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(1)AGH University of Science and Technology Faculty of Computer Science, Electronics and Telecommunications. Ph.D. Dissertation Krzysztof Rusek. Router interface as a queuing system with correlated service times Supervisor: Professor Zdzisław Papir Assistant supervisor: Lucjan Janowski, Ph.D. Eng..

(2) AGH University of Science and Technology Faculty of Computer Science, Electronics and Telecommunications Department of Telecommunications al. Mickiewicza 30, 30-059 Kraków, Poland tel. +48 12 6345582 fax +48 12 6342372 http://www.agh.edu.pl http://www.iet.agh.edu.pl http://www.kt.agh.edu.pl. c Krzysztof Rusek, 2016 Copyright All rights reserved LATEXtemplate by Rafał Stankiewicz.

(3) Acknowledgements. Many people have helped me in my work on this dissertation - I wish to thank all of them. First of all, I would like to thank my supervisors Professor Zdzisław Papir and Lucjan Janowski Ph. D. for their advice, support and patience. I wish also to thank Piotr Guzik for fruitful discussions about statistical data analysis, Piotr Wydrych for his support with LATEXand Olga Białas for language corrections. AGH UST Student Campus administrator Marcin Jakubowski was also very helpful by providing me trace files generated by dormitories..

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(5) Abstract. This dissertation deals with router interface modeling towards queue characteristics calculation. The main focus of the presented analysis is the investigation of the memory architecture in the device and its implications on the analytical models. There is the enormous amount of research on traffic models reported, however, a model of router interface has not gained proper consideration yet. Experiments with a real device reveal that an incorrect model of the router interface can result in a significant disparity between drop probabilities measured on a physical interface and derived from a trace-driven simulation study. In this dissertation an accurate model of the router interface (Cisco IOS-based routers with a non-distributed architecture) is proposed. The interface is modeled as a packet buffer limited to a fixed number of packets regardless of their lengths. The buffer is described as a finite FIFO queuing system fed by the Markovian Arrival Process (MAP) with service times forming a semi-Markov (SM) process (M AP/SM/1/b in Kendall’s notation). Such assumptions allow to obtain new analytical results for the queuing characteristics of the buffer. The following were considered: time to fill the buffer, local loss intensity, loss ratio, and the total number of losses in a given time interval. Predictions provided by the proposed model can be few orders of magnitude closer to the trace-driven simulation results compared to the prediction of the M AP/G/1/b model. Keywords buffer, router interface, Markovian Arrival Process, semi-Markov service time, hidden Markov model, finite-buffer queue, packet loss, first passage time, transient state.

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(7) Streszczenie. Rozprawa dotyczy modelowania interfejsu routera w celu obliczania charakterystyk kolejki. Głównym zadaniem przedstawionej analizy jest zbadanie architektury pamięci urządzenia oraz jej implikacja w modelach analitycznych. Wiele z istniejacych badań odnosi się do modelu samego ruchu. Poprawność tych modeli wraz z modelem interfejsu rutera nie jest jednak tak często weryfikowana. Eksperymenty na rzeczywistym urządzeniu pokazują jednak, że nieprawidłowy model interfejsu może spowodować znaczne różnice między prawdopodobieństwem odrzucenia pakietu mierzonym na interfejsie fizycznym i pochodzącym z symulacji. W rozprawie zaproponowano dokładny model interfejsu routera (router na bazie Cisco IOS o nie-rozproszonej architekturze). Interfejs jest modelowany jako bufor ograniczony określoną liczbą pakietów (niezależnie od ich długości) a nie rozmiarem pamięci. Bufor ten jest opisany jako skończony system kolejkowy FIFO zasilany przez Markowowski proces zgłoszeń (MAP) z semi-Markowowskim (SM) czasem obsługi (M AP/SM/1/b w notacji Kendalla). Takie założenia pozwalają uzyskać nowe wyniki analityczne dla charakterystyk kolejkowania pakietów. Wyznaczone zostały następujące charakterystyki: czas do przepełnienia bufora, lokalne natężenia strat, współczynnik strat oraz łączna liczba strat w danym przedziale czasowym. Przewidywania proponowanego modelu mogą być nawet kilka rzędów wielkości bliżej wyników symulacji w porównaniu do wyników dla systemu M AP/G/1/B. Słowa kluczowe bufor, interfejs rutera, Markowowski proces zgłoszeń, semiMarkowowski czas obsługi, ukryty łańcuch Markowa, kolejka o skończonej pojemności, straty pakietów, czas do przepełnienia bufora.

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(9) Contents. Contents. ix. List of Figures. xi. List of Tables. xiii. List of Symbols. xv. Introduction 1 Packet Buffers 1.1 Buffer Policies . . . . . . . . . . . . 1.2 Experiment . . . . . . . . . . . . . 1.3 Model of a Router Interface . . . . 1.3.1 Buffer Capacity . . . . . . . 1.3.2 Service Process . . . . . . . 1.4 Results . . . . . . . . . . . . . . . . 1.4.1 Drop Function . . . . . . . 1.4.2 Traditional Router Models . 1.4.3 Model verification . . . . . 1.4.4 Measurement Errors . . . . 1.5 Conclusion . . . . . . . . . . . . .. 1. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. 5 5 9 11 11 12 13 13 14 15 17 18. 2 Area of Research 19 2.1 Traffic in telecommunication networks . . . . . . . . . . . . . . . . 19 2.2 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.

(10) x. 3 System Model 3.1 Notation . . . . . . . . . . 3.2 Traffic Model . . . . . . . 3.2.1 Interarrival Times 3.2.2 Packet Lengths . . 3.3 Queue model . . . . . . .. Contents. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. 25 25 26 26 27 29. 4 M AP/SM/1/b Queue Characteristics 4.1 Potential and notation . . . . . . . . 4.2 Time to Buffer Overflow . . . . . . . 4.3 Loss Process . . . . . . . . . . . . . . 4.4 Transient and Stationary Analysis . 4.4.1 Computational aspects . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. 31 31 33 36 38 39. 5 Numerical Results 5.1 Parameter Estimation . . . . . . . . . . 5.2 Model Validation . . . . . . . . . . . . . 5.2.1 Simulated Characteristics . . . . 5.2.2 Validation Results . . . . . . . . 5.2.3 Discussion of Validation Results 5.3 Transient State . . . . . . . . . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. 43 43 47 48 49 56 57. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. Conclusions. 59. A Testbed accuracy. 63. B Sampled Hidden Markov Model. 65. References. 67.

(11) List of Figures. 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8. 3.1. 5.1. Cumulative distribution of packet lenghts [3] – jumps show packet lengths and dashed line shows the average packet length . . . . . . Difference between packet 2 (top) and byte 1 (bottom) oriented buffers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Queue of pointers . . . . . . . . . . . . . . . . . . . . . . . . . . . Concept of the experiment. Drops occurring in the physical router and in the trace-driven simulation were compared . . . . . . . . . . Network used for tests, R1 – tested router, PC2 – traffic generator, PC3 – traffic destination, PC1,PC4 – sniffers . . . . . . . . . . . . Model error for ρ = 0.8 and ρ = 0.9 . . . . . . . . . . . . . . . . . . Error functions. Solid line – error function, dashed line – dispersion measured as 95% confidence interval . . . . . . . . . . . . . . . . . Error function for ρ = 1.1, 20 packet queue, solid line – error function, dashed line – dispersion (95% confidence interval), straight line – linear approximation of dispersion . . . . . . . . . . . . . . .. 6 7 7 10 10 15 16. 17. M AP/SM/1/b queue model: MAP arrivals with m ˇ states, single server with m ˆ states, buffer of size b . . . . . . . . . . . . . . . . . 29. Model fitting. Autocorrelation functions and moments in format: nameE,σ where name is either trace or the name of the model. . . 46 5.2 Cumulative distribution function of time to reach buffer of size 168 packets starting an empty system. The utilization ρ = 0.95, states of the MAP and the server are according to the stationary distribution. Time unit is the average service time. Trace: AGH. . 49.

