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
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

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 evict from a cache to make room for new content
Common/Popular policies – LRU, LFU, FIFO…
Arrivals
Misses

7. 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

8. What is new about CNs? Cache hierarchies Approximate models proposed
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
Add content movement, and evictions. Use green 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 λ(i,j) s(i,j) Vi Exogenous Requests
Replacement Policy

15. 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

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
Add reference to our work ‘10, ‘13

17. 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

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
Distinguish between content (numbers?) V1 V2

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

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

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

24. 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

25. 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

26. 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

27. 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

28. 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

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

30. ({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

31. 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

32. 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

33. 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

34. 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

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

36. 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

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

38. 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

39. 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

40. 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

41. 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)

42. 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

43. 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

44. 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

45. 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

46. 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

47. Questions?

48. Backup Slides

49. 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.

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

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

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

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

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

55. 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

56. 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

57. 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

58. 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

59. 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,