Topics in Stochastic Networks
Performance Scaling
and Algorithmic Challenges
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
Logistics
• Grading policy:
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–
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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:
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–
–
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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