On the Steady-State of Cache Networks

1. On the Steady-State of Cache Networks Elisha J. Rosensweig Daniel S. Menasche Jim Kurose

2. 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

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

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

5. Graphic Notation Content (file) Request for content

6. Caching 101 • Stand-alone caches • Arrival stream is filtered by cache hits. Misses routed towards custodian. • Replacement policy:what to evictfrom a cache to make room for new content • Common/Popular policies – LRU, LFU, FIFO… Arrivals Misses

7. Cache Networks (CN) 101 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

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

9. What is new about CNs? • Cache Networks • Caches & custodians in arbitrary topology v2 v1 v3 v4

10. 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

11. 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

12. 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

13. Modeling Variables Vi s(i,j) Replacement Policy

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

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

16. 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

17. Our work - contributions • Examples where initial state impacts steady-stateof CN • Formulated threeconditions 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

18. 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

19. 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

20. Example #1 V1 V2 V1 V2

21. Example - Performance • Exogenous arrivals • System Behavior FIFO, Cache size = 2 V1 V2

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

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 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}) Random ((2,1,3), (6,3,5)) 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 • Communicating states • Two states communicate if there is a sample path in both directions between them A A t1 t2 > t1 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 samplepath, 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

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 samplepath, we can select any viable request and eviction • Sufficient that transitions are possible A B C

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 ErgodicityPolicy 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)

38. CN ErgodicityPolicy Equivalence Classes LRU Set of States (1,3,2) (2,1,3) Random State (2,3,1) {1,2,3} (1,2,3) (3,1,2) (3,2,1) 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?

44. Backup Slides

45. 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-ergodiccaches • 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 V3 V2 V2 V1 V1

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

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

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