(12) xii. List of Figures. 5.3. 5.4. 5.5. 5.6. 5.7 5.8 5.9. Cumulative distribution function of time to reach buffer of size 20 packets starting an empty system. The utilization ρ = 0.8, states of the MAP arrival process and the server are according to the stationary distribution. Time unit is the average service time. Trace: AGH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Average time to buffer overflow versus the buffer size. The utilization ρ = 0.95, states of the MAP arrival process and the server are according to the stationary distribution. Trace: AGH. . . . . . . . Average time to buffer overflow versus the buffer size. The utilization ρ = 0.8, states of the MAP arrival process and the server are according to the stationary distribution. Trace: AGH. . . . . . . . Stationary packet loss ratio versus buffer size for ρ = 0.7 (a), ρ = 0.8 (b) and ρ = 0.9 (c). Confidence intervals (≈ 2%) are omitted because they are tiny and invisible on a logarithmic scale. Trace: AGH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Characteristics for the trace BC-pAug89, KPC estimator . . . . . . Characteristics for the trace BC-pAug89, EM estimator . . . . . . Local intensity of the loss process for the initially empty (a) and full (b) buffer (b = 62 packets). Time unit is the average service time. Double indexed ∆ is for M AP/G/1/b system (solid line), triple indexed ∆ is for M AP/SM/1/b system (dashed line). . . . .. 50. 51. 51. 52 53 54. 58. A.1 Network used for tests, R1 – tested router, PC2 – traffic generator, PC3 – traffic destination, PC1,PC4 – sniffers . . . . . . . . . . . . 64.

(13) List of Tables. 5.1 5.2 5.3. Time to buffer overflow simulator verification: ρ = 0.9, 50,000,000 samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Loss Ratio for shuffled trace, and HMM packet lengths . . . . . . . 55 Time to buffer overflow for shuffled trace. . . . . . . . . . . . . . . 55.

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(15) List of Symbols. 0. Column vector of zeros. 1. Column vector of ones. b. Buffer capacity. Corr(k) d(i) D(i) Dk. Autocorrelation function at lag k Drop function Cumulative drop function BMAP parametrisation matrix. ∆n,i,j (t). Expected value of number of losses observed in an interval (0, t]. δn,i,j (s). Laplace transform of expected value of number of losses observed in an interval (0, t]. E. Server state emission matrix. Fi (t). Cumulative distribution function of service time in the i-th state. hn,i,j (s). Laplace transform of the tail (1-CDF) of time to buffer overflow. ˇ J(t). State of the arrival process at time t. ˆ J(t). State of the server at time t. λ. Average arrival rate.

(16) xvi. List of Symbols. L(t) LR. Number of losses observed in an interval (0, t] Loss ratio. m ˇ. Number of states in the arrival process. m ˆ. Number of states in the service process. N (t). Number of arrivals until time t. π ˇ. Stationary distribution of the arrival process. π ˆ. Stationary distribution of the service process. Pˇi,j (n, t). ˇ = j|N (0) = 0, J(0) ˇ = i) Counting function P(N (t) = n, J(t). Pˇi,j,k,l (n, t). ˇ = k, J(t) ˆ = Redefined counting function P(N (t) = n, J(t) ˇ = i, J(0) ˆ = j) l|N (0) = 0, J(0). ρ. system utilisation. Rk (s). Potential function. T. Server state transition matrix.

(17) Introduction. With the growth of the Internet, packet networks have become one of the most important elements of the global communication infrastructure. The performance of these networks is heavily influenced by the phenomena associated with queuing packets, such as packet loss caused by buffer overflow or delays associated with queuing. Theoretical understanding of such phenomena is always appreciated as it provides dynamic as well as stationary characteristics of queuing process. Analytical results can also help to optimize device parameters for different traffic types and give some insight how to improve the device design. In order to correctly assess network performance, one has to consider packet storage and packet service in the device. A lot of traffic models and methods for queue modeling can be found in the literature. For a particular hardware architecture, some models are more tightly bounded to the hardware then the others, what affects the accuracy. Traffic models are validated very often, while device models receive much less attention. This dissertation bridges the gap between theory and practice by an experimental study of a physical device and and analytical research towards modeling such device. Results from experimental study allow to develop more realistic simulation environments while the theoretical work increases the understanding on how traffic correlation structure affects queue behaviour for different buffer sizes.. Scope and Thesis This dissertation deals with the single interface of a network switching device like router whose buffer can accommodate a certain number of packets regardless of their length. Service time of each packet is proportional to its length. Such interface is modeled as a queuing system with Markovian input and correlated.

(18) 2. Introduction. service times. Markovian traffic captures interarrival times autocorrelation while service time correlation captures packet length autocorrelation. The following thesis will be supported by both theoretical analysis and experimental verification: The interface of the network device whose buffer can accommodate a certain number of packets regardless of their length can be modeled as a queuing system M AP/SM/1/b with autocorrelated service times. It is possible to solve the model analytically and calculate selected transient and stationary queue characteristics. Using the potential method, developed by Andrzej Chydziński, I solved the M AP/SM/1/b system and derived analytical formulas for packet loss process and time to buffer overflow in both stationary and transient state. The proposed model is compared to the model with independent service times and trace driven simulation in a common evaluation framework. The dissertation is organized as follows. Chapter 1 describes experiments on a real device. Memory model of the interface is defined and validated in series of experiments. The types of buffers defined in this chapter are used later on in the dissertation. This is the key to understand the problem addressed in the dissertation. The area of research is discussed in Chapter 2. The chapter introduces different models of the router interface, and emphasizes how the models require or imply the memory model of the buffer. The chapter ends with the description of the estimation procedures developed for the Markovian traffic models and used in the dissertation for numerical experiments. The formal mathematical model of the interface presented in Chapter 1 is proposed in Chapter 3. The chapter describes the traffic model and the queue model. For the queue model, it is discussed how M AP/SM/1/b system approximates the real interface. Chapter 4 contains the main theoretical result of this dissertation, a derivation in the form of two theorems, of the formula for the Laplace transform of the loss process and the time to buffer overflow in M AP/SM/1/b system. Beyond that, numerical aspects are discussed, in particular an efficient numerical procedure for the case of the service time distribution being the mixture of general distributions is derived. Numerical evaluations of the queue characteristics are collected in Chapter 5. The characteristics are calculated for two systems: one with correlated service times and one with independent service times. Both results are compared to the results of trace driven simulations. The chapter ends with the discussion and statistical analysis of the numerical results. The dissertation concludes with a summary of the proposed contributions, and the perspectives for further research..

(19) Introduction. 3. The dissertation includes two Appendices. Appendix A discusses the accuracy of the experiments with the physical devices described in Chapter 1. It is explained why all the interfaces in the testbed must have the same link speed in order to avoid time measurement error. Appendix B contains derivation of the formula for a transition matrix in a hidden Markov model under independent random sampling. This is an approximation of how packet drops affect service times.. Publications Related to the Dissertation Selected research results have been presented on the international conferences and in the international journals. The list of relevant publications, in chronological order, is as follows: [45] K. Rusek, L. Janowski, and Z. Papir. Correct router interface modeling. In Proceedings of the second joint WOSP/SIPEW international conference on Performance engineering, ICPE ’11, pages 97–102, 2011. ISBN 978-1-45030519-8 [46] K. Rusek, Z. Papir, and L. Janowski. MAP/SM/1/b model of a store and forward router interface. In Performance Evaluation of Computer and Telecommunication Systems (SPECTS), 2012 International Symposium on, pages 1–8, 2012 [47] K. Rusek, L. Janowski, and Z. Papir. Transient and stationary characteristics of a packet buffer modelled as an MAP/SM/1/b system. International Journal of Applied Mathematics and Computer Science, 24(2):429–442, 2014 [44] K. Rusek and Z. Papir. Analiza pojemności bufora i skali czasu autokorelacji ruchu. Przegląd Telekomunikacyjny, Wiadomości Telekomunikacyjne, 8–9/2015:730–732, 2015 The experiments on real devices and different types of memory model of the router interface are reported in [45]. The mathematical model of the queue described in [45] is presented in [46], where the Laplace transform of the loss process is derived. In [47] an analysis of time to buffer overflow is discussed. General discussion about model accuracy is presented in [44]..

