z j (t) = 1 j D(i)) = Π s P(0,i) α s Π j D(i) α j.

27 Probing Schemes Throughout the thesis we concentrated on active network tomography. Two probing schemes, multicast scheme (Caceres et al, 1999) and back-to-back unicast scheme (Coates et al., 2002), were investigated by several groups in the active network tomography literature. Unfortunately both schemes have major drawbacks as discussed in Chapter I. Our work starts with proposing a new class of flexible probing schemes that are capable of overcoming such drawbacks Pure Bicast Experiments for Binary Trees Unicast schemes are appealing because of their simplicity but they contain only one-dimensional information which is not enough to estimate all the internal links. Our goal is to retain the simplicity as much as possible and yet estimate all the internal link loss parameters. The new class of bicast experiments combines bicast pairs with unicast schemes efficiently in order to estimate all the parameters of interest. We restrict attention first to binary trees for which pure bicast experiments, consisting of only bicast pairs, is sufficient. Under a bicast scheme, the packets are transmitted from the sender to pairs of receiver nodes, b = i, j, i, j R, at a time. This is accomplished as follows. For a pair of receiver nodes i, j, define node s to be a splitting node if P(0, s) is the longest common path that i and j share on the tree T. For example, in Figure II.1, the splitting node for the pair 2, 3 is 1, while the splitting node for 8, 11 is 5. Then, the sender transmits a single packet which is duplicated at the splitting node s and both packets are transmitted to the specified pair of receiver nodes. There are several possible choices of bicast experiments for a given tree. For example, for the 3-layer tree in Figure II.2, the bicast experiment may consist of all possible receiver

28 16 pairs or certain selected subsets such as the three pairs 4, 5, 6, 7, and 5, 6. As we will see below, these three pairs are sufficient to estimate all the internal loss rates in this case. One of the advantages of bicast experiments over multicast schemes is data complexity. Specifically, the number of possible outcomes with multicast schemes grows exponentially with the number of layers in the tree. For example, for the 3-layer binary tree in Figure II.2, there are 16 possible outcomes (i, j, k, l). In general, for an L layer symmetric binary tree, the number of outcomes is 2 2L 1 (2 16 = for a 5-layer tree). This problem is especially serious for inference for parameters of the delay distributions (Lo Presti et al., 2002). On the other hand, the number of possible bicast transmission schemes for an L- layer symmetric binary tree is ( ) 2 L 1 = 2 L 2 (2 L 1 1). Thus, the maximum number 2 of outcomes is 4 2 L 2 (2 L 1 1) = 2 L (2 L 1 1) for a complete bicast scheme (all possible pairs of receivers). For example, for a 3-layer tree (Figure II.2) the collection of all possible bicast pairs is C = { 4, 5, 4, 6, 4, 7, 5, 6, 5, 7, 6, 7 }, with a total number of outcomes equal to 24. For a 5-layer tree, the number of possible bicast pairs is 120 with the number of corresponding outcomes equal to 480, two orders of magnitude smaller than that in the equivalent multicast experiment. However, it turns out that we need only a small fraction of all possible bicast pairs. Later in Section 2.4 we prove that the minimal number of bicast pairs needed to identify the link loss probabilities in symmetric binary trees is 2 L Bicast Experiments for General Trees and Other Topologies Pure bicast experiments can be used with more than just binary trees. For example, we can use the following set of bicast pairs in Figure II.4: { 5, 6, 3, 7, and 8, 9 }. These constitute a minimal set, and there is no need for additional unicast schemes.

29 17 On the other hand, for the tree in Figure II.1, any pure bicast experiment will be wasteful. For example, consider the following collection of bicast schemes C = { 2, 3, 6, 12, 13, 14, 8, 15, 9, 10, 10, 11 } that contains a bicast pair that splits at every internal node and covers all the receiver nodes. This is sufficient for estimating all the elements of α. However, it is wasteful since there are two pairs that split at internal node 5: 9, 10 and 10, 11. Consider instead the following collection that combines bicast and unicast schemes: C = { 2, 3, 6, 12, 13, 14, 9, 10, 8, 11, 15 }. It is more efficient in the sense that it minimizes the overlap in coverage of the receiver nodes (see Proposition II.1) So for general trees, we will use a combination of bicast and unicast schemes to estimate the loss rates. 0 α 1 α 1 α 2 4 α3 2 3 α α α α 8 4 α Figure II.4: A General Tree Topology For Which A Pure Bicast Experiment Suffices To Estimate The Link Loss Rates. Note that one could consider a combination of different K cast transmission schemes (as discussed in Section 2.3.3), such as unicast, bicasts, tricasts, etc., that may be optimal (in some appropriate sense) for estimating the loss rates. For

