Bandwidth Allocation in Large Stochastic Networks Mathieu Feuillet - PowerPoint PPT Presentation

Model 1 1 βN files 3 2 2 2 copies/file 3 2 Back-up: λN A file with 0 copies is lost 24

Model 1 1 βN files 3 2 copies/file 3 2 2 Back-up: λN A file with 0 copies is lost 24

Model 1 1 βN files 3 2 copies/file 3 Back-up: λN A file with 0 copies is lost 24

Model 1 1 1 βN files 3 2 copies/file 3 Back-up: λN 24

Model 1 βN files 3 2 copies/file 3 3 Back-up: λN 24

Model 1 βN files 3 3 2 copies/file Back-up: λN 24

Model 1 1 βN files 2 copies/file Back-up: λN 24

Model βN files ∅ 2 copies/file Back-up: λN What is the decay rate of the network? 24

Model X i ( t ) : number of files with i copies at time t . ( X 0 ( t ) , X 1 ( t ) , X 2 ( t )) : a transient Markov Process. X 0 ( t ) + X 1 ( t ) + X 2 ( t ) = βN. A unique absorbing state ( βN, 0 , 0 ) . 2 μx 2 ( X 0 ( t )) ( X 1 ( t )) ( X 2 ( t )) μx 1 λN 1 { x 1 > 0 } 25

Model X i ( t ) : number of files with i copies at time t . ( X 0 ( t ) , X 1 ( t ) , X 2 ( t )) : a transient Markov Process. X 0 ( t ) + X 1 ( t ) + X 2 ( t ) = βN. A unique absorbing state ( βN, 0 , 0 ) . 2 μ ( βN − x 0 − x 1 ) ( X 0 ( t )) ( X 1 ( t )) ( X 2 ( t )) μx 1 λN 1 { x 1 > 0 } 25

Different Behaviors Three time scales:  t → t/N  t → t  t → Nt Three regimes: Overload: 2 β > ρ = λ/μ , Critical load: 2 β = ρ , Underload: 2 β < ρ . 26

Time scale: t → t/N 2 μ ( βN − x 1 − x 0 ) N ( X 0 ( t/N )) ( X 1 ( t/N )) ( X 2 ( t/N )) x 1 λ 1 { x 1 > 0 } μ N 27

Time scale: t → t/N � ergodic if 2 β < ρ, ( L 1 ( t )) : an M/M/ 1 queue transient if 2 β > ρ. 2 μβ ( L 1 ( t )) ∼ Nβ 0 λ 1 { x 1 > 0 } 28

Time scale: t → t/N � ergodic if 2 β < ρ, ( L 1 ( t )) : an M/M/ 1 queue transient if 2 β > ρ. 2 μβ ( L 1 ( t )) ∼ Nβ 0 λ 1 { x 1 > 0 } No loss! 28

Time scale: t → t Overloaded network If 2 β > ρ , ( X 0 ( t ) /N, X 1 ( t ) /N, X 2 ( t ) /N ) converges to a deterministic process ( x 0 ( t ) , x 1 ( t ) , x 2 ( t )) . x 2 ( t ) : 2 copies x 1 ( t ) : 1 copy β x 0 ( t ) : 0 copies λ 2 μ t 29

Time scale: t → t Overloaded network If 2 β > ρ , ( X 0 ( t ) /N, X 1 ( t ) /N, X 2 ( t ) /N ) converges to a deterministic process ( x 0 ( t ) , x 1 ( t ) , x 2 ( t )) . x 2 ( t ) : 2 copies x 1 ( t ) : 1 copy β x 0 ( t ) : 0 copies λ 2 μ t A fraction N ( β − ρ/ 2 ) is lost! 29

Time scale: t → t Underloaded network If 2 β < ρ , ( X 0 ( t ) /N, X 1 ( t ) /N, X 2 ( t ) /N ) converges to  = 0 , x 0 ( t )  = 0 , x 1 ( t )  x 2 ( t ) = β. λ 2 μ x 2 ( t ) : 2 copies β t 30

Time scale: t → t Underloaded network If 2 β < ρ , ( X 0 ( t ) /N, X 1 ( t ) /N, X 2 ( t ) /N ) converges to  = 0 , x 0 ( t )  = 0 , x 1 ( t )  x 2 ( t ) = β. λ 2 μ x 2 ( t ) : 2 copies β t No significant loss! 30

