A calculus for stochastic qos analysis and its application to conformance study

A CALCULUS FOR STOCHASTIC QOS ANALYSIS AND ITS APPLICATION TO CONFORMANCE STUDY LIU YONG (B.Eng. Hunan University, M.Eng. Hunan University) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2005 Acknowledgements I am truly indebted to my supervisors, Assoc. Prof. Tham Chen Khong and Dr. Jiang Yuming, for their continuous guidance and support during this work. Without their guidance, this work would not be possible. I am deeply indebted to the National University of Singapore for the award of a research scholarship. I would also like to give thanks to all the researchers in the Computer Communication Networks Laboratory, who greatly enriched both my knowledge and life with their intelligence and optimism. Lastly, I would like to thank my parents and my wife for their endless love and support. Liu Yong January 2005 i Contents Acknowledgements Summary i vii List of Symbols ix List of Tables xii List of Figures xiii Introduction 1.1 Quality of Service . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Stochastic QoS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Contents 1.3 1.4 iii 1.2.1 Deterministic QoS vs. Stochastic QoS . . . . . . . . . . . . . 1.2.2 Literature Survey on Stochastic QoS . . . . . . . . . . . . . 1.2.3 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Overview of the Solution . . . . . . . . . . . . . . . . . . . . 10 Conformance Study for Networks with Service Level Agreements . . 11 1.3.1 Conformance Study . . . . . . . . . . . . . . . . . . . . . . 11 1.3.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . 12 1.3.3 Overview of the Solution . . . . . . . . . . . . . . . . . . . 12 Structure of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . . 13 Stochastic QoS Bounds Under Deterministic Server 15 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Brief Review of Deterministic Network Calculus . . . . . . . . . . . 16 2.3 Network Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3.1 Traffic Model . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3.2 Server Model . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Stochastic Bounds Under Deterministic Server . . . . . . . . . . . . 29 2.4.1 Single Node Case . . . . . . . . . . . . . . . . . . . . . . . . 29 2.4.2 Multi-Node Case . . . . . . . . . . . . . . . . . . . . . . . . 43 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.4 2.5 Contents Stochastic QoS Bounds Under Stochastic Server iv 48 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2 Server Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.3 Single Node Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.4 Multi-Node Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Per-flow Stochastic QoS Bounds Under Aggregate Scheduling 64 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.2 Aggregate Scheduling with Deterministic Server . . . . . . . . . . . 65 4.3 Aggregate Scheduling with Stochastic Server . . . . . . . . . . . . . 77 4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Conformance Study in Networks with Service Level Agreements 86 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.3 Network Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Contents 5.4 v Conformance Deterioration and Stochastic Burstiness Increase . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.5 Property of Token Bucket Shaper . . . . . . . . . . . . . . . . . . . 94 5.6 Conformance Study of Per-Flow Scheduling Network . . . . . . . . 97 5.6.1 Single Node Case . . . . . . . . . . . . . . . . . . . . . . . . 97 5.6.2 Multi-node Case . . . . . . . . . . . . . . . . . . . . . . . . 99 5.7 5.8 5.9 Conformance Study of Aggregate Scheduling Network . . . . . . . . 100 5.7.1 Per-Flow in Single Node Case . . . . . . . . . . . . . . . . . 101 5.7.2 Per-Flow in Multi-Node Case . . . . . . . . . . . . . . . . . 107 5.7.3 Per-Aggregate Case . . . . . . . . . . . . . . . . . . . . . . . 110 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.8.1 Per-Flow Scheduling Network in Single Node Case . . . . . . 113 5.8.2 Aggregate Scheduling Network in Single Node Case . . . . . 117 5.8.3 Aggregate Scheduling Network in Multi-Node Case . . . . . 119 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Conclusions and Further Research 126 6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 6.2 Contributions of this Thesis . . . . . . . . . . . . . . . . . . . . . . 128 6.2.1 A Calculus for Stochastic QoS Analysis [65] . . . . . . . . . 128 Contents 6.2.2 6.3 vi Conformance Study [66] . . . . . . . . . . . . . . . . . . . . 