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(21) 1. Packet Buffers. The aim of Chapter 1 is to determine how to model a router interface as a single queue in order to accurately predict packet drops. There is an enormous amount of research on traffic models reported, however, a model of router interface has not gained proper consideration yet. Chapter 1 describes a technical packet buffer specification and the experiments performed in order to determine how to model these buffers in order to accurately predict packet drops. The main issue addressed in Chapter 1 is the mechanism of packet storage in the device memory and how it affects queue models.. 1.1. Buffer Policies. Traffic in packet network can be described in terms of packets or memory units such as bits or bytes. Since packet buffer has the storage part and the service part, there are four types of storage buffers possible to construct: 1. byte-byte/s, 2. byte-packet/s, 3. packet-byte/s, 4. packet-packet/s. In order to model a network device one has to consider either how packet is stored and how it is served. But, what does it really mean packet storage or byte service? Packets and bytes are just the units of traffic volume and also units of buffer capacity. Thus, packet or byte per time unit is the unit of service rate. The problem expressed in the language of queuing theory appears when the system has different units of capacity and different units of customers served in one time.

(22) 6. 1. Packet Buffers. unit. For example, let the storage be measured in packets and the service rate be measured in bytes per second. Since packets are not of the same length (Fig 1.1) there is no fixed conversion between packets and bytes. Averaging does not help much since packets of an average length may even not be observed in the traffic. 1 0.8 0.6 0.4 0.2 0. 200. 400. 600. 800. 1,000. 1,200. 1,400. Packet length [B]. Fig. 1.1: Cumulative distribution of packet lenghts [3] – jumps show packet lengths and dashed line shows the average packet length Let us take a closer look at each of the four types of storage buffers. The storage is measured in either packets or bytes since it is the amount of traffic the buffer can hold. Thus, packet can be stored in the device memory in two different ways, named in this dissertation storage policies. Definition 1. Byte oriented policy is memory organisation with buffers size limited by a fixed number of bytes. Definition 2. Packet oriented policy is memory organisation with buffers size limited by a fixed number of packets regardless their lengths. A buffer with byte oriented policy can store a fixed amount of data (in bytes or memory blocks) i.e. a large number of short packets or a small number of long packets. On the other hand, under packet oriented policy, the buffer can store a fixed number of packets regardless of their lengths. Difference between those two policies is visualized in Fig. 1.2. Note that the second policy is easier to implement since a router operating system reads a packet payload into a special memory and then operates only on payload pointers as shown in Fig 1.3. Therefore, a router knows how many packets (pointers) it has in a queue, however, in order to know how many bytes are stored, additional information (packet size) has to be read and processed. On the other hand, a service is the volume of traffic (number of bytes or packets) transferred in time unit. In reality, packet service is impractical since.

(23) 7. 1.1 Buffer Policies. Bytes. Packets. Bytes/Packets. Fig. 1.2: Difference between packet 2 (top) and byte 1 (bottom) oriented buffers Memory. Queue Fig. 1.3: Queue of pointers transmission media have capacity measured in bits per second. Thus only two types of storage buffers are implemented in physical devices: 1. byte-byte/s, 2. packet-byte/s. These two implementations seems to be quite similar but they are totally different when modeled as a queuing system. In order to understand how storage policy affects the model of buffer let us follow the process of transmitting the packet to the destination. One important assumption is that the transmission time of the packet is dominating while other processing times can be neglected. For simplicity let us also assume for a moment that the buffer is full. For the packet oriented buffer, the entire packet has to be.

(24) 8. 1. Packet Buffers. removed so that the next packet can be enqueued. On the other hand, in the byte oriented buffer, if only a fraction of the packet is transmitted, there is a room for a small packet to be enqueued. This small packet would be dropped in the case of packet oriented buffer. Different storage policies would result in different losses, under the same traffic demand, so when modeling a network device, one has to precisely specify which storage model (byte or packet oriented) is used in order to adhere it to a traffic description which comes from network measurements [31, 49]. The question is how to describe those two types of buffers in the language of queuing theory? Let us analyze the byte oriented storage policy first. In practical implementation of this policy, due to performance reasons, memory is organized as a collection of small blocks of constant size (chunks). Each packet is split and each part is stored independently in a different block. Blocks are the resources of the device. In the model of the buffer, the number of blocks is the capacity of the queuing system. Since the size of the block is constant, then the time required to transmit it is also constant. This time is the service time in the model of the system. Real packets have variable lengths, see Fig 1.1. However, it is not a problem, because the arrival of packet consuming n memory blocks can be modeled as the simultaneous arrival of n small packets. This is known as the batch arrival. In this model, the device interface is similar to a bucket leaking liquid. The model becomes a quite simple queuing system of a finite capacity with batch arrivals and deterministic service time. All statistical properties of the traffic are captured by the input process and a single memory block can be interpreted as a job in queuing system. Such model has been often used with various sophisticated random processes as the input. When a buffer is packet oriented, then both an input process and a service rate have to be measured in packets per second. However, in such a case the service rate is variable due to the different sizes of the packets. This problem is discussed in detail in [20]. The model becomes complicated and it has gained much less attention. The queue is still drained at a constant number of bytes per unit time. However, a single memory block can no longer be identified as a job. Since packet lengths are not constant in IP networks, the service time is no longer deterministic. In fact the service process must have the same statistical properties as the packet lengths observed in the traffic traces. This includes the first and the second order statistics together with an autocorrelation function. Now, the entire packet is a job and there is no need for a batch arrival (see Fig. 1.2). Information about packet sizes is removed from the input process and transferred to the service process. Having seen that packet oriented buffers behave so differently than their byte oriented counterparts it is obvious that they require different models. Both storage.

(25) 1.2 Experiment. 9. policies have corresponding mathematical models (referred in this dissertation as byte or packet oriented models). Byte oriented models happen to be far more commonly used by theorists e.g. [6, 9, 49] than packet oriented buffers e.g. [31]. Sometimes the byte oriented model of the buffer is dictated by the traffic model as it is for fluid approximation models (commonly used for self-similar traffic [36]) where a storage model has to be byte oriented in order to obtain a consistent model of a system. A byte oriented approach is correct for some types of high-end routers where packets are stored in chunks of memory [9] (e.g. Cisco ASR and GSR series used by ISPs, where implementations differ significantly from the simple model described in [38]). However, the documentation of a Cisco IOS-based routers with a non-distributed architecture (for example, the ISR series commonly used in small to medium-sized businesses and enterprise branch offices) states that the queue limit is set in packets [11], so a packet policy should be assumed. Also, the default storage policy in the popular network simulator ns-2 is packet oriented [23]. Clearly, there is a gap between theory and practice. In order to bridge this gap an experiment was proposed and performed. The experiment should answer the question if it is possible to mimic the packet policy (suggested by the documentation) by the byte policy so popular among theorists? Or in a more general form, if it is possible to accurately simulate the router interface as a single queue having just the router configuration?. 1.2. Experiment. In order to investigate how accurate the packet loss prediction from the simulation model of a router interface is, a series of two step experiments were performed. The experiments could also verify buffer policy if it were not stated in the documentation. The concept of the experiment is fairly simple. In the first step, artificially generated traffic was transmitted through a lossy interface in the physical router with simultaneous sniffing of incoming and outgoing traffic. In the second step a trace-driven simulation was used to verify if the dropped packets were the same. Buffer size, service rate, and storage policy are the parameters of the simulator that can be set up to mimic the real interface. All simulations is the second step were performed in MATLAB, using custom routines, though the MATLAB simulator was verified by the ns-2 simulator. The concept of the experiment is presented in Fig. 1.4. Each transmitted packet carried a unique ID in its payload, so it was possible to identify dropped packets by checking missing IDs in the output trace file. The experiments took place in a controlled and isolated environment, so there were almost no background traffic..