Time Scale t → Nt � X 0 ( Nt ) � lim = Ψ( t ) , N N → + ∞ where Ψ( t ) is the unique solution of � t 2 μ ( β − Ψ( s )) d s. Ψ( t ) = μ λ − 2 μ ( β − Ψ( s )) 0 31

Time Scale t → Nt � X 0 ( Nt ) � lim = Ψ( t ) , N N → + ∞ where Ψ( t ) unique solution in ( 0 , β ) of ( 1 − Ψ( t ) /β ) ρ/ 2 e Ψ( t )+ t = 1 . β 0 copies t t → Nt is the “correct” time scale to describe 31 decay.

A Stochastic Averaging Phenomenon ∼ N Ψ( t ) X 1 ,t ( ∞ ) ∼ N ( β − Ψ( t )) Fast time scale: At “time” Nt, ( X 1 ( Nt + u / N ) , u ≥ 0 ) : an M / M / 1 with transition rates: + 1 at rate 2 μ ( β − Ψ( t )) − 1 at rate λ. 32

A Stochastic Averaging Phenomenon ∼ N Ψ( t ) X 1 ,t ( ∞ ) ∼ N ( β − Ψ( t )) Slow time scale: ( X 0 ( Nt ) /N ) “sees” only X 1 at equi- librium: � t � t 2 μ ( β − Ψ( s )) Ψ( t )” = ” μ E ( X 1 ,s ( ∞ )) d s = d s. λ − 2 μ ( β − Ψ( s )) 0 0 32

T echnical Corner Step 1 Radon measures: tightness of ( μ N ) with � Nt 1 〈 μ N , g 〉 = � X N � g 1 ( s ) , s d s N 0 33

T echnical Corner Step 1 Radon measures: tightness of ( μ N ) with � Nt 1 〈 μ N , g 〉 = � X N � g 1 ( s ) , s d s N 0 Step 2 Control of limits of ( μ N ) : � Nt � t 1 X N lim 1 ( s ) d s = Ψ( t ) = 〈 π s , I 〉 d s N → ∞ N 0 0 � t π s ( N ) d s = t 0 33

T echnical Corner Step 1 Radon measures: tightness of ( μ N ) with � Nt 1 〈 μ N , g 〉 = � X N � g 1 ( s ) , s d s N 0 Step 2 Control of limits of ( μ N ) : � Nt � t 1 X N lim 1 ( s ) d s = Ψ( t ) = 〈 π s , I 〉 d s N → ∞ N 0 0 � t π s ( N ) d s = t 0 Here: Proof by stochastic domination 33

T echnical Corner Step 1 Radon measures: tightness of ( μ N ) with � Nt 1 〈 μ N , g 〉 = � X N � g 1 ( s ) , s d s N 0 Step 2 Control of limits of ( μ N ) : � Nt � t 1 X N lim 1 ( s ) d s = Ψ( t ) = 〈 π s , I 〉 d s N → ∞ N 0 0 � t π s ( N ) d s = t 0 Here: Proof by stochastic domination Step 3 Identification of π s with martingale techniques and balance equations. 33

Decay Rate of the Network � � t ≥ 0 : X N T N ( δ ) = inf 0 ( t ) ≥ δβN Theorem: T N ( δ ) ρ lim log ( 1 − δ ) − δβ. = − 2 N N → ∞ T N ( δ ) /N δ 34

Conclusion - Three different time scales - A first example of stochastic averaging - Asymptotics on a transitory property. Extensions: - Number of copies: d > 2 ⇒ d − 1 times scales - Decentralized back-up (mean-field) Open problem: - Modeling a DHT: geometrical considerations 35

Example 2: The Law of the Jungle

Context Congestion control: - Rate adjustment to limit packet loss - Retransmission of lost packets 37

Context Congestion control: - Rate adjustment to limit packet loss - Retransmission of lost packets No congestion control: - No rate adjustment - Sources send at their maximum rate - Coding to recover from packet loss 37

Context Congestion control: - Rate adjustment to limit packet loss - Retransmission of lost packets No congestion control: - No rate adjustment - Sources send at their maximum rate - Coding to recover from packet loss Does this bring congestion collapse? 37

Bandwidth Sharing Networks [Massoulié Roberts 00] 0 1 2 - A flow: a stream of packets - Flows are considered as a fluid - Users divided in classes/routes - Poisson arrivals/Exponential sizes - Resource allocation determined by congestion policy 38