129 Further Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Bibliography 132 Appendices 143 A List of Theorems 144 B List of Publications 149 Summary With the advent of the Internet, there is a proliferation of multimedia applications such as video streaming, Voice over IP (VoIP) and network voice- and videoconferencing. These applications need Quality of Service (QoS) guarantees, such as high throughput, low delay and low packet loss for high performance transmission. Many schemes have been proposed for QoS provisioning in a computer network. It is important to evaluate the performance of these QoS provisioning schemes. There is a lot of research work addressing the analysis of deterministic QoS performance. As yet, there has been no general investigation and analysis of end-to-end stochastic QoS performance. In addition, most previous works on stochastic QoS performance analysis only considered a server which provides deterministic service, i.e. deterministically bounded rate service. Few works have vii Summary considered the behavior of a stochastic server providing variable rate service for input flows. In this thesis, a method, referred to as stochastic network calculus, is proposed to systematically investigate the stochastic QoS performance of various deterministic and stochastic servers. The stochastic backlog, delay and output burstiness under deterministic servers are first derived. This is followed by derivation of the corresponding stochastic QoS bounds under a single stochastic server. Then, the input-output characterization of a stochastic server is derived, with which the stochastic end-to-end QoS bounds have also been derived. For studying per-flow stochastic QoS, it is proved in this thesis that a deterministic server offering deterministic service to an aggregate of flows can be regarded as a stochastic server for individual flows in the aggregate. Based on this finding, results on the per-flow stochastic QoS performance are derived under aggregate scheduling. As a practical application of the stochastic network calculus proposed in this thesis, the conformance performance of traffic crossing a network is studied to investigate to what extent a flow is nonconformant to its original traffic specification after crossing a network with Service Level Agreements. In the literatures this problem has only been investigated through simulations, whereas, in this thesis, analytical results on non-conformance probability bounds are derived by applying the proposed stochastic network calculus. viii List of Symbols α : arrival curve β : service curve A (s, t) : amount of traffic arriving in the time interval [s, t) A∗ (s, t) : amount of output traffic in the time interval [s, t) ⊗ : convolution in min-plus algebra : deconvolution in min-plus algebra : conventional convolution B (t) : Backlog at time t d (t) : virtual delay at time t of a system h (α, β) : maximum horizontal distance between α and β ix Bibliography 135 [13] D. Yates, J. Kurose, D. Towsley, and M. Hluchyj, “On per-session end-to-end delay distributions and the call admission problem for real-time applications with QoS requirements,” in ACM SIGCOMM Conference, pp. 2–12, Sept 1998. [14] A. Elwalid and D. 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Appendix A List of Theorems Lemma 2.1: Property of convolution in min-plus algebra Lemma 2.2: Summation of two random processes Theorem 2.1: Deterministic backlog bound for a deterministically bounded flow under a deterministic server Theorem 2.2: Deterministic delay bound for a deterministically bounded flow under a deterministic server Theorem 2.3: Deterministic burstiness bound for a deterministically bounded flow under a deterministic server Theorem 2.4: Concatenation property of a deterministic server Theorem 2.5: Relationship between gSBB and SBB Theorem 2.6: Stochastic backlog bound for a gSBB flow under a deterministic server 144 145 Theorem 2.7: Stochastic delay bound for a gSBB flow under a deterministic server Theorem 2.8: Input-output characterization of a deterministic server with a gSBB flow Theorem 2.9: Stochastic output burstiness bound for a gSBB flow under a deterministic server Theorem 2.10: Stochastic backlog bound for an aggregate flow under a deterministic server Theorem 2.11: Stochastic delay bound for an aggregate flow under a deterministic server Theorem 2.12: Stochastic output characteristics for an aggregate flow under a deterministic server Theorem 2.13: Stochastic output burstiness bound for an aggregate flow under a deterministic server Theorem 2.14: Stochastic end-to-end delay bound for a gSBB flow under a network of deterministic servers in tandem Theorem 2.15: Stochastic end-to-end output characteristic for a gSBB flow under a network of deterministic servers in tandem Theorem 2.16: Stochastic end-to-end output burstiness bound for a gSBB flow under a network of deterministic servers in tandem Theorem 3.1: Stochastic backlog bound for a gSBB flow under a stochastic server Theorem 3.2: Stochastic delay bound for a gSBB flow under a