resource placement and assignment in distributed network topologies
DESCRIPTION
Yuval Rochman, Hanoch Levy, Eli Brosh. Resource Placement and Assignment in Distributed Network Topologies. Accepted to: INFOCOM 2013. Motivation: Video-on-Demand service. Video-on-Demand ( VoD ) internet service Large collection of movies Highly-variable Geo-distributed demand - PowerPoint PPT PresentationTRANSCRIPT
Resource Placement and Assignment in Distributed Network Topologies
Accepted to: INFOCOM 2013
Yuval Rochman, Hanoch Levy, Eli Brosh
2
Motivation: Video-on-Demand service Video-on-Demand (VoD) internet service
Large collection of movies Highly-variable Geo-distributed demand
Use Content Distribution Network
Rochman, Levy, Brosh April 2013
3
Motivation: Content Distribution Network Multi-region server structure (e.g., terminal
based service, cloud) Service costs: intra-region < inter-region <
central
Rochman, Levy, Brosh April 2013
Intra-regionLow cost
Inter-regionMedium cost
CentralHigh cost
Region 2
Central video server
Region 1
User terminals - demand
Request typeDisk
4
System and Objective Players: users + content servers (local, central) Objective: Reduce service costs
Replicating content at regions
Central video server
Region 1
Region 2
User terminals - demand
Low cost Medium cost
High cost
Problem: Which movies to place where?
Rochman, Levy, Brosh April 2013
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Tewari & Kleinrock [2006] Proposed the Proportional Mean Replication.
Zhou, Fu & Chiu [ 2011] Proposed the RLB (Random with Load
Balancing) Replication.
Related Work
Rochman, Levy, Brosh April 2013
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The Multi-Region Placement Problem
Available resourceLocal
storage
Input: Region j storage size: Sj
Stochastic demand distribution Nij ,random variable.
Service costs
Rochman, Levy, Brosh April 2013
S1 =4 S2 =2? ? ? ?
Pr(N11 <=x)Pr(N2
1 <=x) Pr(N12 <=x)Pr(N2
2 <=x)
? ?
Stochastic demand
E.g., high-variability, correlated
Local < Remote < Server
7
Pr(N11<=x)Pr(N2
1<=x) Pr(N12<=x) Pr(N2
2<=x)
S1 =4
The Multi-Region Placement Problem
Local < Remote < Server
Input: Storage Sj , demand Nij , service costs
Allocation: Place resources at regions Cost of allocation: expected cost of optimal assignment
(over all demand realizations)Goal: find allocation with minimal cost
Rochman, Levy, Brosh April 2013
Actual demand
S2 =2? ?
Stochastic demand
Available resource
? ? ? ?Local storage
8
Challenge and principles• Challenge: Combinatorial problem based on multi-dimensional stochastic variables
•Keys of solution: Semi-Separability, Concavity, Reduction to Min-cost Flow problem.
Rochman, Levy, Brosh April 2013
Local storage S1 =4 S2 =2
? ? ? ?
Pr(N11<=x)Pr(N2
1<=x) Pr(N12<=x) Pr(N2
2<=x)
? ?
Stochastic demand
Available resource
Exponential number of allocations
Large database!
Single Region: MatchingDemand realization to resources
Observed Demand
Resources
1 22, 1L L= =
1 21, 2= =n n2
1
profit min( , ) 2i ii
L n
3=S
A profit formula! 9Rochman, Levy, Brosh April 2013
Single region: Revenue Formulation Lemma: optimal matching maximizes revenue of a realization
10
1
Rev({ }) (profit) (min( , ))i
m
i N i ii
L E E L N
Random Demand
Type-i replicas iL
iN
1
profit min( , )m
i ii
L n
Hence: we have to maximize
For any placement and demand
Rochman, Levy, Brosh April 2013
Multi-Region: MatchingMatch local first, then remote, then server.
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Available resource
Multi-Region: Revenue formulation Thm: maximize revenue to find opt placement
{Lij} :
Local revenueGlobal
revenue
1 1 1
max (min( , )) (min( , ))m m k
L j jglo i i loc i i
i i j
R R E L N R E L N
s.t. j j j
iL L s
Type-i resources at region j
12Rochman, Levy, Brosh April 2013
11 1=L 2
1 2=L1 3=L
Local=3
Global=4
Separability and semi-Separability Definition: function is separable iff
Sum of separated marginal components Definition: function is semi-separable iff
“Almost” separated components
1 1
({ }) ( )m k
j j ji i i
i j
f x g x
13Rochman, Levy, Brosh April 2013
1 1 1
({ }) ( ) ( )m k m
j j ji i i i i
i j i
f x g x g x
Where 1
kji i
j
x x
f
f
Key 1: Revenue is Semi-separable Revenue function
Revenue function is semi-separable. Sum of local replicas = # global replicas.