(26) 10. 1. Packet Buffers. Traffic. Router. drops. speed/buffer. Wireshark. same ?. drops Simulator Fig. 1.4: Concept of the experiment. Drops occurring in the physical router and in the trace-driven simulation were compared. R1. R2. C C1. C2. PC1. C2. S. PC2. PC3. PC4. Fig. 1.5: Network used for tests, R1 – tested router, PC2 – traffic generator, PC3 – traffic destination, PC1,PC4 – sniffers. A simple network depicted in Fig. 1.5 was set up for the purpose of the first step of the experiment. In the tests Cisco 2800 Series routers being typical of the ISR family were used. For the connections C1 = C2 = 100 Mbit/s Ethernet was used and the bottleneck was C = 10 Mbit/s link. Such a setup guarantees that drops occur only on a tested interface. The PC2 computer was a traffic generator and the PC3 was a traffic destination. For simplicity UDP was used as a transport protocol. Computers PC1 and PC4 served as sniffers dumping incoming and outgoing traffic, respectively. The Wireshark application was used for traffic capturing. It is crucial for an experiment accuracy that links C1 and C2 have the same speed what is explained in Appendix A. The tested router was R1 and the bottleneck was link C = 10 Mbit between R1 and R2. One may consider.

(27) 1.3 Model of a Router Interface. 11. the router R2 as redundant but such architecture allows replacing the Ethernet link R1R2 by a serial link. The experiments revealed surprising results. In all the simulations of packet oriented buffer with size the same as in the router configuration the number of dropped packets was at least two times greater than the drop number observed in reality. A byte oriented buffer could be parameterized to approximate the real device, but the approximation was dependent on system utilisation, see Section 1.4 for details. Neither model was correct. Section 1.3 presents a model that is able to accurately reproduce observed packet drops.. 1.3. Model of a Router Interface. A FIFO queue system is characterized by three components: input process, service process, and buffer capacity. Each of these three components has to be fitted in order to create an accurate router interface model. In the case of modeling the router, the storage policy is also important. The input process comes from traffic measurements while queue size is obtained from a router configuration. Combining the information from a router configuration (the link speed, the buffer capacity and the storage policy) and traffic measurements (packet lengths) one can derive the service process.. 1.3.1. Buffer Capacity. Each interface in the router has an input and an output queue. When a packet enters the router, it is queued in the input queue of the incoming interface and waits to be processed. If the input queue is full the packet is dropped. Input queue drops generally occur when processing is too slow, so it is a hardware related issue [11]. Output drops on the other hand are caused by a congested interface (the outgoing interface cannot send all the incoming packets) [11]. The output queue is more complicated than a simple FIFO1 buffer [38] as it is split into at least two queues: one hardware queue (tx ring) and one or more software queues. The size of the software queues is set by a hold-queue command whereas hardware queues are managed by the Cisco IOS. The size of the additional hardware queue can be verified by a show controller command in a tx_limit section. The hardware queue is shared between the CPU and the interface chipset. Its size is computed automatically by Cisco IOS (the router tries to minimize the hardware queue size) and depends on the interface speed. For interfaces with FIFO discipline the hardware queue can store up to 128 packets. All queuing mechanisms offered by the Cisco IOS are performed in the software 1 It. is assumed that the interface is configured for FIFO discipline..

(28) 12. 1. Packet Buffers. queue. After packet processing the CPU inserts a packet into the shared hardware queue where it waits for transmission. When the hardware queue is full a packet is queued in the software queue. The interface picks up packets from the hardware queue when it is ready to transmit another one. If the hardware queue is empty, the interface enters an idle state. In an idle state the interface hardware periodically polls the hardware queue and starts the transmission as soon as it finds a new packet. During normal operations the interface interrupts the main CPU when it needs more packets to transmit. On slow interfaces interruption may occur on every successfully transmitted packet. Because packets are transferred from the software queue to the shared hardware queue often it was assumed that hardware and software queues may be modeled as single queue with a size being equal to sum of the sizes of both of the components. Such an assumption happens to be accurate and there is no need to introduce a tandem queue model.. 1.3.2. Service Process. The last but not least part of a router model is the service process. Simulation software does not typically require explicitly service time but rather frame lengths and an interface speed are required. Service time τ is then computed according to following formula l τ= , (1.1) C where l is a layer 2 frame length in bits and C is the interface speed in bits per second. Precise calculation of service times requires precise information about a protocol stack (overhead) and the interface speed. The overhead added by protocols can be easily obtained from a documentation. Unfortunately, some protocols (e.g. MAC in Ethernet [51]) have interframe gaps (IFGs) between frames. In order the equation Eq. (1.1) to be still valid for those protocols, the gap was treated as a part of the frame increasing its length. Two popular protocols MAC in Ethernet (10M) and Cisco HDLC (version of HDLC modified by Cisco) were investigated. The Ethernet frame without VLAN adds a 38 B overhead to an IP packet(including IFG [51]). The Cisco HDLC frame has a smaller overhead (only 8 B per IP packet), but the protocol is more complicated than the Ethernet because each frame starts and ends with one octet frame flag and two frames may share the same flag. When sharing the flag between successive frames, the ending flag in the first frame is also treated as the starting flag of the next frame [18]. Sharing flags has a small impact on a model accuracy in highly congested interfaces when almost all flags are shared. The link speed can be obtained from a router configuration, however, it was decided to measure the service time directly which resulted in slightly better accuracy. The service time was measured as an interdeparture time of the output.

(29) 13. 1.4 Results. from the highly congested interface flooded by packets with known length. The link speed for the simulation software was computed according to Eq. (1.1).. 1.4. Results. Applying a queue parametrization described in Section 1.3 results in an accurate packet loss prediction on a real device. Further on, the experimental results related to the deployed packet storage policy are presented. The experiments were conducted several times with different types of generated traffic. Each experiment was conducted for both fixed and variable packet lengths in order to be certain about the storage policy. In the variable packet length scenarios packet sizes were drawn from a discrete random variable taking values 48 B and 1408 B with equal probability. Size distribution was chosen to mimic size distribution observed in real traffic (smallest and largest jump in Fig. 1.1). Exact packet size distributions can be found e.g. in [49]. The experiment was performed for both Ethernet and serial links at different level of utilization (including an overload state) and different buffer capacities. Because real packets have variable lengths only such results are presented. The most attention is given to Ethernet interfaces but the model is also valid for serial interfaces.. 1.4.1. Drop Function. Suppose that from m input packets n was dropped for both the model and the real queue. Note that it does not indicate that both queues behave similarly. Knowing the total number of dropped packets in one experiment is not enough because these drops can be achieved in many different ways. More information about the drop process is needed, therefore I introduced a drop function d(i), i ∈ N, a more fundamental characteristic describing the drop process at a deeper level. The drop function d(i) takes on two values: ( 0 if the i-th packet was not dropped d(i) = (1.2) 1 if the i-th packet was dropped and fully describes the drop process, but it is hard to visualize on a plot. For an expressive drop presentation the cumulative drop function defined as: D(i) =. i X. d(k). (1.3). k=1. was used. The cumulative drop function is simply the number of dropped packets from the beginning of the observation up to the i-th packet arrival..