stochastic server 146 Theorem 3.3: Stochastic output burstiness bound for a gSBB flow under a stochastic server Theorem 3.4: Input-output characterization of a stochastic server with a gSBB flow Theorem 3.5: Stochastic end-to-end burstiness increase for a gSBB flow under a network of stochastic servers in tandem Theorem 3.6: Stochastic end-to-end delay bound for a gSBB flow under a network of stochastic servers in tandem Theorem 4.1: Per-flow stochastic service curve within an aggregate under a deterministic server Theorem 4.2: Per-flow stochastic backlog bound for a gSBB flow within an aggregate under a deterministic server for general case Theorem 4.3: Per-flow stochastic delay bound for a gSBB flow within an aggregate under a deterministic server for general case Theorem 4.4: Per-flow stochastic output burstiness bound for a gSBB flow within an aggregate under a deterministic server for general case Theorem 4.5: Per-flow stochastic output characteristic for a gSBB flow within an aggregate under a deterministic server for general case Theorem 4.6: Per-flow stochastic backlog bound for a gSBB flow within an aggregate under a deterministic server for independent case Theorem 4.7: Per-flow stochastic delay bound for a gSBB flow within an aggregate 147 under a deterministic server for independent case Theorem 4.8: Per-flow stochastic output burstiness bound for a gSBB flow within an aggregate under a deterministic server for independent case Theorem 4.9: Per-flow stochastic output characteristic for a gSBB flow within an aggregate under a deterministic server for independent case Theorem 4.10: Per-flow stochastic service curve within an aggregate under a stochastic server Theorem 4.11: Per-flow stochastic backlog bound for a gSBB flow within an aggregate under a stochastic server Theorem 4.12: Per-flow stochastic delay bound for a gSBB flow within an aggregate under a stochastic server Theorem 4.13: Per-flow stochastic output burstiness bound for a gSBB flow within an aggregate under a stochastic server Theorem 4.14: Per-flow stochastic output characteristic for a gSBB flow within an aggregate under a stochastic server Lemma 5.1: Per-flow deterministic service curve Corollary 5.1: Non-conformance probability bound under a GR node Corollary 5.2: Non-conformance probability bound for an aggregate Theorem 5.1: Relationship between non-conformance probability and stochastic burstiness increase Theorem 5.2: Property of token bucket shaper for the case where input upper rate 148 equals token generation rate Theorem 5.3: Property of token bucket shaper for the case where input upper rate is less than token generation rate Theorem 5.4: Single node non-conformance probability bound in a per-flow scheduling network Theorem 5.5: Multi-node non-conformance probability bound in a per-flow scheduling network Theorem 5.6: Per-flow stochastic burstiness bound under deterministic service curve Theorem 5.7: Per-flow non-conformance probability bound under deterministic service curve Theorem 5.8: Single node non-conformance probability bound in an aggregate scheduling network for general case Theorem 5.9: Single node non-conformance probability bound in an aggregate scheduling network for independent case Theorem 5.10: Multi-node non-conformance probability bound in an aggregate scheduling network for general case Theorem 5.11: Multi-node non-conformance probability bound in an aggregate scheduling network for independent case Theorem 5.12: Stochastic burstiness bound for an aggregate Appendix B List of Publications 1. Y. Liu, C.K. Tham, Y. Jiang, ”Conformance Study for Networks with Service Level Agreements”, Computer Networks, Elsevier Science, Volume 47, Issue 6, Pages 885-906, April 2005. 2. Y. Liu, C.K. Tham, Y. Jiang, “A Calculus for Stochastic QoS Analysis”. Submitted to Performance Evaluation, Elsevier Science, 2005. 3. Y. Liu, C.K. Tham, Y. Jiang, “A Stochastic Network Calculus”. Technical Report: Department of Electrical and Computer Engineering, National University of Singapore, Nov.2003. http://onelab.ece.nus.edu.sg/ yliu/pubs/snetcalTechRept.pdf 4. C.K. Tham and Y. Liu, “Assured End-to-End QoS through Adaptive Marking 149 150 in Multi-Domain Differentiated Service Networks”, Computer Communications, Special Issue on End-to-End Quality of Service, Elsevier Science, 2004. 5. Y. Liu, C.K. Tham and TCK. Hui, “MAPS: A Localized and Distributed Adaptive Path Selection Scheme in MPLS Networks”, in Proceedings of 2003 IEEE Workshop on High Performance Switching and Routing (HPSR 2003), Torino, Italy, 24-28 June 2003. 6. C.K. Tham and Y. Liu, “Minimizing Transmission Costs through Adaptive Marking in Differentiated Services Networks”, in Proceedings of 5th IFIP/IEEE International Conference on Management of Multimedia Networks and Services (MMNS 2002), Springer Lecture Notes in Computer Science (LNCS) 2496, Heidelberg, Oct 2002, pp. 237-249. [...]