On the Steady-State of Cache Networks

Elisha J. Rosensweig Daniel S. Menasche

Jim Kurose

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Talk Outline

• Introduction – ICN and Cache Networks• Our work – impact of initial state• Motivating Examples• CN Markov model and proof methodology• Equivalence Classes• Discussion• Summary

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Content in the SpotlightHow do I access

XYZ.com?

How do I find

ABC.mp4?

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Recasting ideas from TCP/IP

Host-to-Host communication• Hosts remain fixed• Path unknown and in flux

TCP/IP Specify host addresses

Path determined on-the-fly

Host-to-Content communication• Host and content - fixed• content location in flux

ICN protocolsSpecify content ID

Content located on-the-fly

Content Caching a central feature of new architectures

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Graphic Notation

Content (file) Request for content

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Caching 101

• Stand-alone caches– Arrival stream is

filtered by cache hits. Misses routed towards custodian.

– Replacement policy: what to evict from a cache to make room for new content• Common/Popular policies – LRU, LFU, FIFO…

Arrivals Misses

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Cache Networks (CN) 101• In-network caching

operation for CN1. Consumer requests

content2. Request routed towards

content custodian (exists for each piece of content)

3. En-route to custodian, inspect local cache at router for content copy

4. During content download, store along path

consumer

Cache-router

ContentCustodian

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What is new about CNs?

• Cache hierarchies– Single custodian– Requests flow

upstream, content flows downstream

• Approximate models proposed

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What is new about CNs?

• Cache Networks– Caches & custodians

in arbitrary topologyv1

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v3

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What is new about CNs?

• Cache Networks– Caches & custodians

in arbitrary topology– Introduces cross-

flows – requests in both directions on a link

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v3

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What is new about CNs?

• Cache Networks– Caches & custodians

in arbitrary topology– Introduces cross-

flows – requests in both directions on a link

– Cross-flows create state dependency loops

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Talk Outline

• Introduction – ICN and Cache Networks• Our work – impact of initial state• Motivating Examples• CN Markov model and proof methodology• Equivalence Classes• Discussion• Summary

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Modeling Variables

s(i,j)

Vi

Replacement Policy

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Modeling Variables

consumer

s(i,j)

λ(i,j)

Vi

Replacement Policy

Exogenous Requests

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Modeling Variables

consumer

s(i,j)r(i,j)

λ(i,j)

Vi

V1

V2

….

Vk

Replacement Policy

Exogenous Requests

Miss Routing

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Rosensweig et al 2010, 2013

Our work – the challenge

• Existing models consider the impact of– Request arrival distribution– Network topology and miss routing– Replacement policy and cache size

• Not considered: initial state of caches• Question: Can the initial state affect long term

performance?

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Our work - contributions

• Examples where initial state impacts steady-state of CN

• Formulated three conditions that independently ensure initial state has no impact on steady state– CN ergodicity

• Demonstrated existence of replacement policy equivalence classes– If a member of the class is ergodic , so are all

members of the class

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Talk Outline

• Introduction – ICN and Cache Networks• Our work – impact of initial state• Motivating Examples• CN Markov model and proof methodology• Equivalence Classes• Discussion• Summary

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Motivation

• Why should the initial state impact steady-state of CN?– Arrival pattern for new events determines state– Initial state negligible in many known systems

• However, such CNs exist– Two examples shown in paper– In both, the dependency appears only when

caches are networked

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Example #1

V1 V2

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Example - Performance

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FIFO, Cache size = 2

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Example – single FIFO explained

• Disjoint markov chains, but• Existence probability is identical in both• Conservation of flows

Order matters in FIFO

Example - Performance

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FIFO, Cache size = 2

Example - Performance

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Exogenous arrivals

System BehaviorInitial State Pr(v1 has ) Pr(v1 has )

( , ) 0.46 0.63( , ) 0.33 0.76

λ( ,1)=0.35 λ( ,1)=0.55 λ( ,1)=0.1λ( ,2)=0.05 λ( ,2)=0.15 λ( ,2)=0.8

FIFO, Cache size = 2

Example – Networked FIFO

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• Initial state impacted steady state

• Function of cache networking

when does initial state impact steady-state?

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Sufficient Ergodicity Conditions

• Three independent conditions for CN ergodicity– Initial state does not impact steady-state

• Theorems: The following networks are ergodic– Feed-Forward CNs– CNs with probabilistic caching– Using non-protective replacement policies• Constructive proof for Random Replacement• Equivalence class

Topology

Addmission

Rep. Policy

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Talk Outline

• Introduction – ICN and Cache Networks• Our work – impact of initial state• Motivating Examples• CN Markov model and proof methodology• Equivalence Classes• Discussion• Summary

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Markov Chains for CNs

• CN State = the content of each cache

(c1 state, c2 state,

…)

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Markov Chains for CNs

• State representation depends on replacement policy– Random: set of

content– LRU, FIFO: sequence

of content in cache, represents eviction order

({1,2,3}, {3,5,6})

((2,1,3),

(6,3,5))

Random

LRU / FIFO

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Markov Chain Terminology & Properties - 1

• Recurrent state– If a system is in a recurrent state, it will return to

this state in the (finite) future

• Communicating states– Two states communicate if there is a sample path

in both directions between them

A At1 t2 > t1

A B

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Markov Chain Terminology & Properties - 2