(30) 14. 1. Packet Buffers. Drop function defined in Eq. (1.3) is an easy tool to compare two drop systems. When the difference of two drop functions for the compared systems is equal to zero for all possible inputs and all packets, then those systems are indistinguishable in the sense of the drop process. The delay is omitted since its precise measurement requires accurate and synchronized timers. However, it is unlikely to observe different delays in the queues with the same disciplines when the error functions are perfectly matched. The the queues would have to operate totaly differently. The drop function is a primary tool used to validate the model used in this work. In an ideal case the drop function obtained from router Dr and the one from simulation Ds should be the same for all i (∀i Dr (i) − Ds (i) = 0). In reality, there is always some level of uncertainty, therefore we expect the drop function to be close but not identical. To measure the accuracy of the model, the difference between experimental and simulation cumulative drop functions D∆ named the error function is used: D∆ (i) = Dr (i) − Ds (i),. i ∈ N.. (1.4). Nonzero value of D∆ creates some error in Packet Loss Ratio. After dividing Eq. (1.4) by i, one can assess a relative error function: D∗ (i) =. Dr (i) Ds (i) D∆ (i) = − , i i i. i ∈ N,. (1.5). being a difference between real and simulated Packet Loss Ratios.. 1.4.2. Traditional Router Models. Accurate router interface modeling is not possible without taking into account packet lengths. Traditional router models involving packet lengths are mostly developed for byte oriented storage [9, 49]. The first question to be answered by the experiment was if it was possible to mimic the packet policy by the byte policy? The answer is no and the failure of byte oriented model can be seen in Fig. 1.6, where the error functions for both storage policies are depicted. The plot marked as “byte model” (ρ = 0.8) is the error function for a byte oriented model of a router interface at utilization level 0.8 and buffer size b∗ . The buffer size b∗ (in bytes) is the solution of minimization the error function across the entire trace, given the byte oriented policy of the simulator and the speed estimated during the experiment. The plot marked as “byte model” (ρ = 0.9) is the error function for simulation at utilization level 0.9 and the buffer of the same size b∗ . The error function exhibits a trend and at the end of the experiment reaches the value of almost 4,000. The lines marked as “packet model” are the error function for a model proposed in the dissertation at utilization levels 0.8 and 0.9. Note that both lines presents.

(31) 15. 1.4 Results. 4,000 byte model (ρ = 0.8) byte model (ρ = 0.9) packet model (ρ = 0.8) packet model (ρ = 0.9). D∆ (i). 3,000. 2,000. 1,000. 0 0. 0.2. 0.4. 0.6. 0.8. 1 i. 1.2. 1.4. 1.6. 1.8. 2. ·105. Fig. 1.6: Model error for ρ = 0.8 and ρ = 0.9 much better approximation than the optimized byte model. The lines are close to 0 for all i compared to the line “byte model” (ρ = 0.9) which at the end of the experiment is of two orders of magnitude higher. Clearly, byte oriented models cannot be used for the real device modeling discussed in the dissertation because the required buffer size depends on its utilization, and it is not related to queue sizes for which an interface was configured. The packet policy is more accurate model of a real router interface and even the fitted byte model is less accurate than the packet model. Nevertheless, a small trend in line ‘packet model”(ρ = 0.9) need to be explained. Let us investigate the packet model accuracy in details.. 1.4.3. Model verification. The main proposition is to bind software and hardware queues into one single queue as described in Section 1.3. This proposition was tested for three different software queues for the Ethernet2 interface at the utilization level ρ = 0.8. The error functions are presented in Figs. 1.7(a), 1.7(b) and 1.7(c). The impact of system utilisation ρ is presented in Figs. 1.7(a) and 1.7(d). Each of the presented error charts is an average value of error functions for five 2 Similar. results were observed for a serial interface..

(32) 16. 1. Packet Buffers 100 50. 0 D∆ (i). D∆ (i). 0. −50. −100 −100 −200. 0. 0.5. 1 i. 1.5. −150. 2. 0. 0.5. ·105. (a) 20 packet queue, ρ = 0.8. 1 i. 1.5. 2 ·105. (b) 40 packet queue, ρ = 0.8 100. 100. D∆ (i). D∆ (i). 0. 0. −100. −200 −100 0. 0.5. 1 i. 1.5. 2 ·105. (c) 60 packet queue, ρ = 0.8. −300. 0. 0.5. 1 i. 1.5. 2 ·105. (d) 20 packet queue, ρ = 0.9. Fig. 1.7: Error functions. Solid line – error function, dashed line – dispersion measured as 95% confidence interval. repeated experiments with the same input traffic. Observed error functions are chaotic as shown by the dispersion lines (95% confidence intervals for the observed error functions) depicted in Figs. 1.7(a), 1.7(b) and 1.7(c). Although the error is random, 0 value lies almost always in 95% confidence interval, so statistically obtained results indicate that the chosen model and its parametrization are correct. Besides the random nature of the errors, they have a clear trend. It may be caused by biases in link speed or queue size estimation. Changing the queue size by ±1 leads to a worse model so the error must be caused by a bias of the link speed estimator and other random effects. This also means that it was found the most accurate approximation of buffer capacity. The model accuracy is not clearly dependent on a software queue size, but such a dependency was observed on link utilization. Figs. 1.7(a), 1.7(b) and 1.7(c).

(33) 17. 1.4 Results. present the results of experiments conducted at utilization level ρ = 0.8. Higher utilization level (0.9 and 1.1) results are presented in Figs. 1.7(d) and 1.8. The trend is larger for higher utilisation what supports the hypothesis that the trend is caused by the bias of the link speed estimator.. 0. D∆ (i). −200. −400. −600. D∆ (i) = −0.0028 · i − 22 0. 0.2. 0.4. 0.6. 0.8. 1 i. 1.2. 1.4. 1.6. 1.8 ·10. 2. 5. Fig. 1.8: Error function for ρ = 1.1, 20 packet queue, solid line – error function, dashed line – dispersion (95% confidence interval), straight line – linear approximation of dispersion Each of the presented results contains some uncertainty, but the question is if it is acceptable. A simple measure of accuracy is the relative error D∗ function which estimates the bias of the Packet Loss Ratio. Let us consider the worst scenario shown in Fig. 1.8. The maximum of an error function can be approximated (with a trend) by a linear function. Such an approximation is also depicted in Fig. 1.8. The relative error function D∗ for a large number of packets is a tangent of the fitted line and is equal to 0.0028. It is a small error considering that the observed Packet Loss Ratio was equal to 0.189, the model error is less than 1.5% of the observed drop probability.. 1.4.4. Measurement Errors. Each measurement has an uncertainty error. In the experiments errors appeared in two points: traffic sniffing and interface speed calculation. Instead of the expensive DAG card a simple Wireshark application was used in the experiments to dump.

(34) 18. 1. Packet Buffers. the traffic. This is another source of noise in the results due to packet processing in sniffer. Because Wireshark trace files were used to estimate link speed, the speed measurement is also uncertain. The question is how these uncertainties affect the proposed model? In order to answer this question another experiment, similar to the original one, was conducted. In the main experiment I compared drop functions measured in a router with ones obtained from the trace-driven simulation. To estimate an error, I compared drop functions obtained from two simulations run with different parameters. The changed parameters were the arrival time of each packet (introducing the jitter), link speed, and buffer length. The observation is that the error in the arrival time measurement has the lowest impact on a drop function. This type of error is mostly important in highly congested links. Simulation results were confirmed by real experiments where an error increases when link utilization increases (compare Figs. 1.7(d) and 1.8). The accuracy of link speed and buffer length estimations is far more significant then arrival time accuracy. Perfect fit of link speed is crucial. The simulations showed, that in a real world example, the relative error in a link speed of order 10−5 causes an error function reaching up to 60 packets in a 200,000 packet long simulation. Equally important is that observed error patterns are similar to those observed in the experiments. Based on simulation results it can be assumed that link speed error is the main source of the total error in the model.. 1.5. Conclusion. The results indicate that a single finite FIFO queue with buffer counted in packets (packet policy) is an accurate model for a single interface in a typical router used in LAN networks. Despite its complicated internal structure consisting of several buffers, it is possible to tune the queue size in packets and the link capacity in bits per second to mimic drops occurring in a real device with acceptable errors. The accuracy of a model highly depends on the accuracy of the link speed estimation. The protocol stack and the overhead resulting from lower level protocols is also crucial for accuracy. The proposed parametrization of a standard model leads to more realistic simulations and may be used for network dimensioning or router performance evaluation..