... Under this framework, various QoS provisioning schemes can be analyzed to determine their abilities in stochastic QoS provisioning It will be a general and effective theoretical tool for the analysis of end -to- end stochastic QoS performance 9 1.2 Stochastic QoS for a flow over a network It also can be used for admission control to provide a stochastic QoS guarantee In addition, it may be used for network... after passing through a stochastic server However, it is not clear from [34] how to derive the input-output characterization of a stochastic server which is important for end -to- end stochastic QoS analysis Comparing with Cruz’s work [34], the input-output characterization of a stochastic server is derived in this thesis and is applied to analyze various end -to- end stochastic QoS performance In 8 1.2 Stochastic. .. input-output characterization of a stochastic server is derived, thus providing an effective way for end -to- end stochastic QoS analysis In addition, it is proved that a server serving an aggregate of flows can be regarded as a stochastic server for individual flows within the aggregate Based on this, the proposed framework is further applied to analyze per-flow stochastic QoS performance in an aggregate scheduling... performance analysis have been developed into an elegant theory under the name of network calculus [6] However, the worst case bounds are often far away from practical results and QoS provisioning based on the worst case analysis will thus usually lead to low utilization of network resources To address this issue, some researchers have paid attention to stochastic QoS analysis since most multimedia applications... server are extended to a network of deterministic servers in tandem This chapter not only derives the stochastic QoS performance for a single flow, but also considers the corresponding stochastic QoS performance for an aggregate of flows In Chapter 3, the stochastic QoS performance under a single stochastic server is derived Then, a network of stochastic servers in tandem is considered for which the stochastic. .. 1.3 Conformance Study for Networks with Service Level Agreements 1.3 Conformance Study for Networks with Service Level Agreements 1.3.1 Conformance Study As an application of the stochastic network calculus proposed in this thesis, conformance performance of traffic crossing a network has been studied To achieve some level of QoS assurance, a network will have Service Level Agreements (SLAs) with its. .. dimensioning to determine the amount of network resources needed for a flow to meet its stochastic end -to- end QoS requirements 1.2.4 Overview of the Solution To address the problem stated above, a stochastic network calculus is proposed to analyze the end -to- end stochastic QoS performance of a system with stochastic bounded input traffic over a series of deterministic and stochastic servers The input-output characterization... considered the impact of interactions within the same service class ignoring those caused by intra-class traffic Therefore, analytical results in conformance study are needed for a thorough understanding of this problem With the analytical results, it will be possible to evaluate the effect of different network parameters on the conformance deterioration and to solve or alleviate the conformance deterioration problem... thesis is to systematically evaluate stochastic QoS over a computer network 1.2.2 Literature Survey on Stochastic QoS There is a lot of research work addressing the analysis of stochastic QoS performances under different network scenarios Generally speaking, most previous works on stochastic QoS performance analysis can be classified into four scenarios A Deterministic Traffic under Deterministic Server... stochastic end -to- end QoS bounds are also derived In Chapter 4, the per-flow stochastic QoS performance under aggregate scheduling is studied In particular, the server providing service to an aggregate under aggregate scheduling is proved to be a stochastic server to each individual flow and the corresponding per-flow stochastic service is derived Then the per-flow stochastic QoS performance under aggregate scheduling . A CALCULUS FOR STOCHASTIC QOS ANALYSIS AND ITS APPLICATION TO CONFORMANCE STUDY LIU YONG (B.Eng. Hunan University, M.Eng. Hunan University) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF. the analysis of deterministic QoS performance. As yet, there has been no general investigation and analysis of end -to- end stochastic QoS performance. In addition, most previous works on stochastic. individual flows in the aggregate. Based on this finding, results on the per-flow stochastic QoS performance are derived under aggregate scheduling. As a practical application of the stochastic network