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
2
Content in the Spotlight
How do I
access
XYZ.com?
How do I
find
ABC.mp4?
3
Recasting ideas from TCP/IP
Host-to-Host communication
• Hosts remain fixed
• Path unknown and in flux
Host-to-Content communication
TCP/IP
Specify host addresses
Path determined on-the-fly
ICN protocols
Specify content ID
Content located on-the-fly
• Host and content - fixed
• content location in flux
Content Caching a central
feature of new architectures
4
Graphic Notation
Content (file)
Request for content
5
Caching 101
• Stand-alone caches
Arrivals
Misses
– 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…
6
Cache Networks (CN) 101
consumer
• In-network caching
operation for CN
1. Consumer requests
content
2. 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
Content
Custodian
Cacherouter
7
What is new about CNs?
• Cache hierarchies
– Single custodian
– Requests flow
upstream, content
flows downstream
• Approximate models
proposed
8
What is new about CNs?
• Cache Networks
– Caches & custodians
in arbitrary topology
v2
v1
v3
v4
9
What is new about CNs?
• Cache Networks
– Caches & custodians
in arbitrary topology
– Introduces crossflows – requests in
both directions on a
link
v2
v1
v3
v4
10
What is new about CNs?
• Cache Networks
– Caches & custodians
in arbitrary topology
– Introduces crossflows – requests in
both directions on a
link
– Cross-flows create
state dependency
loops
v2
v1
v3
v4
11
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
12
Modeling Variables
Vi
s(i,j)
Replacement
Policy
13
Modeling Variables
consumer
Exogenous Requests
λ(i,j)
Vi
s(i,j)
Replacement
Policy
14
Modeling Variables
V1
consumer
Exogenous Requests
λ(i,j)
V2
Vi
….
s(i,j)
r(i,j)
Vk
Miss
Routing
Replacement
Policy
15
Our work – the challenge
• Existing models consider the impact of
– Request arrival distribution
– Network topology and miss routing
– Replacement policy and cache size
Rosensweig et al
2010, 2013
• Not considered: initial state of caches
• Question: Can the initial state affect long term
performance?
16
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
17
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
18
Motivation
• Why should the initial state impact steadystate 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
19
Example #1
V1
V2
V1
V2
20
Example - Performance
Exogenous arrivals
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
System Behavior
Initial State
Pr(v1 has
)
Pr(v1 has )
V2
( , )
( , )
0.46
0.33
0.63
0.76
Example – Networked FIFO
• Initial state
impacted steady
state
• Function of cache
networking
when does initial state
impact steady-state?
V1
V2
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})
((2,1,3),
(6,3,5))
Random
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
A
t1
A
t2 > t1
• Communicating states
– Two states communicate if there is a sample path
in both directions between them
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 sample path, we can select any
viable request and eviction
– Sufficient that transitions are possible
1,2
Evict file 2
1,3
Request file 3
Evict file 1
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 sample path, we can select any
viable request and eviction
– Sufficient that transitions are possible
A
C
B
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 Ergodicity
Policy Equivalence Classes
LRU
Set of States
(1,3,2)
Random State
{1,2,3}
(2,1,3)
(2,3,1)
(1,2,3)
(3,2,1)
(3,1,2)
38
CN Ergodicity
Policy Equivalence Classes
LRU
Set of States
(1,3,2)
(2,1,3)
Random State
{1,2,3}
For LRU, each file in
the cache might be the
next to be evicted
(2,3,1)
(1,2,3)
(3,2,1)
(3,1,2)
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 nonergodic CNs?
– Many implications if realistic such CNs exist
– How does structure impact behavior, in general
43
Questions?
Backup Slides
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
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
V2
V3
V2
V1
V1
48
Random Replacement CNs - 2
V4
V4
V3
V2
V3
V2
V1
V1
49
Random Replacement CNs - 2
V4
V4
V3
V2
V3
V2
V1
V1
50
Random Replacement CNs - 2
V4
V4
V3
V2
V3
V2
V1
V1
51
Random Replacement CNs - 2
Identical state
V4
V4
V3
V2
V3
V2
V1
V1
52
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
53
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
54
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 ccncaching trees” , IFIP Networking 2011
55
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
56
Example – single FIFO explained
Order matters in
FIFO
• Disjoint markov chains, but
• Existence probability is identical in both
• Conservation of flows
57
58