1 1 1
( ) (min( , )) (min( , ))jii
m k mL j j
loc i i glo N i iNi j i
E R R E L N R E L N
14Rochman, Levy, Brosh April 2013
Local replicas Global
replicas
( ) (min( , ))
( ) (min( , ))i
i
j ji loc N i
i glo N i
g a R E a N
g a R E a N
1 1 1
( ) ( ) ( )m k m
L j ji i i i
i j i
E R g L g L
11 1=L 2
1 2=L1 3=L
1
kji i
j
L L
15
Key 2: Concavity Partial expectation
Partial Expectations are concave! Cumulative(cdf) is monotonic
Thus, Partial expectation is concave
( ) Pr( )XF x X x
1
1
( ) 1 (1 ( ))a
Xx
PE a CDF F x
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Tail formula:
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Placement Optimization Problem Find {Li
j} allocation of type-i movie at region j ({Li
j} ) maximizing:
Under: capacity bound in each region
1 1 1 1 1
( ) Pr( ) Pr( )j
i iL Lm m kL j
glo i loc ii t i j t
E R R N t R N t
1
mj j ji
i
L L s
Concave in placement vars
{Lij}
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The Multi-Region Problems Symmetric bounded– QEST 2012, low
complexity Greedy algorithm, max-percentile based
Asymmetric bounded– INFOCOM 2013, higher complexity Reduction to min-cost flow problem
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Key 3: Min cost flow
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s t
11/13
12/12
15/20
1/44/9
7/7
4/48/13
11/14
0 2Flow/Capacity0Weight
1
0
0
0 0
0
April 2013
Flow value 11 8 19f
Flow weight (cost) ( ) 4*1 15*2 34W f
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The Min-Cost Flow Problem Input:
A positive capacity function C on the edges, C: ER+
A positive weight function W on the edges, W: ER+
Required Flow value r Output: : an s-t flow f, with flow value= r, which minimizes weight Σf(e) W(e) .
s t
11/13 15/20
1/4 4/97/7
4/48/13 11/14
1 212/12
April 2013
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Main theoremTheorem : Assume: - concave & semi-separable Then, there is effective solution for
Solution uses min cost flow algorithm On 7-layer graph!
Correctness at the paper. April 2013
1
max ({ })
s.t
ji
mj ji
i
f L
L s
f
Rochman, Levy, Brosh 21
7-layer graph: Local part
S
1a
2a
1 1,a t
1 2,a t
2 1,a t
2 2,a t
1 1, ,1a t
1 1, , 2a t
1 2, ,1a t
2 1, ,1a t
Region Region, Movie type Region, Movie, #
replicas
0,
0,
0,
0,2 0,s
1 0,s
11Pr(, )1 1locR N
11Pr(, )1 2locR N
Capacity, Weight
12Pr(, )1 1locR N
21Pr(, )1 1locR N
April 2013
Capacity of region
Local weight
22
7-layer graph: Global part
1 1, ,1a t
1 1, , 2a t
1 2, ,1a t
2 1, ,1a t
1t
2t
1,1t
1, 2t
2 ,1t
2 , 2t
t
Region, Movie type, # items
Movie type, # items
0,
0,
0,
0,
0,
0,
0,
0,
1Pr( 1, )1 gloR N
1Pr( 2, )1 gloR N
2Pr( 1, )1 gloR N
2Pr( 2, )1 gloR N Movie type
April 2013 Rochman, Levy, Brosh
Global weight
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Min-Cost Flows
Standard solution to min-cost flow using Successive Shortest Path (SSP).
Complexity of SSP (standard solution) is
s= total storage in the system k= # regions m= # movie types
High complexity!
3 2 2( )O s k m
312
skm
April 2013
Rochman, Levy, Brosh 24
Other proposed algorithms
Bipartite algorithm (INFOCOM 2013) in complexity of
(instead of ) Idea: use only Region and movie type nodes
Clique algorithm -complexity of
Online algorithm.
3 2 2( )O s k m
312
skm
( ( ) )O s k m km
2( ( log ))O sk k m
s= total storage in the systemk= # regionsm= # movie types
April 2013
Conclusions Algorithms for resource placement and
assignment Geared for distributed network settings Arbitrary demand pattern (e.g., highly-variable,
correlated)
Joint placement-assignment problem Multi-dimensional stochastic demand New solution techniques
25Rochman, Levy, Brosh April 2013
Questions?
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27
An alternative allocation: Proportional mean Allocate movies proportion to mean of
distribution
How good are the results?
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1
2
3
( ) 4( ) 12( ) 8
E NE NE N
28
Two resource-types. Single region, capacity n
Proportional Mean: Expected profit= 2*n/(k+1)
Optimal allocation: n replicas to red . Expected profit=n.
Proportional Mean Not optimal
0k
0
1
n
Pr(N=x)
1-1/k
1/k
nk2x=
Rochman, Levy, Brosh April 2013
demand
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Reduction to single region
S t
1t1,1t1 1, Pr(1 )N
2t
1, 2t
1,t s
2 ,1t
2 , 2t
.
.
3t
Capacity, Weight
1 2,Pr(1 )N
1 ),Pr(1 N s
2 1,Pr(1 )N
2 2, Pr(1 )N
0,
0,
0,
0,
0,
0,
0,
0,
Movie type Movie type, # replicasApril 2013
Flow value= s
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Convert max to min
Correctness
1 1
Pr( )iLm
ii j
N j
1
m
ii
L s
1{ }
maxm
i iL = 1 1
Pr( )iLm
ii j
N j
1
m
ii
L s
1{ }
minm
i iL =
.s t.s t
Original New
If solution is 1
m
ii
L s
0iL
April 2013
Rochman, Levy, Brosh 31
Correctness
S t
1t1,1t1 1, Pr(1 )N
2t
1, 2t
1,t s
2 ,1t
2 , 2t
.
.
Capacity, Weight
1 2,Pr(1 )N
1 ),Pr(1 N s
2 1,Pr(1 )N
2 2, Pr(1 )N
0,
0,
0,
0,
0,
0,
0,
Movie type, # replicas
1
1 1
min Pr( )iLm
ii t
N t= =
<å å
April 2013
1 1 2Pr( 1) Pr( 2) Pr( 1)N N N= < + < + <1
s.t m
ii
L s=
=å
<<
Concavity!
<
Rochman, Levy, Brosh 32
Reduction to multi regionConvert max to min:
1 1 1 1 1
min ( ) Pr( ) Pr( )j
i iL Lm m kL j
glo i loc ii t i j t
E C R N t R N t= = = = =
= < + <å å å å å
1
s.t m
j ji
i
L s=
=åGlobal
Local
April 2013
Semi-Separability!