Topics in Stochastic Networks

1. Topics in Stochastic Networks Performance Scaling and Algorithmic Challenges

2. Logistics • Instructor: Yuan Zhong; yz2561@columbia.edu • 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

3. 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 topicsand 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.

4. 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!

5. Overview • Scope: processing networks Diff. entities arrive to be processed Leave after being processed System that processes them

6. Overview • Scope: processing networks Diff. entities arrive to be processed • Coupled processing activities • Constrained capacity Leave after being processed Network!

7. Overview Call operator assignment Investment Savings Chinese English, etc Spanish

8. Overview

9. Overview • Examples abound • Manufacturing: wafer fabrication, production • Services: call centers, cloud computing, healthcare • Communications: wireless networks, routers, Internet

10. Overview • Loss system: lose entities if demands cannot be satisfied instantly • Loss probability • Queueing system: queue up entities if demands cannot be satisfied instantly • Delay/queue size

11. Overview • Important questions to address • Also the pricing and economic aspect (not covered) Long-term capacity management and planning Performance: Loss prob,queueing delay, etc Day-to-day operations and controls

12. Overview • Important questions to address • Also the pricing and economic aspect (not covered) Design of networks: hiring of personnel, Bandwidth capacity, etc Call drops, time to download files, etc Routing and scheduling of customers/entities

13. Overview • Important questions to address • Engineering: design • and optimize network • ≈ More modern Long-term capacity management and planning Performance: Loss prob,queueing delay, etc Day-to-day operations and controls • Science: analysis of network • and compute perf. metrics • ≈ More classical

14. Overview • Important questions to address • Simple design, easy control Long-term capacity management and planning Performance: Loss prob,queueing delay, etc Day-to-day operations and controls • Good performance

15. Overview • Important questions to address • Simple design, easy control Achieve jointly? • Good performance

16. Simple Teaser 1 Non-empty Queue O(n) memory

17. Simple Teaser 1 Random Queue Zero memory

18. 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

19. 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

20. 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

21. 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

22. 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

23. Some Important Omissions • Fluid models of queueing networks • Mean-field analysis • Heavy-traffic analysis; diffusion approximation • Large-deviations analysis • Simulation methods

24. Takeaways from the class • Appreciation of good modeling – an “art” • Asking good research questions • Good use of elementary and simple tools