On the Steady-State of Cache Networks

Elisha J. Rosensweig Daniel S. Menasche Jim Kurose

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 2

Content in the Spotlight

How do I access XYZ.com

?

How do I find ABC.mp4

?

3

Recasting ideas from TCP/IP

• •

Host-to-Host communication

Hosts remain fixed Path unknown and in flux

Host-to-Content communication

• • Host and content - fixed content

location

in flux

TCP/IP

Specify host addresses

Path determined on-the-fly

ICN protocols

Specify content ID

Content located on-the-fly

Content Caching a central feature of new architectures

4

Graphic Notation

Content (file) Request for content 5

Caching 101

• Stand-alone caches –

filtered

by cache hits. Misses routed towards custodian.

–

Replacement policy: what to evict

make room for new content • Common/Popular policies – LRU, LFU, FIFO… from a cache to 6

•

Cache Networks (CN) 101

consumer In-network caching operation for CN 1. Consumer

requests content

2. 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

Content Custodian Cache router

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What is new about CNs?

• • Cache hierarchies – Single custodian – Requests flow upstream, content flows

downstream

Approximate models proposed 8

What is new about CNs?

• Cache Networks – Caches & custodians in

arbitrary

topology v 2 v 1 v 4 v 3 9

What is new about CNs?

• Cache Networks – Caches & custodians in

arbitrary

topology – Introduces

cross flows

– requests in both directions on a link v 1 v 2 v 3 v 4 10

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

v 1 v 4 v 2 v 3 11

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 12

Modeling Variables

V i s(i,j) Replacement Policy

13

Modeling Variables

consumer

λ(i,j) Exogenous Requests V i s(i,j) Replacement Policy

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V 1 V 2 ….

V k

Modeling Variables

consumer

λ(i,j) Exogenous Requests V i s(i,j) r(i,j) Miss Routing Replacement Policy

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Our work – the challenge

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

Rosensweig et al 2010, 2013

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 17

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 18

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 V1 V2 20

Example - Performance

Exogenous arrivals λ( ,1)=0.35

λ( ,2)=0.05

λ( ,1)=0.55

λ( ,2)=0.15

FIFO, Cache size = 2 λ( ,1)=0.1

λ( ,2)=0.8

System Behavior

Initial State Pr(v 1 has ) Pr(v 1 has )

( , ) ( , ) 0.46

0.33

0.63

0.76

V1 V2

Example – Networked FIFO

• • Initial state

impacted steady state

Function of cache

networking

V1

when does initial state impact steady-state?

V2

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 Topology – CNs with probabilistic caching Addmission – Using

non-protective

replacement policies • Constructive proof for Random Replacement • Equivalence class Rep. Policy 24

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 25

Markov Chains for CNs

• CN State = the content of each cache

(c 1 c 2 state, state, …)

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Markov Chains for CNs

• State representation depends on replacement policy – Random:

set

content of – LRU, FIFO:

sequence

of content in cache, represents eviction order

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

A t 1 A t 2 > t 1

• Communicating states – Two states communicate if there is a sample path in both directions between them

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 29

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 30

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

Request file 3

1,2

Evict file 2

1,3

Evict file 1

2,3

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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 C B

34

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 35

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 36

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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Random State

CN Ergodicity

Policy Equivalence Classes LRU Set of States (1,3,2) (2,1,3) (2,3,1) {1,2,3} (1,2,3) (3,2,1) (3,1,2) 38

Random State

CN Ergodicity

Policy Equivalence Classes LRU Set of States (1,3,2) (2,1,3) (2,3,1) {1,2,3} (1,2,3)

For LRU, each file in the cache might be the next to be evicted

(3,2,1) (3,1,2) 39

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

40

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 41

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 42

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:

ergodic CNs?

– – What structures yield non Many implications if realistic such CNs exist How does structure impact behavior, in general 43

Questions?

Backup Slides

Assumptions

• • •

Independence Reference Model

exogenous requests (IRM) for –

Pr(X j = f i | X 1 ,..,X j-1 ) = Pr(X j =f i )

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 46

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

V4 V2 V3 V1 V4 V2 V3 V1 48

Random Replacement CNs - 2

V4 V2 V3 V1 V4 V2 V3 V1 49

Random Replacement CNs - 2

V4 V2 V3 V1 V4 V2 V3 V1 50

Random Replacement CNs - 2

V4 V2 V3 V1 V4 V2 V3 V1 51

Random Replacement CNs - 2

V4 V2 V3 V1

Identical state

V4 V2 V3 V1 52

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 53

Probabilistic Caching

• • • Admission control policy Each content i that passes through cache j is cached locally with probability p

ij

– Can be different for each i and j.

Theorem: when using probabilistic caching, the system is ergodic 54

a-NET, Net Calculus & Ergodicity

Related 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 55

a-NET, Net Calculus & Ergodicity

Related 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 56

Example – single FIFO explained

Order matters in FIFO • • Disjoint markov chains, but Existence probability is identical in both • Conservation of flows 57

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