Bandwidth Sharing Networks [Massoulié Roberts 00] λ 0 μ 0 ϕ 0 λ 1 λ 2 μ 1 ϕ 1 μ 2 ϕ 2 - A flow: a stream of packets - Flows are considered as a fluid - Users divided in classes/routes - Poisson arrivals/Exponential sizes - Resource allocation determined by congestion policy 38

Resource Allocation Usually, α -fair policies are considered [MW00] . Here: - Sources send at their maximum rate (1 or a ) - T ail dropping: At each link, output rates are proportional to input rates 39

Resource Allocation Usually, α -fair policies are considered [MW00] . Here: - Sources send at their maximum rate (1 or a ) - T ail dropping: At each link, output rates are proportional to input rates x 0 x 1 x 2 39

Resource Allocation Usually, α -fair policies are considered [MW00] . Here: - Sources send at their maximum rate (1 or a ) - T ail dropping: At each link, output rates are proportional to input rates x 0 α = x 0 + x 1 x 0 x 1 x 2 x 1 ϕ 1 = x 0 + x 1 39

Resource Allocation Usually, α -fair policies are considered [MW00] . Here: - Sources send at their maximum rate (1 or a ) - T ail dropping: At each link, output rates are proportional to input rates x 0 α = x 0 + x 1 x 0 � � α ϕ 0 = min α, x 2 a + α x 1 x 2 � � x 2 a ϕ 2 = min x 2 a , x 1 ϕ 1 = α + x 2 a x 0 + x 1 39

Ergodicity Condition 0 1 2 Optimal ergodicity condition: ρ 0 + ρ 1 < 1 , ρ 0 + ρ 2 < 1 where ρ i = λ i /μ i . We know α -fair policies are optimal [BM02] . 40

Ergodicity Condition 0 1 2 Optimal ergodicity condition: ρ 0 + ρ 1 < 1 , ρ 0 + ρ 2 < 1 where ρ i = λ i /μ i . We know α -fair policies are optimal [BM02] . What about our policy? 40

Fluid Limits x 0 α = x 0 + x 1 � � x 0 α ϕ 0 = min α, x 2 a + α x 1 x 2 � � x 2 a ϕ 2 = min x 2 a , x 1 ϕ 1 = α + x 2 a x 0 + x 1 If x 2 ≫ 0, class 2 uses virtually all the second link. If ( z 0 ( t ) , z 1 ( t ) , z 2 ( t )) is a fluid limit with z 2 ( 0 ) > 0,  ˙ z 0 ( t ) = λ 0 ,  z 1 ( t ) ˙ z 1 ( t ) = λ 1 − μ 1 z 0 ( t )+ z 1 ( t ) ,  ˙ z 2 ( t ) = λ 2 − μ 2 . 41

Fluid Limits x 0 α = x 0 + x 1 � � x 0 α ϕ 0 = min α, x 2 a + α x 1 x 2 � � x 2 a ϕ 2 = min x 2 a , x 1 ϕ 1 = α + x 2 a x 0 + x 1 If x 2 ≫ 0, class 2 uses virtually all the second link. If ( z 0 ( t ) , z 1 ( t ) , z 2 ( t )) is a fluid limit with z 2 ( 0 ) > 0,  ˙ z 0 ( t ) = λ 0 ,  z 1 ( t ) ˙ z 1 ( t ) = λ 1 − μ 1 z 0 ( t )+ z 1 ( t ) ,  ˙ z 2 ( t ) = λ 2 − μ 2 . If ρ 2 < 1, ( z 2 ( t )) reaches 0 in finite time. 41

Fluid Limits α x 0 � � α ϕ 0 = min α, x 2 a + α x 1 x 2 � � x 2 a ϕ 2 = min x 2 a , x 1 ϕ 1 = α + x 2 a x 0 + x 1 Classes 0 and 1 are frozen: π α 2 is the stationary distribution of class 2 α � � �� ¯ Φ 0 ( α ) = E π α Φ 0 α, . 2 x 2 a + α 42

Fluid Limits When z 2 ( t ) = 0:  � � z 0 ( t ) = λ 0 − μ 0 ¯ ˙ z 0 ( t ) ϕ 0 ,  z 0 ( t ) + z 1 ( t )     z 1 ( t ) ˙ z 1 ( t ) = λ 1 − μ 1 ,   z 0 ( t ) + z 1 ( t )    ˙ = 0 . z 2 ( t ) with α � � �� ¯ Φ 0 ( α ) = E π α Φ 0 α, . 2 x 2 a + α 43

