On the Steady-State of Cache Networks
description
Transcript of On the Steady-State of Cache Networks
![Page 1: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/1.jpg)
On the Steady-State of Cache Networks
Elisha J. Rosensweig Daniel S. Menasche
Jim Kurose
![Page 2: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/2.jpg)
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
![Page 3: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/3.jpg)
3
Content in the SpotlightHow do I access
XYZ.com?
How do I find
ABC.mp4?
![Page 4: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/4.jpg)
4
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 protocolsSpecify content ID
Content located on-the-fly
Content Caching a central feature of new architectures
![Page 5: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/5.jpg)
5
Graphic Notation
Content (file) Request for content
![Page 6: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/6.jpg)
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
![Page 7: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/7.jpg)
7
Cache Networks (CN) 101• In-network caching
operation for CN1. Consumer requests
content2. 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
ContentCustodian
![Page 8: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/8.jpg)
8
What is new about CNs?
• Cache hierarchies– Single custodian– Requests flow
upstream, content flows downstream
• Approximate models proposed
![Page 9: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/9.jpg)
9
What is new about CNs?
• Cache Networks– Caches & custodians
in arbitrary topologyv1
v2
v4
v3
![Page 10: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/10.jpg)
10
What is new about CNs?
• Cache Networks– Caches & custodians
in arbitrary topology– Introduces cross-
flows – requests in both directions on a link
v1
v2
v4
v3
![Page 11: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/11.jpg)
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
v1
v2
v4
v3
![Page 12: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/12.jpg)
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
![Page 13: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/13.jpg)
13
Modeling Variables
s(i,j)
Vi
Replacement Policy
![Page 14: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/14.jpg)
14
Modeling Variables
consumer
s(i,j)
λ(i,j)
Vi
Replacement Policy
Exogenous Requests
![Page 15: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/15.jpg)
15
Modeling Variables
consumer
s(i,j)r(i,j)
λ(i,j)
Vi
V1
V2
….
Vk
Replacement Policy
Exogenous Requests
Miss Routing
![Page 16: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/16.jpg)
16
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?
![Page 17: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/17.jpg)
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
![Page 18: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/18.jpg)
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
![Page 19: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/19.jpg)
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
![Page 20: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/20.jpg)
20
Example #1
V1 V2
V1 V2
![Page 21: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/21.jpg)
Example - Performance
V1
V2
FIFO, Cache size = 2
![Page 22: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/22.jpg)
22
Example – single FIFO explained
• Disjoint markov chains, but• Existence probability is identical in both• Conservation of flows
Order matters in FIFO
![Page 23: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/23.jpg)
Example - Performance
V1
V2
FIFO, Cache size = 2
![Page 24: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/24.jpg)
Example - Performance
V1
V2
Exogenous arrivals
System BehaviorInitial State Pr(v1 has ) Pr(v1 has )
( , ) 0.46 0.63( , ) 0.33 0.76
λ( ,1)=0.35 λ( ,1)=0.55 λ( ,1)=0.1λ( ,2)=0.05 λ( ,2)=0.15 λ( ,2)=0.8
FIFO, Cache size = 2
![Page 25: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/25.jpg)
Example – Networked FIFO
V1
V2
• Initial state impacted steady state
• Function of cache networking
when does initial state impact steady-state?
![Page 26: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/26.jpg)
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
![Page 27: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/27.jpg)
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
![Page 28: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/28.jpg)
29
Markov Chains for CNs
• CN State = the content of each cache
(c1 state, c2 state,
…)
![Page 29: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/29.jpg)
30
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
![Page 30: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/30.jpg)
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 At1 t2 > t1
A B
![Page 31: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/31.jpg)
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
![Page 32: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/32.jpg)
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
![Page 33: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/33.jpg)
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
1,2
1,3 2,3
Request file 3
Evict file 1Evict file 2
![Page 34: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/34.jpg)
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
![Page 35: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/35.jpg)
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
![Page 36: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/36.jpg)
37
Ergodicity proof - reminder
• In charting a sample path, we can select any viable request and eviction– Sufficient that transitions are possible
A BC
![Page 37: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/37.jpg)
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
![Page 38: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/38.jpg)
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
![Page 39: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/39.jpg)
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
![Page 40: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/40.jpg)
41
CN ErgodicityPolicy 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
![Page 41: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/41.jpg)
42
CN ErgodicityPolicy 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
![Page 42: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/42.jpg)
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
![Page 43: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/43.jpg)
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
![Page 44: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/44.jpg)
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
![Page 45: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/45.jpg)
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
![Page 46: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/46.jpg)
Questions?
![Page 47: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/47.jpg)
Backup Slides
![Page 48: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/48.jpg)
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.
![Page 49: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/49.jpg)
50
Random Replacement CNs - 2
V1
V2
V3
V4
V1
V2
V3
V4
![Page 50: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/50.jpg)
51
Random Replacement CNs - 2
V1
V2
V3
V4
V1
V2
V3
V4
![Page 51: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/51.jpg)
52
Random Replacement CNs - 2
V1
V2
V3
V4
V1
V2
V3
V4
![Page 52: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/52.jpg)
53
Random Replacement CNs - 2
V1
V2
V3
V4
V1
V2
V3
V4
![Page 53: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/53.jpg)
54
Random Replacement CNs - 2
V1
V2
V3
V4
V1
V2
V3
V4Identical state
![Page 54: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/54.jpg)
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
![Page 55: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/55.jpg)
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
![Page 56: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/56.jpg)
57
a-NET, Net Calculus & ErgodicityRelated 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
![Page 57: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/57.jpg)
58
a-NET, Net Calculus & ErgodicityRelated 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
![Page 58: On the Steady-State of Cache Networks](https://reader035.fdocuments.in/reader035/viewer/2022062521/5681691f550346895de04c8f/html5/thumbnails/58.jpg)
59
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