# Probability-Model based Network Traffic Matrix Estimation

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1 Computer Science and Information Systems (): DOI: 0.98/CSIS3000T Probability-Model based Network Traffic Matrix Estimation Hui Tian, Yingpeng Sang, Hong Shen 3,4, and Chunyue Zhou School of Electronics and Information Engineering Beijing Jiaotong University, China School of Computer Science Beijing Jiaotong University, China 3 School of Information Science and Technology Sun Yat-sen University, China 4 School of Computer Science University of Adelaide, Australia Abstract. Traffic matrix is of great help in many network applications. However, it is very difficult to estimate the traffic matrix for a large-scale network. This is because the estimation problem from limited link measurements is highly underconstrained. We propose a simple probability model for a large-scale practical network. The probability model is then generalized to a general model by including random traffic data. Traffic matrix estimation is then conducted under these two models by two minimization methods. It is shown that the Normalized Root Mean Square Errors of these estimates under our model assumption are very small. For a large-scale network, the traffic matrix estimation methods also perform well. The comparison of two minimization methods shown in the simulation results complies with the analysis. Keywords: traffic matrix estimation, probability model,.. Introduction In an IP network, the traffic matrix (TM) describes the traffic volumes traversing the network from the input nodes to the exit nodes over a measured period. Such a matrix is very helpful in many network applications such as capacity plan, anomaly detection, traffic engineering, and network reliability analysis []. In these application scenarios, TMs act as an important input. The outputs of various network engineering tasks are directly relevant to the input. So plenty of work aim to conduct measurements on TMs [5] or indirect inference from readily available link measurements [5, 6]. Whichever way is used, it is extremely hard to obtain the traffic matrix for a large network. The link measurements which are related to traffic matrix is more realistic to be obtained in practice. Denote the link measurements by Y and the traffic matrix by X. Y is an n l by vector and X is n p by vector where n l << n p. Thus, the relationship between link measurements and the traffic matrix is represented by Y = AX where A is The corresponding author is Hong Shen.

2 30 Hui Tian et al. the routing matrix. Given a set of link measurements, we aim to infer an overall view of the traffic matrix. This is a classic under-constrained, linear-inverse problem. we cannot obtain a solution if we do not introduce any side information such as the prior model of a traffic matrix. Therefore, paper [, 4] assumed the origin-destination-demands follows a Poisson distribution. In [], Origin-Destination (OD) pairs are modeled according to a Gaussian distribution. Medina et al. assumed a logit-choice model in [7] and M. Roughan et al. proposed a gravity model [9, 0]. There are plenty of methods proposed based on these prior traffic model assumption such as Bayesian inference, information theoretic approach, maximum likelihood estimation, Linear programming and multiresolution analysis method []. These methods perform very differently for different traffic data. There is no method which can work well on all traffic data. Compressed Sensing (CS) is recently greatly developed in many applications. It shows that any sufficiently compressible signal can be accurately recovered from a small number of non-adaptive, randomized linear projection samples [6]. It can also be applied in traffic matrix estimation due to the sparsity of the traffic matrix such as the work in [3, 4, 8, 7]. In this paper, we look for a sparsity model which can be used for efficient traffic matrix estimation, resulted in a probability model [3]. This model is simple but efficient. From the simulation results, we will see a generalized model based on our probability model can work well in Origin-Destination pair traffic estimation. This generalized probability-based model can not only help to generate traffic data for large-scale network simulations, but also help to study sparse traffic matrix.. Model A large-scale network usually involves multiple routers/switches in multiple levels. In each level, many routers/switches are located for redundancy and reliability. If we simplify the network topology without changing basic networking function, we will get a tree-like network. In the tree-like network, there is one parent node and N children. The parent node is considered as a node connecting outside Internet and the child nodes are servers which may be composed of a set of children. Our model starts with a single star network. This star network is assumed to include one parent node and several child nodes which are connected through one router. This star may be the lowest level star and all of similar stars can be aggregated to form the original network. The traffic estimation on this simple star will be formed into the whole network traffic matrix. By doing so, estimating a N N traffic matrix by N link measurements which is highly under-constrained though, can be largely reduced in the dimensionality of the problem to a simple star network. We assume a network model which is composed of n servers which are connected to the Internet node via router. Given the link measurements Y observed on these edge links which are immediate link connecting the end nodes, the problem is to estimate the end-to-end traffic matrix X. Y = AX, A is an n l by n p routing matrix. In the simple single star model we assume that all servers are the same. We will relax this assumption later on, but this allows us to build a very simple traffic model firstly. This model only needs three probability parameters which is thus named as the probability model. In our probability model, it is assumed any pairs of servers send traffic to each other with the same probability p, any server sends traffic to the Internet node with the

