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

Content in the Spotlight How do I access XYZ.com? How do I find ABC.mp4?

Recasting ideas from TCP/IP Host-to-Host communication Host-to-Content communication Hosts remain fixed Path unknown and in flux TCP/IP Specify host addresses Path determined on-the-fly Host and content - fixed content location in flux ICN protocols Specify content ID Content located on-the-fly Emphasize “on the fly” Content Caching a central feature of new architectures

Graphic Notation 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

Cache Networks (CN) 101 In-network caching operation for CN consumer In-network caching operation for CN Consumer requests content Request routed towards content custodian (exists for each piece of content) En-route to custodian, inspect local cache at router for content copy During content download, store along path Cache-router Content Custodian

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

What is new about CNs? Cache Networks Caches & custodians in arbitrary topology v2 v1 Add content movement, and evictions. Use green v3 v4

What is new about CNs? Cache Networks Caches & custodians in arbitrary topology Introduces cross-flows – requests in both directions on a link v2 v1 v3 v4

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 v2 v1 v3 v4

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

Modeling Variables Vi s(i,j) Replacement Policy

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

Modeling Variables V1 V2 …. Vk consumer λ(i,j) s(i,j) r(i,j) Vi Exogenous Requests λ(i,j) Vi s(i,j) r(i,j) Miss Routing Replacement Policy

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? Rosensweig et al 2010, 2013 Add reference to our work ‘10, ‘13

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

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

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

Example #1 V1 V2 Distinguish between content (numbers?) V1 V2

Example - Performance FIFO, Cache size = 2 V1 V2 Remove 21-22, use example 0, fix notation V2

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

Example - Performance FIFO, Cache size = 2 V1 V2 Remove 21-22, use example 0, fix notation V2

Example - Performance Exogenous arrivals System Behavior λ( ,1)=0.35 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 Initial State Pr(v1 has ) Pr(v1 has ) ( , ) 0.46 0.63 0.33 0.76 Remove 21-22, use example 0, fix notation V2

Example – Networked FIFO Initial state impacted steady state Function of cache networking V1 Distinguish between content (numbers?) when does initial state impact steady-state? V2

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 Change graphics – three boxes side-by-side

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 Change graphics – three boxes side-by-side

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

Markov Chains for CNs CN State = the content of each cache (c1 state, c2 state, …)

({1,2,3}, {3,5,6}) Markov Chains for CNs ((2,1,3), (6,3,5)) 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}) Random ((2,1,3), (6,3,5)) 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 A A t1 t2 > t1 A 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

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

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 Evict file 1 1,3 2,3

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

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

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 Add comment: positive request rate for all files

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

CN Ergodicity Policy Equivalence Classes LRU Set of States (1,3,2) (2,1,3) Random State (2,3,1) {1,2,3} (1,2,3) (3,2,1) (3,1,2)

CN Ergodicity Policy Equivalence Classes LRU Set of States (1,3,2) (2,1,3) Random State (2,3,1) {1,2,3} (1,2,3) Make it clear that this contracted state has transitions to other contracted states (3,2,1) (3,1,2) For LRU, each file in the cache might be the next to be evicted

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

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

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 Consider if this should be mentioned earlier in the slides, as part of motivation

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

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.

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

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

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

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

Random Replacement CNs - 2 Identical state V4 V4 V3 V3 V2 V2 V1 V1

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 Might want to add some points on proof mechanism. The fade-out at the end of this slide is designed for this discussion

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

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 Citation format consistency and brevity

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

Pr(Xj = fi | X1,..,Xj-1) = Pr(Xj=fi) 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 Backup slide: independence of nodes visualized,