Tim Roughgarden Stanford CS
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Transcript of Tim Roughgarden Stanford CS
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Quantifying Trade-Offs via Competitive Analysis
(Clean Slate Seminar)
Tim RoughgardenStanford CS
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Clean Slate Trade-Offs
• Clean Slate design fraught with trade-offs between competing objectives
• "There is not likely to be a unique answer for the list of requirements, and every requirement has some cost. The cost of a particular requirement may become apparent only after exploration of the architectural consequences of meeting that objective in conjunction with others...it there requires an iterative process..."– NewArch Intro paper, 2000.
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Clean Slate Trade-Offs
E.g., overprovisioning: good or bad?• Nick: inefficient, motivates Valiant load-
balancing in backbone network• Bernd: good, QoS becomes easy
Theme in my research:• rigorously quantify trade-offs between
competing objectives– e.g., excess capacity vs. performance
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Plan for Talk
Goals:• illustrate this idea with several examples:
routing, protocol design, pricing, capacity installation– models vary in direct relevance to clean slate
• emphasize commonalities + flexibility of analysis approach, qualitative insights via quantitative analysis
• illustrate my own interests/expertise
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Example #1: Routing
Motivating example:
• low capacity, prop delay vs. high capacity, prop delay
• d how close arrival rate is to “knee” of delay curve
Conges-tion [secs]
Rate R
s t
c(x) = xd
c(x) = 1
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Example #1: Routing
Motivating example:
• low capacity, prop delay vs. high capacity, prop delay• d how close arrival rate is to “knee” of delay curve• dumb routing (source, delay-based, etc) = all on top
Conges-tion [secs]
Rate R
s t
c(x) = xd
c(x) = 10
1
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Example #1: Routing
Motivating example:
• low capacity, prop delay vs. high capacity, prop delay• d how close arrival rate is to “knee” of delay curve• dumb routing (source, delay-based, etc) = all on top• smart routing = offload some to bottom
Conges-tion [secs]
Rate R
s t
c(x) = xd
c(x) = 10
1 1-Є
Є
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Trade-offs in Routing
Summary:• constraint: can’t/don’t want to implement
smart routing• trade-off: excess capacity vs. performance
(avg delay relative to optimal routing)
Next: two related approaches for quantifying this trade-off. – [Roughgarden/Tardos 00], [Roughgarden 02]
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Quantifying the Trade-Off
Approach #1 (the ratio):• as a function of the excess capacity,
what is the ratio: avg delay of delay-based routing vs. avg delay of optimal routing– at least 1, the closer to 1 the better– “competitive ratio”, “price of anarchy”
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Quantifying the Trade-Off
Approach #1 (the ratio):• as a function of the excess capacity, what is
the ratio: avg delay of delay-based routing vs. avg delay of optimal routing– at least 1, the closer to 1 the better– “competitive ratio”, “price of anarchy”
Answer: grows as d/ln d• small as long as there’s
some overprovisioning s t
c(x) = xd
c(x) = 1
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Qualitative Insights
Insight #1:• advocates overprovisioning but...
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Qualitative Insights
Insight #1:• advocates overprovisioning but...• even (say) 20% works wonders
– both Nick and Bernd are right!
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Qualitative Insights
Insight #1:• advocates overprovisioning but...• even (say) 20% works wonders
– both Nick and Bernd are right!
Insight #2: worst-case = trivial topology• worst-case ratio does not degrade with
more complex topologies, traffic matrices
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Quantifying the Trade-Off
Approach #2 (match the old optimum):• how much overprovisioning need before
delay-based routing as good as optimal?
with overprovisioning without overprovisioning
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Quantifying the Trade-Off
Approach #2 (match the old optimum):• how much overprovisioning need before
delay-based routing as good as optimal?
Answer: 100% (double the capacity)• cf., “switch speedup results” by Ashish,
Nick, Balaji
with overprovisioning without overprovisioning
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Bigger Picture
• had one or more constraints– not feasible to route traffic optimally
• two competing objectives– minimize both overprovisioning + average delay
• two ways to quantify trade-off– competitive ratio, min capacity to simulate opt
• precise answers, qualitative insights– small amount of overprovisioning helps– trivial worst-case topologies
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Ex #2: Protocols for Bandwidth Allocation
Setup: [Johari/Tsitsiklis 04] + [Johari 04]• goal is to partition bandwidth (e.g. 1 link) to
maximize sum of heterogeneous utilities
rk
uk
Equal-slope “Pareto condition”
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Trade-Offs for a Bandwidth Allocation Protocol
Constraint: can’t directly implement optimum (e.g., don’t know utility functions); want decentralized protocol to do this
• [Kelly] simple such protocol exists if no user “large” (has non-negligible “market power”)
• [JT04] quantify trade-off between protocol performance, max market power of a player– at most 25% efficiency loss
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Kelly mechanism still optimal
Qual Insight #1: market power not a big deal.
Idea: use efficiency loss as novel metric to compare different protocols.
Theorem: [J04] Kelly mechanism the best one! – all protocols in a certain class have > 25% eff loss
Qual Insight #2: Kelly mechanism designed for no market-power setting, but still optimal (in above sense) more generally.
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Ex #3: Pricing a Service
Motivating question: how do we price a service (e.g. a movie broadcast) so that it is (at least somewhat) economically viable?
Constraint: "fairness" = every customer's cost can only go down as more customers served
• economies of scale• connected to "collusion-resistance"
sserver
n potential clients with valuations
edge cost = 1
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Ex #3: Trade-offs
Trade-off: want to charge enough to cover costs, but also want "good solution"– easy to cover costs of the empty set!
• max "surplus" = benefit to served customers - cost of serving them
sserver
n potential clients with valuations
edge cost = 1
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Ex #3: Trade-offs
Trade-off: want to charge enough to cover costs, but also want "good solution"– easy to cover costs of the empty set!
• max "surplus" = benefit to served customers - cost of serving them
Old result: can't have both [Moulin/Shenker].New result (w/Sundararajan): quantify trade-off
curve between them.
sserver
n potential clients with valuations
edge cost = 1
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Ex #3: Insights
Qualitative insight #1: can have approximate versions of both goals.– approximate cost recovery + nearly maximum-
possible surplus
#2: trivial examples exhibit worst-case behavior (like in routing, complex topology doesn't make things worse)
Open issue: trade-offs when economic viability a constraint, "fairness" an objective
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Example #4: Valiant Load-Balancing
Constraint: [Zhang-Shen/Mckeown 04,05]: allocate edge capacity w/out knowing traffic matrix
Assume: know amount of traffic out of each node in backbone network (say R each)– linear # of parameters instead of quadratic
• want sufficient capacity to route any traffic matrix respecting these node constraints
Intuitively: lack of knowledge need more capacity. But how much more?
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Example #4: VLB
Theorem: [ZM 04,05]: only a factor 2!• know matrix just do one-hop routing
need at most nR capacity (n = # nodes)
• VLB: two-hop routing suffices, at most 2R/n capacity on each of n2 links
• extensions (node-varying R, failures,...)
• future: avg prop delay vs. capacity trade-offs (w.r.t. underyling physical network)
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Summary
• much of the clean slate work will be struggling with different trade-offs
• quantitative analysis flexible, often tractable, often offers new qualitative insights
• always looking for new problems to tackle...
• future: evaluate the e2e principle?– has suggestive "smart" vs. "dumb" flavor...