3 Probability-Model based Network Traffic Matrix Estimation 3 same probability p and the Internet node sends the traffic to any server with the same probability p 3. Obviously, (n )p + p =, np 3 =. () 3. Traffic matrix estimation Firstly, we consider a network composed of servers, node denoting Internet and router as Figure shows. Thus we have p + p = and p 3 = /. As we consider that a server will either send the packets to other servers or to the Internet node. There will not be any traffic to itself. Thus the routing matrix A is A = () Fig.. A network with servers and Internet node Assume the total traffic on each node are X = {x, x, x 3}, where x i, i {,,..., n+ } denotes the total traffic generated by node i. The elements in Y are the traffic on outgoing link from node, incoming link to node, and then node, and 3 with the same order. Thus, p p 3 Y = BX = 0 0 p 0 p 3 X (3) 0 0 p p 0 Normally we don t have the full set of the measurements on Y. For example, we only have 3 links measurements which is denoted by Y s, Y s = B s X. If we choose 3 links, for example, incoming link to node, incoming link to node, incoming link to Internet

4 3 Hui Tian et al. node. We denote the selected link measurements as Y s, and the selection matrix is B s. Then we have B s = 0 p p 3 p 0 p (4) p p 0 Since det(b s ) 0, we have X = Bs Y s. Because the relationship between the OD traffic matrix X and traffic generating by end nodes X can be described by the assumed probability matrix as follows p 0 0 p p 0 X = P X = X (5) 0 p p p Therefore we have X = P Bs Y s. P and B s can both be calculated as above, we can get a solution to X in this probability model and let it be X p. X p = p p p p p p p p p p p p p p p p Y s (6) Therefore, we can obtain a solution related to one simple parameter under the assumption of the probability model. If p can be set an appropriate value according to the realistic networks, Y s is observed measurements on selected links, we can have a determined X p. Now our problem is to find a solution X which is the closest to X p under the constraint Y = AX. That is, Minimize f(x) = X X p, s.t. Y = AX, where is the L norm. For simplicity in comparison, we call this to be Minimization (Min ) method. This method is difficult to find an efficient solution if there are errors in measurements on Y. Therefore, we can employ Lagrangian function. The aim is thus to minimize L(X, λ). L(X, λ) = X X p +λ AX Y (7)

5 Probability-Model based Network Traffic Matrix Estimation 33 Again, for simplicity in comparison, we denote this by Min method. In Min method, L norm is not easy to write in the form of quadratic X and using derivative, we solve the problem of Minimize g(x) = X X p, s.t. Y = AX, and construct the Lagrangian function as follows. Thus, L(X, λ) = X X p + λ AX Y (8) L(X, λ) = λx T A T AX + λy T Y λy T AX + X T X X T X p + X T p X p (9) The Karush-Kuhn-Tucker conditions for this local minimum are given as follows. L(X, λ) X 0, (0) L(X, λ) λ = 0, () L(X, λ) X = 0, () λ L(X, λ) λ = 0, (3) λ and X 0, λ 0. Deriving from above, the KKT conditions are simplified into: (λa T AX + I)X = λa T Y + X p, (4) X T A T AX Y T AX + Y T Y = 0, (5) and X 0, λ 0. Hereby we can solve this Minimization problem by quadratic programming. Intuitively, in this quadratic programming, the more links measurements are considered, the better the result is obtained for X.