Fluid Limits When z 2 ( t ) = 0:  � � z 0 ( t ) = λ 0 − μ 0 ¯ ˙ z 0 ( t ) ϕ 0 ,  z 0 ( t ) + z 1 ( t )     z 1 ( t ) ˙ z 1 ( t ) = λ 1 − μ 1 ,   z 0 ( t ) + z 1 ( t )    ˙ = 0 . z 2 ( t ) with α � � �� ¯ Φ 0 ( α ) = E π α Φ 0 α, . 2 x 2 a + α Stochastic averaging 43

Ergodicity Conditions Ergodicity conditions: ρ 1 < 1 , ρ 2 < 1 , ρ 0 < ¯ ϕ 0 ( 1 − ρ 1 ) Optimal conditions: ρ 1 < 1 , ρ 2 < 1 , ρ 0 < min ( 1 − ρ 1 , 1 − ρ 2 ) 44

Ergodicity Conditions Ergodicity conditions: ρ 1 < 1 , ρ 2 < 1 , ρ 0 < ¯ ϕ 0 ( 1 − ρ 1 ) Optimal conditions: ρ 1 < 1 , ρ 2 < 1 , ρ 0 < min ( 1 − ρ 1 , 1 − ρ 2 ) But: ¯ ϕ 0 ( 1 − ρ 1 ) < min ( 1 − ρ 2 , 1 − ρ 1 ) 44

Ergodicity Conditions Ergodicity conditions: ρ 1 < 1 , ρ 2 < 1 , ρ 0 < ¯ ϕ 0 ( 1 − ρ 1 ) Optimal conditions: ρ 1 < 1 , ρ 2 < 1 , ρ 0 < min ( 1 − ρ 1 , 1 − ρ 2 ) But: ¯ ϕ 0 ( 1 − ρ 1 ) < min ( 1 − ρ 2 , 1 − ρ 1 ) Not optimal! 44

Impact of Maximum Rate a Optimal a = 1 Class 0: ρ 0 a = 0 . 1 a = 0 . 01 Class 1: ρ 1 45

Impact of Maximum Rate a Optimal a = 1 Class 0: ρ 0 a = 0 . 1 a = 0 . 01 Class 1: ρ 1 What happens when a → 0 ? 45

Scaling the Maximum Rate a We freeze α and consider the process ( X S 2 ( t )) with Q -matrix: q ( x 2 , x 2 + 1 ) = λ 2 , � x 2 a � q ( x 2 , x 2 − 1 ) = μ 2 min x 2 a , α + x 2 a 46

Scaling the Maximum Rate a We freeze α and consider the process ( X S 2 ( t )) with Q -matrix: q ( x 2 , x 2 + 1 ) = λ 2 , � a x 2 a / S � q ( x 2 , x 2 − 1 ) = μ 2 min x 2 , S α + x 2 a/S Time-scale: t �→ St 46

Scaling the Maximum Rate a We freeze α and consider the process ( X S 2 ( t )) with Q -matrix: q ( x 2 , x 2 + 1 ) = λ 2 , � a x 2 a / S � q ( x 2 , x 2 − 1 ) = μ 2 min x 2 , S α + x 2 a/S Time-scale: t �→ St ( X S 2 ( St ) /S ) ⇒ ( x 2 ( t )) with � x 2 ( t ) a � ˙ x 2 ( t ) = λ 2 − μ 2 min ax 2 ( t ) , α + x 2 ( t ) a 46

Scaling the Maximum Rate a We freeze α and consider the process ( X S 2 ( t )) with Q -matrix: q ( x 2 , x 2 + 1 ) = λ 2 , � a x 2 a / S � q ( x 2 , x 2 − 1 ) = μ 2 min x 2 , S α + x 2 a/S Time-scale: t �→ St ( X S 2 ( St ) /S ) ⇒ ( x 2 ( t )) with � x 2 ( t ) a � ˙ x 2 ( t ) = λ 2 − μ 2 min ax 2 ( t ) , α + x 2 ( t ) a Fixed point: � � ρ 2 α max 1 , x 2 = 1 − ρ 2 a 46