(35) 2. Area of Research. The goal of Chapter 2 is to survey the properties of network traffic, the requirements of traffic models, and the relation between the buffer policy and the queue model. The related work on queuing the Markovian traffic is described and compared to the model proposed in the dissertation. Chapter 2 ends with a discussion of the existing estimators of models parameters.. 2.1. Traffic in telecommunication networks. The pioneering work of Agner Krarup Erlang began the applications of queuing theory in telecommunications. Since then an enormous progress has been observed, though the idea to compute performance metrics of the telecommunications system given the traffic, remains the same. The traffic in telecommunication will be understood as a flow of information in the communication network. At the very beginning only analog signals were transmitted over dedicated coper links. Currently data of many types (including voice, video and text) is transmitted. Quite often the transition happens in a real time. Instead of building dedicated infrastructure for each service, packet commutation was developed, to transmit different types of data in a uniform way. Each packet contains the network related information(overhead) and the payload. Traffic in packet network has complicated nature. Despite of being generated by humans and machines, often it must be described like a natural phenomenon. The complexity of the network topology is one of the factors of traffic complexity. Another factor is diversity of link capacity and physical transmission media (coper, fiber), each having different reliability and connectivity characteristics. Despite the operation of single component is known, a group of components behaves in more complicated way, and sometimes direct measurements are the only way of getting the characteristics of interest..

(36) 20. 2. Area of Research. The key concept in telecommunications networks is the existence of protocols and their encapsulation. The idea can be illustrated on the following example. The internet protocol (IP) is used to transfer the information through the network between the endpoints. This protocol operates on addresses and delivers datagrams to the destination. However, it can only transfer the information, without any acknowledgment and verification [39]. This, together with retransmissions of the lost packets is the task of the higher level Transmission Control Protocol (TCP) [40]. On top of the protocol stack there is an application protocol such as HTTP, widely used in the internet. Because of this encapsulation, HTTP „sees” only text, graphics and other multimedia content of the web page. Reliable transmission is provided by TCP protocol while network details (topology and addresses) are hidden inside the IP protocol. This stack of protocols imposes hierarchical structure of the network and complexity of the traffic. Each level has its own behaviour, dynamics and time scale. For example, HTTP sessions are initiated on highly large time scale compared to transfer times of IP datagrams. Telecommunication traffic exhibits several levels of complexity. The three most general can be distinguished [1] and will be shortly discussed hereinafter. Geographical complexity plays an important role. Despite the internet is built of high speed core links and fast routers, with traffic sources located at the edge, the distance from edge to core changes noticeably from point to point. Access link speed also varies from slow modem to 10 Gb/s for modern Local Area Network (LAN). In addition, mobile users generate traffic with variable spatial characteristic. Traffic sources are not distributed homogenously. Establishments like universities or offices have far more different traffic characteristic than housing estates. Furthermore, the traffic is split and merged in routers in different ways. What can be seen as superposition of many sources at the edge of the network, inside the core becomes just an another type of traffic. Complexity of the offered traffic relates to the nature of traffic sources as well. For example, users of WWW arrive and leave randomly. The time user spends on particular task in the web is different for different users. The activity of users also changes with time. Each user uses slightly different applications and their protocols generate different traffic. Finally, the content like text, or multimedia have their own properties which affect properties of the traffic. Time complexity of the traffic is omnipresent. All the traffic aspects mentioned above depend on time and are present on many time scales beginning from microseconds for protocols operating on packets in LAN networks through daily and weakly trends up to the evolution over months and years. The range of time scales and link capacities observed in telecommunication networks results in many time scale dependent phenomena. They were discovered in the early nineties, when precise measurements and analyses of the network.

(37) 2.1 Traffic in telecommunication networks. 21. began [37]. It appeared that network traffic is far from being smooth and dominated by the main trend. It is independent of the time scale. The concept of timescale independence of the traffic can be expressed as lack of the characteristic timescale describing traffic fluctuations. Instead, the transition between timescales should be described. In case of telecommunication traffic, such transition was discovered empirically and the property of scale independence is present in the form of burstiness. The burstiness is tendency of packets to form dense and sparse regions over a time axis. Burstiness has a huge negative impact on queue characteristics [34]. The queuing system fed by bursty traffic has to be operated at a lower utilisation in order to offer acceptable level of QoS [1]. Burstiness is caused by two phenomena [1]. The first one is a heavy-tailed distribution of interarrival times. This causes so-called amplitude burstiness. The second and the most important phenomenon is high correlation between distant events. This causes so-called time burstiness. Dependence between time distant events is described by the autocorrelation function which is used to classify random process as Long-Range Dependent process (LRD) or Short-Range Dependent (SRD) process. If the autocorrelation function decays more slowly than an exponential decay, or equivalently its sum is infinite, the process is Long-Range Dependent. Otherwise the process is ShortRange Dependent. The most general description of scale invariance of the traffic requires the notion of self-similarity. Random process X(t) is self-similar if X(t) and its scaled version cH X(t/c) are statistically identical [1]. The parameter c determines change in the time-scale, while H is Hurst exponent – a quantitative measure of self-similarity. Self-similarity or LRD properties packet interarrival times are just one aspect of a rich correlation structure of IP traffic. Similar behavior was observed for the packet lengths [16]. The packet lengths can also be correlated with the interarrival times [49]. All these correlations will considerably affect the accuracy of any switching device model. Self-similar process (as well as other time series models) is an accurate description of the traffic in packet network, but it is hard to find analytical formulas for queue characteristics with such process at the input. Among many attempts to model the correlation structure of IP traffic in respect to its queuing, the Markovian models have been proven to be a powerful tool. Despite the fact they cannot accurately model self-similar and long-range dependent behavior over all time scales, they can approximate them up to an arbitrary time scale [42], which is often sufficient. What is more important however, in most cases Markovian traffic models lead to analytically solvable queuing models. This is why they are still attractive and actively explored, see e.g. [6–8, 15]..

(38) 22. 2.2. 2. Area of Research. Related work. Efficient dimensioning and effective management of packet network resources require both traffic descriptions and network devices (in particular an interface of a router) modeling. Surprisingly, while traffic models are validated very often, device models receive much less attention. The research on the topic, reported in [2, 45, 50] and described in the Chapter 1, showed that packet policy is quite often used in real devices. Yet, the common models of variable packet length traffic do not fit the packet policy buffer. The main modeling problem for real devices addressed in the dissertation is that they have packet oriented buffers (Definition 2) which are drained at a constant rate in bytes per second, not packets per second (Chapter 1). Since the time required to complete packet service depends on the actual packet length (link speed is constant), the service process model has to reflect all the statistical properties of packet lengths, including their correlation. The model of packet oriented buffer is complicated, but it can be approximated by a system with Markovian arrivals and a semi-Markovian service process. Queues with Markovian input have been studied extensively since the beginning of computer networks and a lot of theoretical work was done towards the end of the last century. At the time of ATM, packet lengths were not important, since an ATM cell has a constant length. The typical approach was to focus mainly on the arrival process and use deterministic service time. Models, like M M P P/D/1 and M AP/D/1 (together with the finite buffer versions) have been developed for interfaces of network devices [4]. The majority of models designed for ATM devices are not suitable for IP devices because IP packets are not of a constant length. In order to properly model IP traffic, the models have to be modified to fit the real IP network more precisely [43]. The simplest approach is application of a general service distribution to model packet lengths. Correct packet lengths distribution indeed improves the accuracy of the model [30, 52, 53]. However, the correlation structure of packet lenghts is not captured by this model. In order to make a flexible model, the concept of a batch arrival was introduced. An arrival of the l bytes length packets is interpreted as a simultaneous arrival of l pieces each of size 1 B. This allows to model the correlation of packet lengths and interarrival times as well as the autocorrelation of packet lengths. However, there is no general model suitable for all storage policies, since often the policy is implied by the model. The most flexible traffic model from the Markov family with batch arrivals is the Batch Markovian Arrival Process (BMAP). This model can handle packet lengths and all the correlations observed in real traffic. It is also possible to construct an analytically solvable queue model with BMAP at the input [6, 14]..

