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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 2

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Content in the Spotlight 3 How 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 protocols Specify content ID Content located on-the-fly 4 Content Caching a central feature of new architectures

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Graphic Notation 5 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 6

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Cache Networks (CN) 101 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 consumer Cache- router 7 Content Custodian

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What is new about CNs? Cache hierarchies – Single custodian – Requests flow upstream, content flows downstream Approximate models proposed 8

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What is new about CNs? Cache Networks – Caches & custodians in arbitrary topology 9 v1v1 v2v2 v4v4 v3v3

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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 10 v1v1 v2v2 v4v4 v3v3

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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 11 v1v1 v2v2 v4v4 v3v3

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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 12

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Modeling Variables 13 s(i,j) ViVi Replacement Policy

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Modeling Variables 14 consumer s(i,j) λ(i,j) ViVi Replacement Policy Exogenous Requests

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Modeling Variables 15 consumer s(i,j) r(i,j) λ(i,j) ViVi V1V1 V2V2 …. V k 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? 16

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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

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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 18

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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 19

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Example #1 V1 V2 20 V1 V2

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Example - Performance V1 V2 FIFO, Cache size = 2

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Example – single FIFO explained 22 Disjoint markov chains, but Existence probability is identical in both Conservation of flows Order matters in FIFO

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Example - Performance V1 V2 FIFO, Cache size = 2

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Example - Performance V1 V2 Exogenous arrivals System Behavior Initial StatePr(v 1 has ) (, )0.460.63 (, )0.330.76 λ(,1)=0.35λ(,1)=0.55λ(,1)=0.1 λ(,2)=0.05λ(,2)=0.15λ(,2)=0.8 FIFO, Cache size = 2

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Example – Networked FIFO V1 V2 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 Theorem: Feed-Forward CNs are ergodic – Topology Theorem: CNs with probabilistic caching - ergodic – Admission Control Theorem: Non-protective replacement policies – Constructive proof for Random Replacement – Equivalence class 26

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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 27 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 28

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Markov Chains for CNs CN State = the content of each cache 29 (c 1 state, c 2 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 30 ( {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 31 A A A A t1t1 t 2 > t 1 A A B 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 32

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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 33

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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 34 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 35 A A B 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 36 A A C C B 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 37 A A B B C C

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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 38

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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 39

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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 40

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CN Ergodicity Policy 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 41

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CN Ergodicity Policy 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 42

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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 43

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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 44

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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 45

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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 46

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Questions?

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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. 49

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Random Replacement CNs - 2 50 V1 V2 V3 V4 V1 V2 V3 V4

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Random Replacement CNs - 2 51 V1 V2 V3 V4 V1 V2 V3 V4

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Random Replacement CNs - 2 52 V1 V2 V3 V4 V1 V2 V3 V4

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Random Replacement CNs - 2 53 V1 V2 V3 V4 V1 V2 V3 V4

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Random Replacement CNs - 2 54 V1 V2 V3 V4 V1 V2 V3 V4 Identical 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 55

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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 56

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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 57

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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 58

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Assumptions Independence Reference Model (IRM) for exogenous requests 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 59

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