6 34 Hui Tian et al. 4. Simulations 4.. Metrics As we discussed in the above section, there are two regularization methods which obtain the traffic matrix closest to the matrix X p in the prior assumed model (also called the probability model). To compare these two methods, we will use Normalized Root Mean Square Error () to show the performance of estimated traffic matrices. Here the is defined as e. i ( ˆx i x i ) /(N (N + )) e = Max(x i ) Min(x i ). (6) Since quadratic distances are acceptable distance metric and lead to simple quadratic optimizing problem, it is easy to obtain solutions to both minimization problems. As we have denoted for simplicity, the Minimization problem Min{d( ˆX, X p )} subject to Y = AX and X 0 is denoted by Min method and the other Minimization problem by Min method. Firstly, we simulate both methods on a network with N = 8 servers and one Internet node. Thus in total (N + ) = 8 links measurements are available. The traffic matrix to estimate is a 9 by 9 matrix, i.e. an 8 by vector. We obtain a cumulative distribution function (CDF) of to observe the performance of these estimates. We then have a look at the sensitivity of s to λ. As the number of links used in measurements increases, we will check the trend of s. 4.. A General Model The simple prior model leads to all servers having identical traffic. This may not be accordance with the practical cases where there are differences on measurements to different servers. We thus generalize the above simple model to a more practical model which combines this simple model with a random traffic model. This general model is described to be X gen = αx p + ( α)x random. In this general model, it is seen that the parameter α defines a random traffic matrix to be estimated which is still related to the prior assumed probability model. The changes of α give different weights of the prior model in this general model. Intuitively, we can conclude that the performance of estimates would be affected by this parameter α. Our simulation will check how differs as α increases. We also introduce a bias factor β which counts the ratio of wrong measurements to correct measurements. This factor is helpful in testing the performance of estimation in both the probability model and the general model Experiments The Figure (a) is obtained by monitoring 0 links measurements. The other two important parameters are set as a = 0.5 and λ = 0. We checked the CDF of of the

7 Probability-Model based Network Traffic Matrix Estimation 35 estimates by two minimization methods in two models separately. As shown in Figure (a), the CDF of of estimated results by Min method (represented by two red curves) are worse than that by Min method. However Min method gives very small whose CDF approaches to with an of in the general model. In the probability model, both methods work very competitively and Min is only a bit better than Min method. We also note that by using Min method, the estimation process gives a much better performance in the probability model than in the general model. Min method does not show this superiority in probability model to general model which shall be shown in a more large-scale network. Figure (b) shows the results when using the full set of 8 links to estimate the original traffic matrix. It is obvious both methods work out CDFs approaching to with smaller s, especially when we look at the performance of Min method in the general model. In the probability model, Min method works slightly better than Min method. This is because more links measurements give more constraints in Y = AX which limits the searching space of finding a minimal distance from ˆX to X p. This outperformance will be more obvious when there are errors in measurements. This also gives the reason that we prefer Min method to Min method in a large-scale network with more possible errors in measurements. Now we have a look at how varies as λ increases when using Min method. Again, we use measurements on 0 links in this 9-node network simulation and let α = 0.5. Figure 3 shows that the does not change when λ takes a value greater than a threshold. It is found λ does not affect the estimation performance if it is set a value greater than 5. Therefore, the value of λ being 0 in above simulations provides a good initialization for the traffic estimation process to run. In the general model, α is a parameter which adjusts the weights of the traffic data under the probability model assumption and the random traffic data. Figure 4(a) shows how this parameter affects the performance of estimation. It is obvious that as α increases which means the random data weights less, both methods work better. In Figure 4 (a), since all measurements are correct, both methods can work out the same TM which is closest to the practical TM. The curves of both methods are overlapped. When there are errors in measurements, two methods have different performances. We set the bias factor to be 0. in Figure 4(b), that is, there are 0% links which have errors in measurements. Min method shows a better performance than Min method. When the bias factor increases which means there are more errors in measurements, the outperformance of Min method to Min method shall be greater. In Figure 5, the read curve denoted by data gives the result of Min method in the general model. As the number of links used in estimation increases, data shows an improved estimation performance of Min method in the general model. The other three cases do not show obvious change as the number of links used in estimation varies. It is predicted in a large-scale network, using more link measurements would be more beneficial except the case where Min method is used and errors in measurements exist. In practice, the network size is much larger than the above 9-node network simulated above. How will these two estimation methods perform as the network size N is increasing? Figure 6 gives the normalized RMSE by two methods in two models as N changes. We let α = 0.5, λ = 0 and the number of links used to estimate be three. Network size is changing from 4 nodes to a reasonably large size 53 nodes. It is seen that the

8 36 Hui Tian et al Min in general model Min in general model Min in probability model Min in probability model CDF of Min in general model Min in general model Min in probability model Min in probability model (a) 0.7 CDF of (b) Fig.. CDF of Normalized Root Mean Square Error by using both Minimization methods for two models in a 9-node network Min method in Probability model Min method in General model lamda Fig. 3. Normalized Root Mean Square Error as λ changes by Min method in two different models

9 Probability-Model based Network Traffic Matrix Estimation Min method Min method alpha (a) Min method Min method alpha (b) Fig. 4. CDF of Normalized Root Mean Square Error by using both Minimization methods for two models in a 9-node network

