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Topics in Stochastic Networks Performance Scaling and Algorithmic Challenges Logistics • Instructor: Yuan Zhong; [email protected] • Class: Mudd 627, MW 2:40 – 3:55pm • Office hour: Fri 4 – 6pm; Mudd 344 (or by appointment) • Class homepage: http://www.columbia.edu/~yz2561/teaching.html Logistics • Grading policy: – – – – 4 hw sets; 40% in total Handout/return: L3/8, L8/13, L13/18, L18/23 Extensions will be allowed as per instructor’s permission Project: 60% • Project: – Critical survey of literature (2-3 papers) + suggestions for future work. Possible topics and references coming soon. – Model formulation and analysis/simulations. – Presentation last week of classes; short paper before. – Final versions due Dec 10; proposals due Nov 9. Overview • Stochastic networks: broadly speaking, systems of interacting components + stochasticity • Some examples: – – – – Ideal gas, Ising models Social and economic networks Epidemic networks Etc… • This course is about none of the above! Overview • Scope: processing networks Diff. entities arrive to be processed Leave after being processed System that processes them Overview • Scope: processing networks Diff. entities arrive to be processed • Coupled processing activities • Constrained capacity Network! Leave after being processed Overview Call operator assignment Investment Chinese English, etc Savings Spanish Overview Overview • Examples abound – Manufacturing: wafer fabrication, production – Services: call centers, cloud computing, healthcare – Communications: wireless networks, routers, Internet Overview • Loss system: lose entities if demands cannot be satisfied instantly • Queueing system: queue up entities if demands cannot be satisfied instantly • Loss probability • Delay/queue size Overview • Important questions to address Long-term capacity management and planning Performance: Loss prob, queueing delay, etc Day-to-day operations and controls • Also the pricing and economic aspect (not covered) Overview • Important questions to address Design of networks: hiring of personnel, Bandwidth capacity, etc Call drops, time to download files, etc Routing and scheduling of customers/entities • Also the pricing and economic aspect (not covered) Overview • Important questions to address • Engineering: design Long-term capacity management and planning and optimize network • ≈ More modern Performance: Loss prob, queueing delay, etc • Science: analysis of network and compute perf. metrics • ≈ More classical Day-to-day operations and controls Overview • Important questions to address • Simple design, Long-term capacity management and planning easy control Performance: Loss prob, queueing delay, etc • Good performance Day-to-day operations and controls Overview • Important questions to address • Simple design, easy control Achieve jointly? • Good performance Simple Teaser r 1 n r n r n Non-empty Queue 1 Delay » 1- r O(n) memory Simple Teaser r 1 n r n r n Random Queue n Delay » 1- r Zero memory Part I(a): Loss Networks • Examples: telephone networks, workforce management, hotel room mgmt., etc; also abundant applications in communications • Control-less system: loss probability computation • Key insight: loss probabilities are hard to compute, but simple approximations work well – Limit theorems, Erlang’s fixed point approximation • Tools: Markov processes, cvx opt, some analysis • “Loss networks” by F. Kelly, AAP 1991. “Lecture notes on stochastic networks”, by Kelly and Yudovina Part I(b): Network of Queues • Mostly control-less systems: Jackson networks, Kelly networks, Whittle networks • Manufacturing and production; communications • Key insight: for a broad range of systems, queue-size distributions have product form – Product of independent components – Simple description; good for provisioning and optimization • Main tool: Markov processes (time reversal) • “Fundamentals of queueing networks” by H. Chen and D. D. Yao “Reversibility and stochastic networks” by Kelly for examples Part 2(a): Switched Networks • Wireless networks, Internet routers, call centers • Operation and control of networks – Queue size difficult to compute; focus on system stablity – Q: how can I keep queue size finite? • Key insight: a simple, wide applicable class of control policies that ensure system stability – Q1: queue size bounds under these policies? – Q2: Low-complexity approximation of these policies? • Tools: Markov chains, Lyapunov functions, graph theory, optimization, randomized algorithms • No textbook, research papers Part 2(b): Flow-Level Networks • Main application: congestion control in the Internet – a major achievement of stoc. net. over the last 10 – 20 years – Ideas found in operations management as well • Main question: how to fairly and efficiently allocate resources? – A framework that successfully explains TCP of the Internet • Tools: Markov processes, Lyapunov functions, convex optimization, (a little bit of econ) • No textbook, research papers • Also connections with product-form networks Part 3: Decentralized Opt. • Algorithmic in nature; perhaps of more interest to electrical engineers and computer scientists • Main question: in a large-scale network, how to ensure good performance without a central coordinator/controller? • Applications: road networks, the Internet, wireless networks • Tools: convex optimization, mixing time of Markov chains, graph theory, Markov processes • Very recent research results Some Important Omissions • Fluid models of queueing networks • Mean-field analysis • Heavy-traffic analysis; diffusion approximation • Large-deviations analysis • Simulation methods Takeaways from the class • Appreciation of good modeling – an “art” • Asking good research questions • Good use of elementary and simple tools