30 18 example, for the tree in Figure II.1, one might use a combination of bicast schemes with a 4-cast that splits at node 5 with four receiver nodes 8, 9, 10, and 11. However, we pay special attention to combinations of unicast and bicast schemes since they are simple, easy to analyze, and sufficiently flexible k cast Schemes As noted before, we will be considering a general probing experiment that is based on a combination of k cast schemes for different values of k. So we begin with a description of a k cast scheme. A k cast scheme sends a probe simultaneously to a given subset k of the receivers in R. The scheme is completely specified by the k-tuple of receiver nodes: r 1, r 2,..., r k, r j R, j = 1,...k. For example, two possible 4-cast schemes for the general tree topology shown in Figure II.1 are 12, 13, 14, 15 and 2, 3, 6, 12. The common unicast scheme corresponds to k = 1, while for a given tree with R receiver nodes, the multicast scheme [1] becomes an R cast scheme. In general, however, the size of the multicast scheme varies with the size of the tree of interest. We have introduced the notion of a splitting node for a bicast scheme with receiver nodes r 1, r 2 in Section For a k cast scheme, the splitting nodes can be defined in terms of the splitting nodes of pairs of receiver nodes. Consider the k cast scheme with receiver nodes r 1, r 2,..., r k, and let {r i, r j } be any subset of them. Then, the internal node s s(r i, r j ) is a splitting node for this particular pair if P(0, s) is the common path that {r i, r j } share on the tree. Note that the number of splitting nodes for a k cast scheme can range from 1 to (k 1). For example, for the pair 6, 12 for the tree depicted in the left panel of Figure II.1, there is only one splitting node: 4. For the 4-cast scheme 2, 3, 6, 12, however, there are three splitting nodes ({1, 4, 7}).

31 19 A probe packet for a k cast scheme has 2 k possible outcomes, each of dimension k. These correspond to whether the outcome for the receiver node is one or zero (the node receives the transmission or not). For example, for the 4-cast scheme 12, 13, 14, 15, there are 16 possible outcomes with (Z 12 = 0, Z 13 = 1, Z 14 = 0, Z 15 = 1) indicating that the packet was successfully received by receivers 13 and 15 but not by 12 and 14. Suppose we send N probes using a k cast. This can be viewed as a multinomial experiment with 2 k outcomes; the success probability for each outcome is a complex function of the underlying link successful transmission rates α s. Consider, for example, the probability of the event (Z 12 = 0, Z 13 = 1, Z 14 = 0, Z 15 = 1). Under the posited stochastic model it is given by a sum of products of α i s and (1 α i ) s. In the next chapter, we consider a transformation of the outcome space for which the success probabilities are simple products of the link loss parameters. A collection of unicast (1 cast) schemes, bicast (2 cast) schemes,..., up to k cast schemes is named a k cast experiment. 2.4 Identifiability In this section we first establish the sufficient and necessary identifiability condition for single-source topologies. The problem for multi-source topologies is slightly more complicated. An internal node may have multiple parent nodes on an multisource topology, unlike the single-source topology case. This requires a different definition of parent nodes. We then introduce a sufficient identifiability condition for multi-source topologies.

32 Single-Source Topology Identifiability Condition We need some additional notation. Denote the probability of a successful transmission of a packet over the path P(s, l) by π(s, l). So, (2.1) π(s, l) = α i. i P(s,l) Suppose the bicast probe b is sent to the pair of receiver nodes i b, j b, i b, j b R, with splitting node s b. The observed outcome can take one of the following four values: (Z ib (t), Z jb (t)) = (0, 0) or (0, 1) or (1, 0) or (1, 1) depending on whether the packet was received by none, one or both of the intended receivers. Let γij b denote the corresponding probability of any of these events. Then, (2.2) (2.3) (2.4) (2.5) γ11 b = π(0, s b )π(s b, i b )π(s b, j b ), γ10 b = π(0, s b )[1 π(s b, i b )]π(s b, j b ), γ01 b = π(0, s b )π(s b, i b )[1 π(s b, j b )], γ00 b = [1 π(0, s b )] + π(0, s b )[1 π(s b, i b )][1 π(s b, j b )], Since γ1,1 b + γ1,0 b + γ0,1 b + γ0,0 b = 1, there are only 3 free probabilities for each pair of receiver nodes b. Notice also that π(0, i b ) = γ1,1 b + γ1,0 b and π(0, j b ) = γ1,1 b + γ0,1. b The following proposition characterizes a minimal combination of bicast and unicast schemes that lead to identifiability of all the loss parameters in a general tree topology. Define B to denote the collection of all bicast pairs used in the experiment and let U denote the collection of unicast schemes. Let Γ b = {γ1,1, b γ1,0, b γ0,1, b γ0,0} b denote the set of probabilities from a pair of receiver nodes b = i, j and denote by Γ = {Γ b : b B}, the probabilities generated by all bicast pairs in B. Let u = {δ u 1, δ u 0 } denote the probabilities of the two outcomes for unicast scheme u and denote by = { u : u U}.

