Resource Placement and Assignment in Distributed Network Topologies

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


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?


5

Tewari & Kleinrock [2006] Proposed the Proportional Mean Replication.

Zhou, Fu & Chiu [ 2011] Proposed the RLB (Random with Load

Balancing) Replication.

Related Work


6

The Multi-Region Placement Problem

Available resourceLocal

storage

Input: Region j storage size: Sj

Stochastic demand distribution Nij ,random variable.

Service costs


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


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.


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


Multi-Region: MatchingMatch local first, then remote, then server.

11Rochman, Levy, Brosh April 2013

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


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


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


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


Tail formula:

16

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}


17

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


18

Key 3: Min cost flow

Rochman, Levy, Brosh

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

Rochman, Levy, Brosh 19

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


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


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


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


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


Questions?


27

An alternative allocation: Proportional mean Allocate movies proportion to mean of

distribution

How good are the results?


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=


demand


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


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


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!

<


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!

Documents

Resource Placement and Assignment in Distributed Network Topologies