Topics in Stochastic Networks

Performance Scaling and Algorithmic Challenges

• 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

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.

Logistics

• 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

Overview

Diff. entities arrive to be processed

System that processes them

Leave after being processed

• Scope: processing networks

Overview

Diff. entities arrive to be processed

• Coupled processing activities

• Constrained capacityLeave after being processed

Network!

Call operator assignment

English,etc

InvestmentChinese

Spanish

Savings

Overview

Overview

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

Overview

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

Overview

• Important questions to address

• Also the pricing and economic aspect (not covered)

Performance: Loss prob, queueing delay,

etc

Long-term capacity management and planning

Day-to-day operationsand controls

Overview

• Important questions to address

• Also the pricing and economic aspect (not covered)

Call drops, time to download files,

etc

Design of networks: hiring of personnel,

Bandwidth capacity, etc

Routing and scheduling of customers/entities

Overview

• Important questions to address

Performance: Loss prob, queueing delay,

etc

Long-term capacity management and planning

Day-to-day operationsand controls

• Science: analysis of network and compute perf. metrics• ≈ More classical

• Engineering: design and optimize network• ≈ More modern

Overview

• Important questions to address

Performance: Loss prob, queueing delay,

etc

Long-term capacity management and planning

Day-to-day operationsand controls

• Good performance

• Simple design, easy control

Overview

• Important questions to address

• Good performance

• Simple design, easy control

Achieve jointly?

Non-empty Queue

1

Simple Teaser

O(n) memory

Random Queue

1

Simple Teaser

Zero memory

• 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(a): Loss Networks

• 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 I(b): Network of Queues

• 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(a): Switched 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 2(b): Flow-Level Networks

• 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

Part 3: Decentralized Opt.

• Fluid models of queueing networks

• Mean-field analysis

• Heavy-traffic analysis; diffusion approximation

• Large-deviations analysis

• Simulation methods

Some Important Omissions

• Appreciation of good modeling – an “art”

• Asking good research questions

• Good use of elementary and simple tools

Takeaways from the class