10 38 Hui Tian et al data data data 3 data Number of links used in estimation Fig. 5. Normalized Root Mean Square Error as the number of links used in estimation varies by two methods in two different models trends to be smaller as the network size increases. Therefore, we can predict when the network size is greater, say hundreds or thousands, the of these two estimation methods would work fairly well. Figure 6 also shows two methods work better in probability model than in general model. As we only use 3 links to estimate the total traffic matrix, Min method performs a bit better than Min method. When we have more links measurements, Min will show its superiority to Min method as shown in above simulation. Note also both methods work well and are convergent when only 3 links are used in estimation. When there are more links measurements available, both methods will perform better Min method in general model Min method in general model Min method in probability model Min method in probability model Network size N Fig. 6. Normalized Root Mean Square Error as N changes by two methods in two different models 5. Conclusion Traffic matrices in a large-scale network are difficult to be estimated because the estimation problem from limited link measurements is highly under-constrained. We propose a simple probability model for a star-like network which is easy to be aggregated to more

11 Probability-Model based Network Traffic Matrix Estimation 39 star-like networks together to form a large-scale practical network. In order to enable the probability model to adapt to more realistic data in the network, we generalize the probability model to include the random traffic data. Traffic matrix estimation is then conducted under these two models by two minimization methods. It is shown that the Normalized Root Mean Square Errors of these estimates under our model assumption are very small. For a large-scale network, the traffic matrix estimation methods also perform well. The comparisons between two minimization methods showed that Min method is more robust to network measurements errors and thus preferred in practice. Acknowledgments. This work is supported by National Natural Science Foundation of China No and 6703, the Fundamental Research Funds for the Central Universities No. 0JBM06 and No. 0JBM0, and Beijing Natural Science Foundation No References. Alderson, D.L., Chang, H., Roughan, M., Uhlig, S., Willinger, W.: The many facets of internet topology and traffic. Networks and Heterogeneous Media (4), (Dec 006). Cao, J., Davis, D., Wiel, S.V., Yu, B.: Time-varying network tomography. J. Amer. Statis. Assoc 95(45), (000) 3. Coates, M., Pointurier, Y., Rabbat, M.: Compressed network monitoring for ip and all-optical networks. In: ACM SIGCOMM Internet Measurement Conference (IMC). pp. 4 5 (007) 4. Crovella, M., Kolaczyk, E.: Graph wavelets for spatial traffic analysis. In: Proc. of IEEE Infocom. pp (003) 5. Feldmann, A., Greenberg, A., Lund, C., Reingold, N., Rexford, J., True, F.: Deriving traffic demands for operational ip networks: Methodology and experience. IEEE/ACM Transactions on Networking pp (Jun 00) 6. Haupt, J., Bajwa, W.U., Rabbat, M., Nowak, R.: Compressed sensing for networked data (Dec 007) 7. Medina, A., Taft, N., Salamatian, K., Bhattacharyya, S., Diot, C.: Traffic matrix estimation: existing techniques and new directions. SIGCOMM Comput. Commun. Rev. 3(4), 6 74 (00) 8. Rincon, D., Roughan, M., Willinger, W.: Towards a meaningful mra analysis of traffic matrices. In: ACM SIGCOMM Internet Measurement Conference. pp Vouliagmeni, Greece (Oct 0-008) 9. Roughan, M., Greenberg, A., Kalmanek, C., Rumsewicz, M., Yates, J., Zhang, Y.: Experience in measuring internet backbone traffic variability: Models, metrics, measurements and meaning. In: Proc. of the International Teletrafic Congress (ITC-8). pp Berlin Germany (003) 0. Roughan, M., Thorup, M., Zhang, Y.: Traffic egnieeging with estimated traffic trsffic matrices. In: Proc. ACM SIGCOMM Internet Measurement Conference (IMC). pp Miami Beach, FL, USA (003). Tebaldi, C., West, M.: Bayesian inference on network traffic using link count data. J. Am. Statist. Assoc. 93(44), (998). Tian, H., Roughan, M., Sang, Y., Shen, H.: Diffusion wavelets-based analysis on traffic matrices. In: Proc. of th International Conference on Parallel and Distributed Computing, Applications and Technologies. Gwangju, South Korea (Oct 0-, 0) 3. Tian, H., Sang, Y., Shen, H.: New methods for network traffic matrix estimation based on a probability model. In: Proc. of IEEE conference on Networks. Singapore (Dec 0) 4. Vardi, Y.: Network tomography: estimating source-destination traffic intensities from link data. Journal of the American Statistical Association 9, (996)

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