(39) 2.3 Estimation. 23. However, the model implicitly assumes that a buffer is byte-oriented. Detailed explanation of this fact is given in Chapters 1 and 3. To the author’s best knowledge a model of a finite packet oriented buffer fed by the BMAP has not been derived yet. This is not a surprise because this system is quite complicated. However, under the assumption that packet lengths are independent of interarrival times, the model gets simplified. The arrival process is reduced to the MAP process since no batch arrivals are required. Autocorrelation of packet lengths can be represented by the service time autocorrelation. The semi-Markov process (SM), being a direct generalization of the Markov process [21], is often used as a model of the correlated service process in a queue. Queues with semi-Markov service times have been studied for many years [33]. Most of the theoretical results were obtained for an infinite capacity system. Recently, Dudin et al. [14] explored the general finite system BM AP/SM/1/b, using the embedded Markov chain approach. However, none of the analytical results provides the possibility of a transient state analysis of a finite capacity system. In most cases, the stationary analysis is sufficient. However, transient state analysis reveals more details about the system as was argued for in [25, 28, 35]. The loss process in a general M AP/G/1/b system was analyzed in [8] using a novel approach called potential method. In this method the system of integral equations describing a queue is transformed into a system of linear equations by the Laplace transform. The resulting system of linear equations is solved by a recursive function called the potential [7]. Chydziński et al. [7–10] showed how useful this approach is by using it for derivation of different queue characteristics such as queue size distribution, loss process, time to buffer overflow, and many others in either the stationary or the transient state.. 2.3. Estimation. Among many advantages, Markovian traffic models have one disadvantage: a lot of parameters. Often, those parameters do not have intuitive interpretation. Multitude of Markovian traffic model parameters makes it hard to properly fit the model and it is a research task on its own. As a result many fitting algorithms and estimators have been designed e.g. [5, 19, 24, 31]. In the theory of estimators, maximum-likelihood estimation (MLE) has many advantages and is widely used for estimating parameters of statistical model [32]. The expectation maximization (EM) algorithm is a popular realization of MLE in the presence of latent variables [13]. Ryden was the first to derive the EM algorithm for parameter estimation of Markov modulated Poisson process (MMPP) [48]. Later this algorithm was extended to BMAP by Klemm et al. [24]. This version of the estimator is used in the dissertation. The model estimated by the EM algorithm can approximate first and second.

(40) 24. 2. Area of Research. order statistics of the traffic quite accurately. The model is also quite simple (dimensionality is a parameter of the estimator) and numerically tractable. However, in some cases the model produced by the EM estimator is not very accurate for capturing self-similar behavior of the traffic [31]. Moreover, the algorithm is computationally expensive since it requires matrix exponentiation. More recently, improved estimation procedures were proposed [5, 31]. Those new estimators minimize the error of an autocorrelation function fit over multiple time scales, thus resulting in a better self-similar approximation. Numerical implementation is also simple and the estimation is faster. However, quite often the estimated model has large dimensionality and leads to numerical errors when calculating queue characteristics. Since interarrival times and packet lengths are modeled independently, so two different estimators have to be deployed. The first is the previously described MAP estimator and the second is the estimator of Hidden Markov Model of packet lengths. Hidden Markov Model, in its nature is quite similar to Markov traffic models, thus the estimator is also similar. The commonly used estimator is the BaumWelsh algorithm [41], which was also used in the dissertation. This is yet another estimator from the EM family. The main estimator used in numerical examples is the EM estimator [24] (the approach followed [7, 8]). The estimator based on Kronecker product (KPC) [5] is used as a second estimator for comparison..

(41) 3. System Model. Chapter 1 described a physical device and how it works. Chapter 3 presents the mathematical model of the system which allows to calculate queue characteristics. Description of the system consists of two parts: a traffic model and a queue model. The traffic model describes properties of interarrival times and packet lengths. The queue model on the other hand describes the process of packet service.. 3.1. Notation. In this dissertation, both the input process and the service process are related to some Markov process. In order to have an unambiguous notation, each property x of the input process will be denoted by x ˇ while the corresponding property of the service process will be denoted by x ˆ. Note that this convention applies only to properties having counterparts, like the number of states. Block matrices appear L quite often. Therefore, some shorthand notation will be used. The symbol denotes the matrix direct sum (not to be confused with the Kronecker sum) - a shorthand notation for a block diagonal matrix. The Kronecker product will be denoted by ⊗. Transposition of a matrix X will be traditionally denoted by X T . In order to denote the block transposition in a block matrix, the following property of the transposition operation will be used: . AT CT. BT DT. T. . A = B. C . D. Column vectors of ones and zeros will be denoted by 1 and 0 respectively. Traditionally I is the identity matrix and Ix is the identity matrix of size x..

(42) 26. 3.2. 3. System Model. Traffic Model. The traffic model (interarrival times and packet lengths) used in this dissertation is derived from the BMAP process. In order to make the model analytically tractable for a packet-oriented queue I assumed no correlation between the interarrival times and the packet lengths. Implication of this assumption on queue characteristics as well as the level of the correlation in the real traffic are discussed in Chapter 5, where numerical results are presented.. 3.2.1. Interarrival Times. Formally the BMAP process is defined as a two dimensional Markov process ˇ (N (t), J(t)) on the state space {(i, j) : i ≥ 0, 1 ≤ j ≤ m}, ˇ with infinitesimal ˇ generator for (N (t), J(t)) equal to:   D0 D1 D2 D3 · · ·  0 D0 D1 D2 · · ·   Q=0 , 0 D0 D1 · · ·   .. .. .. .. .. . . . . . where Dk are m ˇ ×m ˇ matrices, elements in Dk , k ≥ 1 are nonnegative, D0 has nonnegative off-diagonal P∞ elements and negative diagonal elements. It is also assumed that D = k=0 Dk is an infinitesimal generator and D 6= D0 . N (t) ˇ denotes the state of the underlying denotes the number of arrivals in (0, t] and J(t) (modulating) Markov chain with the infinitesimal generator D. The BMAP can also be defined in a so-called ’constructive’ way. Assume that the underlying Markov chain has just entered a state i. Transition to another state occurs after exponentially distributed time with parameter λi . Transition to state j without a batch arrival happens with probability pi (0, j). With probability pi (k, j) there is a transition to state j and simultaneously k jobs arrive. It is assumed that: ∞ X m ˇ X pi (0, i) = 0, pi (k, j) = 1 (3.1) k=0 j=1. and λi = −(D0 )ii , (D0 )ij , λi (Dk )ij pi (k, j) = λi pi (0, j) =. 1 ≤ i ≤ m, ˇ 1 ≤ i, j, i 6= j ≤ m, ˇ 1 ≤ i, j ≤ m. ˇ.