• Ergodic set– A set of recurrent states where all states

communicate with one another• Quasi-ergodic system– A system with a single ergodic set

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Markov Chain Terminology & Properties - 3

• Property: a quasi-ergodic system has a single steady-state– i.e. Steady state not affected by initial state

• Goal: prove that given CN is quasi-ergodic

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Ergodicity proof methodology

• Need to construct sample path between states• In charting a sample path, we can select any viable

request and eviction– Sufficient that transitions are possible

1,2

1,3 2,3

Request file 3

Evict file 1Evict file 2

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Ergodicity proof methodology

• Given any pair of recurrent states, we design a sample path between them– sequence of requests, and corresponding evictions

A B

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Ergodicity proof methodology

• Sufficient condition: for each pair of recurrent states A,B, find state C both can reach

• Basis– Recurrency ensures there is also a path from this

third state to each, so A and B communicate

A C B

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Ergodicity proof - reminder

• In charting a sample path, we can select any viable request and eviction– Sufficient that transitions are possible

A BC

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Talk Outline

• Introduction – ICN and Cache Networks• Our work – impact of initial state• Motivating Examples• CN Markov model and proof methodology• Equivalence Classes• Discussion• Summary

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Rep. Policy Equivalence Classes

• In paper, we constructively prove Random replacement is Ergodic– Assuming positive request probability for each file

• Additionally, we show many replacement policies are equivalent to Random replacement in this respect

• Definition: non-protective policies– Each file in the cache might be the next to be evicted

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Rep. Policy Equivalence Classes

• Proof sketch – Construct Markov chain for non-protective policy – Contract transitions for exogenous cache hits• i.e., transitions between states where stored content

does not change– Prove the contracted chain is same Markov chain

as for Random replacement• Transitions might have different weights, but chain has

same structure

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CN ErgodicityPolicy Equivalence Classes

{1,2,3}

(1,3,2) (2,1,3)

(2,3,1)(1,2,3)

(3,1,2)(3,2,1)

Random State

LRU Set of States

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CN ErgodicityPolicy Equivalence Classes

{1,2,3}

(1,3,2) (2,1,3)

(2,3,1)(1,2,3)

(3,1,2)(3,2,1)

Random State

LRU Set of States

For LRU, each file in the cache might be the

next to be evicted

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Talk Outline

• Introduction – ICN and Cache Networks• Our work – impact of initial state• Motivating Examples• CN Markov model and proof methodology• Equivalence Classes• Discussion• Summary

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Ramifications - 1

• Results apply also to heterogeneous networks– Any combination of non-protective policies

• Simulations– What parameters to vary

• Power of structural arguments– Structure of the network is what determines

ergodicity– Edge weights irrelevant; no need to solve system

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Ramifications - 2

• With non-ergodic CNs, new set of challenges– Initial state has long term impact, and so– Seeding of state can modify global behavior at low

cost– Impact on system management, analysis and

architecture

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Summary

• CNs might be affected by initial state• For certain topologies, admission control and/or

replacement policies a CN is shown to be ergodic• Proof methodology– Structural arguments

• Open question: What structures yield non-ergodic CNs?– Many implications if realistic such CNs exist– How does structure impact behavior, in general

Questions?

Backup Slides

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Random Replacement CNs - 1

• Two copies A,B of the same CN, different state– Same topology, exogenous request patterns,

replacement policy– Different content stored in some caches

• Sample Path Construction– Requests: single sequence of exogenous requests,

applied to both copies– Evictions: different for each copy, ensures reaching

the same state from both.

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Random Replacement CNs - 2

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Random Replacement CNs - 2

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Random Replacement CNs - 2

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Random Replacement CNs - 2

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Random Replacement CNs - 2

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V4Identical state

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Feed-Forward CNs

• In Feed-forward networks, requests flow in only one direction one each link– Content flows in the

opposite direction• Theorem: FF networks

are always Ergodic

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Probabilistic Caching

• Admission control policy• Each content i that passes through cache j is

cached locally with probability pij

– Can be different for each i and j.• Theorem: when using probabilistic caching,

the system is ergodic

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a-NET, Net Calculus & ErgodicityRelated Work

• Hierarchy Modeling & Evaluation– P. Rodriguez;“Scalable Content Distribution in the

Internet”, PhD thesis, Universidad Publica de Navarra, 2000

– H. Che et al; “Analysis and design of hierarchical web caching systems”, INFOCOM 2001

– S. Borst et al; “Distributed caching algorithms for content distribution networks” , INFOCOM 2010

– I. Psaras et al; “Modeling and evaluation of ccn-caching trees” , IFIP Networking 2011

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a-NET, Net Calculus & ErgodicityRelated Work

• (Hybrid) P2P systems– S. Ioannidis and P. Marbach, “On the design of

hybrid peer-to-peer systems”, SIGMETRICS 2008.– S. Tewari and L. Kleinrock, “Proportional

replication in peer-to-peer networks”, INFOCOM 2006.

• Similar, but differences exist– Overlay P2P topology not used for download

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Assumptions

• Independence Reference Model (IRM) for exogenous requests

Pr(Xj = fi | X1,..,Xj-1) = Pr(Xj=fi)– Standard in the literature

• Assume positive request pattern at each cache– Each file is requested exogenously with non-zero

probability• Consider only individually-ergodic caches– The behavior of each cache alone is independent of its

initial state