Report

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 TCP/IP Specify host addresses Path determined on-the-fly ICN protocols Specify content ID Content located on-the-fly • Host and content - fixed • content location in flux Content Caching a central feature of new architectures 4 Graphic Notation Content (file) Request for content 5 Caching 101 • Stand-alone caches Arrivals Misses – 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… 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 Cacherouter 7 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 v2 v1 v3 v4 9 What is new about CNs? • Cache Networks – Caches & custodians in arbitrary topology – Introduces crossflows – requests in both directions on a link v2 v1 v3 v4 10 What is new about CNs? • Cache Networks – Caches & custodians in arbitrary topology – Introduces crossflows – requests in both directions on a link – Cross-flows create state dependency loops v2 v1 v3 v4 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 Vi s(i,j) Replacement Policy 13 Modeling Variables consumer Exogenous Requests λ(i,j) Vi s(i,j) Replacement Policy 14 Modeling Variables V1 consumer Exogenous Requests λ(i,j) V2 Vi …. s(i,j) r(i,j) Vk Miss Routing Replacement Policy 15 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? 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 steadystate 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 V1 V2 20 Example - Performance Exogenous arrivals FIFO, Cache size = 2 λ( ,1)=0.35 λ( ,1)=0.55 λ( ,1)=0.1 λ( ,2)=0.05 λ( ,2)=0.15 λ( ,2)=0.8 V1 System Behavior Initial State Pr(v1 has ) Pr(v1 has ) V2 ( , ) ( , ) 0.46 0.33 0.63 0.76 Example – Networked FIFO • Initial state impacted steady state • Function of cache networking when does initial state impact steady-state? V1 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 – CNs with probabilistic caching – Using non-protective replacement policies • Constructive proof for Random Replacement • Equivalence class Topology Addmission 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 (c1 state, c2 state, …) 26 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 27 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 t1 A t2 > t1 • Communicating states – Two states communicate if there is a sample path in both directions between them A B 28 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 1,2 Evict file 2 1,3 Request file 3 Evict file 1 2,3 31 Ergodicity proof methodology • Given any pair of recurrent states, we design a sample path between them – sequence of requests, and corresponding evictions A B 32 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 33 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 37 CN Ergodicity Policy Equivalence Classes LRU Set of States (1,3,2) Random State {1,2,3} (2,1,3) (2,3,1) (1,2,3) (3,2,1) (3,1,2) 38 CN Ergodicity Policy Equivalence Classes LRU Set of States (1,3,2) (2,1,3) Random State {1,2,3} For LRU, each file in the cache might be the next to be evicted (2,3,1) (1,2,3) (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: What structures yield nonergodic 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(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 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 - 2 V4 V4 V3 V2 V3 V2 V1 V1 48 Random Replacement CNs - 2 V4 V4 V3 V2 V3 V2 V1 V1 49 Random Replacement CNs - 2 V4 V4 V3 V2 V3 V2 V1 V1 50 Random Replacement CNs - 2 V4 V4 V3 V2 V3 V2 V1 V1 51 Random Replacement CNs - 2 Identical state V4 V4 V3 V2 V3 V2 V1 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 pij – 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 ccncaching 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 58