33 21 Proposition II.1. Let C = B U be a collection of bicast (b B) and unicast (u U) schemes. Suppose B and U are chosen as follows: (I) for each internal node s in T, there is exactly one bicast pair b B whose splitting node is s; (II) the bicast pairs in B are chosen to maximize the number of receiver nodes r R that are covered; and (III) the unicast schemes in U are chosen to cover the remaining receiver nodes r R that are not covered by the bicast pairs in B. Then, there exists a bijection between α and Γ. Proof: Sufficiency: First notice from equations (2.2)-(2.5) that the γ b i,j s are uniquely determined by the path probabilities π b (s, t) of the bicast pair b which in turn are determined by the {α}s. Therefore, Γ = {Γ b ; b B} is uniquely determined by α. An analogous line of reasoning establishes the result for the collection of probabilities = { u ; u U} obtained from the unicast schemes u. We will show next that the elements of α are also uniquely determined by Γ (i.e. by the collection of {Γ b } for which b = i b, j b have all internal nodes as splitting nodes and by the { u } that cover receiver nodes not covered by bicast schemes). Recall that a node i V {0} belongs to the k-th layer L k of T if its shortest path from the root node has k links. We need to consider the following three cases: (i) the splitting node for bicast pair b is node 1 (i.e., belongs to layer L 1 see Figure II.2), (ii) the splitting node is any internal node (i.e. s I), and finally (iii) the case of receiver nodes r R (e.g. nodes 14 or 15 in Figure II.1). Case 1: For bicast pair b 0 = i b, j b with splitting node 1 in the canonical numbering scheme adopted in the paper, we have that α 1 = π b 0 (0, 1) = (γb γ b 0 10)(γ b γ b 0 γ b ),

34 22 and therefore determined by the elements of Γ (see left panel of Figure II.5). Case 2: We now proceed with the induction on the remaining internal nodes s I. According to the statement of the Proposition, every internal node s is the splitting node for one bicast scheme in the collection C. Assume that for all internal nodes s such that s L 1 L 2 L k 1, α s s have been determined by Γ. We show next that α t, t L k is also determined by Γ. Since t is an internal node, there exists a bicast scheme b 0 C whose splitting node corresponds to t. We have that π b 0 (0, t) = π b 0 (0, f(t))α t, with all the members of π b 0 (0, f(t)) already determined. As before we have that π b 0 (0, t) = (γb γ b 0 10)(γ b γ b 0 which combined with the previous observation establishes the identifiability of α t γ b 0 11 from the elements of Γ (see right panel of Figure II.5). Case 3: We now deal with the receiver nodes r R. A receiver node can be covered either by a unicast scheme or a bicast scheme. Notice that due to the induction hypothesis in the previous step all α s, with s I have been identified. Suppose that the receiver node is covered by a unicast scheme u 0. We then have that π u 0 (0, r) = π u 0 (0, f(r))α r with all elements of π u 0 (0, f(r)) already identified by the induction. We also have that δ u 0 1 = π u 0 (0, r), which combined with the previous observation, establishes the identifiability of α r from elements of. Suppose that the receiver node r is covered by a bicast scheme b 0 = i b0, j b0 with splitting node s b0. We have that 01), π(s b0, i b0 ) = γ b 0 11 γ b γ b 0 01 and π(s b0, j b0 ) = γ b 0 11 γ b γ b But π b 0 (0, r) = π b 0 (0, s b0 )π b 0 (s b0, r) = π b 0 (0, f(r))α r, with r being either i b0 or j b0 and the result follows as above.

