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 3 How do I access XYZ.com? How do I find ABC.mp4?

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

Graphic Notation 5 Content (file) Request for content

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

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

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 9 v1v1 v2v2 v4v4 v3v3

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

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

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 13 s(i,j) ViVi Replacement Policy

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

Modeling Variables 15 consumer s(i,j) r(i,j) λ(i,j) ViVi V1V1 V2V2 …. V k Replacement Policy Exogenous Requests Miss Routing

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

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 19

Example #1 V1 V2 20 V1 V2

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

Example – Networked FIFO V1 V2 Initial state impacted steady state Function of cache networking when does initial state impact steady-state?

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 23

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 24 Topology Addmission Rep. Policy

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 26 (c 1 state, c 2 state, …)

Markov Chains for CNs State representation depends on replacement policy – Random: set of content – LRU, FIFO: sequence of content in cache, represents eviction order 27 ( {1,2,3}, {3,5,6} ) ( (2,1,3), (6,3,5) ) Random LRU / FIFO

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 28 A A A A t1t1 t 2 > t 1 A A B B

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 31 1,2 1,3 2,3 Request file 3 Evict file 1Evict file 2

Ergodicity proof methodology Given any pair of recurrent states, we design a sample path between them – sequence of requests, and corresponding evictions 32 A A B B

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 33 A A C C B B

Ergodicity proof - reminder In charting a sample path, we can select any viable request and eviction – Sufficient that transitions are possible 34 A A B B C C

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 37

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 38

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

Questions?

Backup Slides

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 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. 47

Random Replacement CNs V1 V2 V3 V4 V1 V2 V3 V4

Random Replacement CNs V1 V2 V3 V4 V1 V2 V3 V4

Random Replacement CNs V1 V2 V3 V4 V1 V2 V3 V4

Random Replacement CNs V1 V2 V3 V4 V1 V2 V3 V4

Random Replacement CNs V1 V2 V3 V4 V1 V2 V3 V4 Identical state

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

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 – S. Tewari and L. Kleinrock, “Proportional replication in peer-to-peer networks”, INFOCOM Similar, but differences exist – Overlay P2P topology not used for download 56

Example – single FIFO explained 57 Disjoint markov chains, but Existence probability is identical in both Conservation of flows Order matters in FIFO

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