On the Steady-State of Cache Networks Elisha J. Rosensweig Daniel S. Menasche Jim Kurose.
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Transcript of On the Steady-State of Cache Networks Elisha J. Rosensweig Daniel S. Menasche Jim Kurose.
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• 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
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 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
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
v1
v2
v4
v3
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
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
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
s(i,j)
Vi
Replacement Policy
14
Modeling Variables
consumer
s(i,j)
λ(i,j)
Vi
Replacement Policy
Exogenous Requests
15
Modeling Variables
consumer
s(i,j)r(i,j)
λ(i,j)
Vi
V1
V2
….
Vk
Replacement Policy
Exogenous Requests
Miss Routing
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?
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
V1 V2
Example - Performance
V1
V2
FIFO, Cache size = 2
22
Example – single FIFO explained
• Disjoint markov chains, but• Existence probability is identical in both• Conservation of flows
Order matters in FIFO
Example - Performance
V1
V2
FIFO, Cache size = 2
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
Example – Networked FIFO
V1
V2
• Initial state impacted steady state
• Function of cache networking
when does initial state impact steady-state?
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
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
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
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
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
1,2
1,3 2,3
Request file 3
Evict file 1Evict file 2
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 BC
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
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 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
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
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
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
Questions?
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 - 2
V1
V2
V3
V4
V1
V2
V3
V4
51
Random Replacement CNs - 2
V1
V2
V3
V4
V1
V2
V3
V4
52
Random Replacement CNs - 2
V1
V2
V3
V4
V1
V2
V3
V4
53
Random Replacement CNs - 2
V1
V2
V3
V4
V1
V2
V3
V4
54
Random Replacement CNs - 2
V1
V2
V3
V4
V1
V2
V3
V4Identical state
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
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 & 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
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
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