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Providing performance guarantees is one of the most important issues for future telecommunication networks. This book describes theoretical developments in performance guarantees for telecommunication networks from the last decade. Written for the benefit of graduate students and scientists interested in telecommunications-network performance this book consists of two parts. The first introduces the recently-developed filtering theory for providing deterministic (hard) guarantees, such as bounded delay and queue length. The filtering theory is developed under the min-plus algebra, where one replaces the usual addition with the min operator and the usual multiplication with the addition operator. As in the classical linear system theory, the filtering theory treats an arrival process (or a departure process ) as a signal and a network element as a system. Network elements, including traffic regulators and servers, can be modelled as linear filters under the min-plus algebra, and they can be joined by concatenation, "filter bank summation", and feedback to form a composite network element. The problem of providing deterministic guarantees is equivalent to finding the impulse response of composite network elements. This section contains material on: - (s, r)-calculus - Filtering theory for deterministic traffic regulation, service guarantees and networks with variable-length packets - Traffic specification - Networks with multiple inputs and outputs - Constrained traffic regulation The second part of the book addresses stochastic (soft) guarantees, focusing mainly on tail distributions of queue lengths and packet loss probabilities and contains material on: - (s(q), r(q))-calculus and q-envelope rates - The large deviation principle - The theory of effective bandwidth The mathematical theory for stochastic guarantees is the theory of effective bandwidth. Based on the large deviation principle, the theory of effective bandwidth provides approximations for the bandwidths required to meet stochastic guarantees for both short-range dependent inputs and long-range dependent inputs.

Part I. Deterministic Guarantees
Chapter 1. (sigma , ho)-calculus
1.1 (sigma , ho)-traffic characterization
1.2 Multiplexing
1.3 Work conserving links
1.4 Output burstiness
1.5 Routing
1.6 Multi-class networks with feedforward routing
1.7 Single-class networks with nonfeedforward routing
1.8 General traffic characterization
1.9 Notes
Chapter 2. Filtering Theory for Deterministic Traffic Regulation and Service Guarantees
2.1 Filtering theory under the min-plus algebra
2.1.1 Min-plus algebra
2.1.2 Subadditive closure
2.2 Traffic regulation
2.2.1 Maximal f-regulator
2.2.2 Realizations of leaky buckets under the (min,+)-algebra
2.2.3 Traffic regulation for periodic constraint functions
2.3 Service guarantees
2.3.1 f-servers
2.3.2 Work conserving links with priorities
2.3.3 Work conserving links with vacations
2.3.4 GPS links
2.3.5 SCED links
2.3.6 Jitter control
2.3.7 Window flow control
2.3.8 Service curve allocation
2.4 Extensions to networks with variable length packets
2.4.1 L-packetizer
2.4.2 Work conserving links with nonpre-emptive priorities
2.4.3 PGPS links
2.4.4 SCED with nonpre-emptive priority
2.4.5 Window flow control with variable length packets
2.5 Notes
Chapter 3. Traffic Specification
3.1 Projections under the min-plus algebra
3.2 Ordered orthogonal bases under the min-plus algebra
3.3 C-transform under the min-plus algebra
3.4 Notes
Chapter 4. Networks with Multiple Inputs and Outputs
4.1 Min-plus matrix algebra
4.2 Traffic regulation for multiple inputs
4.3 Service guarantees for multiple inputs
4.4 Notes
Chapter 5. Constrained Traffic Regulation and Dynamic Service Guarantees
5.1 Time varying filtering theory under the min-plus algebra
5.2 Maximal dynamic F-regulator
5.3 Maximal dynamic F-clipper
5.4 Constrained traffic regulation
5.5 Dynamic F-servers
5.6 The dynamic SCED scheduling algorithm
5.7 General system theory
5.8 Notes
Chapter 6. Filtering Theory for Networks with Variable Length Packets
6.1 Preliminaries on the max-plus algebra
6.2 Traffic regulation for marked point processes
6.2.1 Minimal g-regulator
6.2.2 Minimal g-regulators in parallel
6.2.3 Inversion formula and superposition of g-regular traffic
6.2.4 Segmentation and reassembly
6.3 Service guarantees for marked point processes
6.3.1 g-server
6.3.2 g-servers in tandem
6.3.3 g-servers in parallel
6.3.4 g-server with feedback
6.4 Scheduling
6.4.1 Nonpre-emptive servers with multiple priorities
6.4.2 The SCED scheduling algorithm
6.5 Notes
Part II. Stochastic Guarantees
Chapter 7. (sigma ( heta ), ho ( heta ))-calculus and heta-envelope Rates
7.1 Convexity and related inequalities
7.2 (sigma ( heta ), ho ( heta ))-traffic characterization
7.3 Multiplexing
7.4 Work conserving links
7.5 Routing
7.6 Acyclic networks and intree networks
7.7 Notes
Chapter 8. Introduction of the Large Deviation Principle
8.1 Legendre transform
8.2 Cram'er's theorem
8.3 The G"artner-Ellis theorem
8.4 Sanov's theorem
8.5 Mogulskii's theorem
8.6 The contraction principle
Chapter 9. The Theory of Effective Bandwidth
9.1 Effective bandwidth at a work conserving link
9.2 Multiplexing independent arrivals
9.3 Routing
9.4 Intree networks
9.4.1 Sample path large deviations
9.4.2 Closure properties of sample path large deviations
9.4.3 The proof for the lower bound
9.5 Work conserving links with priorities
9.6 Conjugate processes
9.6.1 Finite-state Markov arrival processes
9.6.2 Autoregressive processes
9.6.3 Properties of conjugate processes
9.7 Fast simulations
9.7.1 Change of measures and importance sampling
9.7.2 Simulation methodology for steady state probabilities
9.8 Martingale bounds
9.9 Traffic descriptors
9.9.1 A four-parameter traffic descriptor
9.9.2 A two-state Markov fluid model
9.9.3 Closed-form approximations
9.10 Fuzzy reasoning for the theory of effective bandwidth
9.10.1 Work conserving links
9.10.2 Multiplexing independent arrivals
9.10.3 Routing
9.10.4 Output characterization from a work conserving link
9.11 Fractional Gaussian noise
9.12 M/G/infty inputs
9.13 Notes
References
Index
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