35 23 Thus, we have established that all the elements of α are uniquely determined by the elements of Γ ; thus, there is a bijection between α and Γ. Necessity: We argue by contradiction. The crucial aspect in the Proposition is the existence of a bicast pair whose splitting node is an internal node in T. We consider a combination of bicast schemes in the collection C that includes all possible pairs, except the collection of pairs B 0 = {b C : splitting node is s b0 I} that have as their splitting node s b0, an internal node of T. We will show that this combination of bicast pairs fails to identify all the elements of α; thus, no bijection between α and Γ exists. Let f(s b0 ) and d(s b0 ) denote the parent and any child node of node s b0, respectively (see Figure II.6). From the previous derivations it is easy to see that the following relationships hold: (2.6) (2.7) π b (0, d(s b0 )) = π b (0, f(s b0 )) α sb0 α d(sb0 ) π b (f(s b0 ), l) = α sb0 α d(sb0 ) π b (d(s b0 ), l) for some b C and where l corresponds to some receiver node for pair b. Notice that as argued in the sufficiency part of the proof, only these π s are uniquely determined by their Γ b s. Further notice, that both (2.6) and (2.7) correspond to two actual equations, since node s b0 has two children nodes for any bicast schemes in B 0. A straightforward calculation shows that only the values of the products α sb0 α d(sb0 ) can be calculated uniquely from the elements of Γ by taking the appropriate ratios, but the individual parameters can not be disentangled. Hence, C fails to identify all the elements of α. This completes the proof of the Proposition.

36 f( S b0 ) α 1 1 Sb 0 i b j b i b j b Figure II.5: Cases 1 and 2 of the sufficiency of Proposition II.1. 0 f( S b0 ) Sb 0 d( S b0 ) i b j b Figure II.6: Demonstration of (2.6)-(2.7).

37 25 To see why condition (II) is needed, consider the following subtree of Figure II.1 with nodes 0, 1, 2, 3, 4, 6 and 7. The receiver nodes are 2, 3, 6 and 7. The following experiment with two bicast pairs 2, 6 and 6, 7 and a unicast scheme 2 satisfies conditions (I) and (III). However, this is wasteful as it covers receiver node 2 twice. The pure bicast experiment with two pairs 2, 3 and 6, 7 satisfies (II) and is the the minimal scheme. It is easy to obtain an algorithm that constructs the minimal scheme based on the set covering problem (Chvatal, 1979). This can be formulated as an integer program as follows. Given the edge set E of the tree topology T and the set of all possible bicast schemes B whose paths traverse E, find a minimum size subset B B whose members satisfy the first two conditions of Proposition II.1. The conditions show up as the constraints in the integer program formulation. The following corollary gives a necessary and sufficient condition for the link loss probabilities to be identifiable from a set of bicast probes in a binary tree. We refer to such a scheme as a minimal pure bicast experiment. The proof is a special case of that for Proposition II.1. Corollary II.1. Let B = {b = i, j : i, j, R} denote the subset of the bicast schemes to be used in a probing experiment. Let Γ b = {γ1,1, b γ1,0, b γ0,1, b γ0,0} b denote the set of probabilities from a pair of receiver nodes b = i, j under a bicast transmission scheme and Γ = {Γ b : b B}, the probabilities generated by all bicast pairs in B. There exists a bijection between α and Γ if and only if for every internal node s I, there exists a pair b = i, j B such that s is the splitting node for the corresponding pair. In other words, a minimal bicast experiment for a binary tree has as many pairs as the number of internal nodes with one pair splitting at each node. So for a symmetric

38 26 binary tree, the minimal number of pairs is 2 L 1 1. For the 3-layer symmetric tree in Figure II.2, the three pairs 4, 5, 5, 6 and 6, 7 constitute a minimal bicast experiment. So do the three pairs 4, 5, 4, 7 and 6, 7. For a 5-layer symmetric tree, the minimal number of pairs is 15, so there are only 4 15 = 60 possible outcomes (three orders of magnitude smaller than a multicast scheme). Bicast experiments also work in more general topologies such as acyclic graphs. Figure II.7 depicts a single source acyclic graph topology. Under a multicast transmission scheme the above graph would give rise to a 5-layer tree structure comprised of 12 receivers as shown on Figure II.8, since the multicast probing packets would be replicated at all the forking nodes on the graph. This is clearly inefficient. On the other hand, according to Proposition II.1, six bicast pairs whose splitting nodes correspond to the internal nodes 1, 2, 3, 4, 5 and 7 suffice to identify the loss rates. 0 α(1,2) α(0,1) 1 α(1,3) 2 α(2,5) α(3,4) 3 α(2,4) α(3,5) α(4,6) 4 5 α(4,7) α(5,7) α(5,8) α(7,9) α(7,10) 9 10 Figure II.7: A Single Source Acyclic Graph Topology. A generalization of Proposition II.1 gives the sufficient and necessary condition