(43) 27. 3.2 Traffic Model. In relation to IP traffic modeling, batches (simultaneous arrivals of multiple jobs) represent packets of different lengths (in bytes or some other memory units). Thus the memory block implicitly becomes a job in a queue model. According to discussion from Chapter 1, in the case of packet oriented buffer, there is no need for batch arrivals. The special case of a BMAP process, with single batch size (when Dk = 0, k > 1) is called a Markovian Arrival Process (MAP) and it is the arrival model in the dissertation. A special case of MAP, with D1 being a diagonal matrix, is known as the Markov Modulated Poisson Process (MMPP) [17]. Thus, all the results in the dissertation are also valid for the MMPP. The MMPP is a popular traffic model. However, the MAP is more tunable to traffic characteristics and more general. Because of the Markov nature of BMAP process (and its special cases, MAP in particular), it is possible to analytically calculate first and second order statistics of the interarrival times. In this dissertation only the average arrival rate is required and for clarity the rest are omitted. The average arrival rate in MAP is given by: λ=π ˇ D1 1, (3.2) where π ˇ is the stationary vector for D: π ˇ D = 0T ,. π ˇ 1 = 1.. This formula is required to obtain the stationary packet loss ratio.. 3.2.2. Packet Lengths. The constructive definition of the BMAP process is the starting point for a derivation of a packet length model. According to this definition, in each state i of the underlying Markov chain, there is a bivariate joint transition probability distribution pi (k, j) of the next state j and the the batch size k. This distribution determines packet lengths and the next state of the Markov modulator. As a joint distribution it allows the model to the correlation between packet lengths and the interarrival times [24], as well as the autocorrelation of both. In this dissertation, I use an simplification and assume that packet lengths and interarrival times are independent. However, I assume that either of them could be autocorrelated. In such a case, the packet length and the next state are described by the marginal distributions of pi (k, j). To be more precise, let tn denote the n-th state transition time of the BMAP process. The marginal distributions of pi (k, j) are given by X k. ˇ n+1 ) = j|J(t ˇ n ) = i) = Ti,j , pi (k, j) = P(J(t.

(44) 28. 3. System Model. and X. ˇ n ) = i) = Ei,k , pi (k, j) = P(l(tn ) = k|J(t. j. where l(tn ) is the observable batch size. This outcome represents packet lengths and, indirectly, the time service times. Independent description of packet lengths and the next state lead to a packet lengths model known as a Hidden Markov Model (HMM) [41]. Such model has already been proposed as an accurate model of packet lengths [12]. Time evolution of the HMM is governed by the Markov chain on the state space S = {1, . . . , m}. ˆ However, the state of this chain (state at time t is denoted as ˆjt ) is hidden from the observer. In each state only the probabilistic outcome l ∈ L = {l1 , . . . , ldˆ} is observable. If all state transitions were accompanied by an arrival, such a model could be used as a packet model. However, due to transitions without arrival (k = 0), probabilities Ti,j and Ei,k have to be modified to remove this unobserved emission. This can easily be done using the total probability theorem in a similar way to that presented in the Appendix B. However, it is more practical to estimate model parameters directly from the trace file. The resulting HMM model gives additional flexibility in parameter estimation. In summary, the proposed HMM packet length model is parameterized by the following: 1. m, ˆ the number of states in the model. This parameter corresponds to the number of states in BMAP. 2. L, distinct observation symbols (an alphabet). This parameter corresponds to batch sizes in BMAP and is the vector of distinct packet lengths. 3. The state transition probabilities T : Ti,k = P(jt+1 = k|jt = i). Transition matrix of the underlying Markov chain. 4. The emission probabilities E: Ei,k = P(lk at t|jt = i). A discrete distribution of packet lengths emitted in each state. Note that the initial state distribution πi = P(j1 = i) is not required here because all results are conditioned on an initial state..

(45) 29. 3.3 Queue model. 3.3. Queue model. Router interface with packet oriented buffer is modeled as a single server FIFO queuing system Fig. 3.1 fed by the Markovian Arrival Process, and the service times are assumed to be autocorrelated. The system capacity is finite and equal to b (including a customer currently being served).. sm ˇ .. . s1. sm ˆ Buffer b. .. . s1. C. semi-Markov (m, ˆ T, E, L) MAP (m, ˇ D0 , D1 ) Fig. 3.1: M AP/SM/1/b queue model: MAP arrivals with m ˇ states, single server with m ˆ states, buffer of size b. Since the service times are simply scaled packet lengths, it is straightforward to derive the service process from the model presented in Section 3.2.2. The states of the HMM are mapped to the states of the server. In the i-th server state, the service time is a random variable with CDF Fi (t), i = 1, . . . , m. ˆ After each service completion, the state of the server is changed in such a way that the state sequence forms a homogenous Markov chain. Let {Jˆn }, n ≥ 0 denote the state after n − 1 transitions and Gn , n ≥ 0 be the (n − 1)th service time. Supposing G0 = 0, the described service process (Jˆn , Gn ) is a semi-Markov process (SM) [21] defined on a finite set of states S = {1, 2 . . . , m} ˆ by the matrix PJˆn−1 ,j (t) = P(Jˆn = j, Gn ≤ t|(Jˆn−1 , Gn−1 )). Note that the semiMarkov process is constructed only of service times. The idle periods are ignored. ˆ ∈ S as being the server state at time t including the idle However, I define J(t) periods. It is assumed that the time origin corresponds to a departure epoch. Definition of the semi-Markov process allows dependence of sojourn time on the current state and the future state as well. However, for practical purposes we can assume the sojourn time as independent of the future state. Under this assumption, the HMM model of packet lengths can be interpreted as a service.

(46) 30. 3. System Model. process. In such a case Pi,j (t) = Ti,j Fi (t), where T is the state transition matrix of the underlying Markov chain1 . Because of packet loss, the state transition matrix T of the server is not exactly the same as the transition matrix T in a HMM estimated from traffic. The quantitative analysis of the problem is presented in the Appendix B. However, for small loss probabilities the difference is negligible. In summary, the model is parameterized by follows: D0 , D1 , T, E, L, C, b. The last two parameters: link speed C and buffer capacity b are the properties of the system. Rest of the parameters describe the traffic. Dimensionality of matrices D0 , D1 and T, E relates to complexity of the traffic model. It is also related to autocorrelation of the traffic. Vector L contains most typical packet lengths. If the traffic will change in the future (e.g. some packet length will become frequent), this vector will also change.. 1 In. general T = [Pi,j (∞)]i,j ..

(47) 4. M AP/SM/1/b Queue Characteristics. Chydziński reported numerous queue characteristics of nontrivial queuing systems expressed in terms of ’potential’, a recurrent matrix series [6–8]. All his results were derived for a system with uncorrelated service times. Chapter 4 shows how to generalize the potential method for systems with correlated service times forming a semi-Markov process. Two important characteristics are considered. The first is the time to reach the buffer capacity, also known as ’first passage time’ and abbreviated as ttbo which stands for ’time to buffer overflow’. The second characteristic is the loss process described by the expected number of losses in a given time interval or the loss ratio. Note that the formula for ttbo is also valid for an infinite system. In such a case, ttbo means the time when the queue exceeds a given occupancy level.. 4.1. Potential and notation. Potential method originates from the study of an integer random walk by Korolyuk and was used in queuing theory by Chydziński. The paper [7] can be used as a reference of the method. In this approach, we begin with the system of integral equations obtained from the total probability theorem. In the Laplace transform domain, this system is transformed to a system of linear equations and can be solved using typical numerical methods. However, due to special structure of the system, the solution can be expressed in terms of potential Rk – a recurrent matrix series by virtue of the following lemma. Lemma 1 (Chydziński [7]). Assume that A0 , A1 , A2 , . . . is a sequence of m × m matrices such that det(A0 ) 6= 0 and ψ1 , ψ2 , ψ3 , . . . is a sequence of column vectors.