Processor sharing queues motivated by P2P networks · Processor sharing queues motivated by P2P networks Andres Ferragut Universidad ORT Uruguay Collaborators: F. Paganini, M. Jonckheere

Processor sharing queues motivated by P2Pnetworks

Andres Ferragut

Universidad ORT Uruguay

Collaborators: F. Paganini, M. Jonckheere

Stochastic Networks 2014

Agenda

Motivation

Queueing analysis of P2P systems

A general reversibility result for processor sharing

Fluid models and PDEs

Ongoing work

Andres Ferragut PS queues motivated by P2P Stochastic Networks 2014 2 / 69

Agenda

Motivation




Ongoing work


P2P systems

I A large proportion of Internettraffic is due to peer-to-peer (P2P)file exchange systems.

I Paradigm shift: replaces thetraditional client-server model.

I Why? Scalability.I The system leverages capacity

from participating peers.I As demand becomes large, so

does the available supply.


P2P systems

I A large proportion of Internettraffic is due to peer-to-peer (P2P)file exchange systems.

I Paradigm shift: replaces thetraditional client-server model.

I Why? Scalability.I The system leverages capacity

from participating peers.I As demand becomes large, so

does the available supply.


P2P systemsHow do they work?

I The file of interest is divided in small chunks.I Some terminology...

I Seeders: Own all the chunks, want to redistribute.I Leechers: Those who are still downloading, and also contribute.

I The chunks are exchanged according to some simple rules:


P2P systemsHow do they work?

I The file of interest is divided in small chunks.I Some terminology...

I Seeders: Own all the chunks, want to redistribute.I Leechers: Those who are still downloading, and also contribute.

I The chunks are exchanged according to some simple rules:


Homogeneous P2P networkA macroscopic view...

I Consider a swarm of peers around a single content.

I Peers arrive at a certain rate λ.

I Each contribute a fixed bandwidth µ in files/sec.

I x(t) := leecher population, y(t) := seeder population.

I Total uplink bandwidth:

R(x, y) = µ(x+ y)

How does leecher and seeder populations evolve?



λ

Leecher queue

x(t)

R(x, y)

Seeder queue

y(t)

γ(y)

Assumptions:

I The whole uplink bandwidth is allocated (efficiency).

I Each peer receives an equal part of the bandwidth(processor-sharing).



λ

Leecher queue

x(t)

R(x, y)

Seeder queue

y(t)

γ(y)

Assumptions:

I The whole uplink bandwidth is allocated (efficiency).

I Each peer receives an equal part of the bandwidth(processor-sharing).


Related work

Population dynamics:

I First queueing model: [Yang and De Veciana, 2004].Memoryless assumptions.

I Fluid model: [Qiu and Srikant, 2004], we build up on thesemodels.

I Other related models: [Clevenot and Nain, 2004],[Simatos et al., 2008], [Leskela et al., 2010].

Chunk exchange dynamics:

I [Kesidis et al., 2007, Massoulie and Vojnovic, 2008] analyzethe system as coupon exchange.

I State-space explosion makes the system difficult to analyze.

I [Hajek and Zhu, 2011, Zhu and Hajek, 2012] identifyproblems with chunk distribution on some extreme scenarios.


Related work

Population dynamics:

I First queueing model: [Yang and De Veciana, 2004].Memoryless assumptions.

I Fluid model: [Qiu and Srikant, 2004], we build up on thesemodels.

I Other related models: [Clevenot and Nain, 2004],[Simatos et al., 2008], [Leskela et al., 2010].

Chunk exchange dynamics:

I [Kesidis et al., 2007, Massoulie and Vojnovic, 2008] analyzethe system as coupon exchange.

I State-space explosion makes the system difficult to analyze.

I [Hajek and Zhu, 2011, Zhu and Hajek, 2012] identifyproblems with chunk distribution on some extreme scenarios.


Agenda

Motivation




Ongoing work


Queueing model[Yang and De Veciana, 2004, Qiu and Srikant, 2004]

If we assume job sizes are exponentially distributed we have:

λ

Leecher queue

x(t)

µ(x+ y)

Seeder queue

y(t)

γy

(x(t), y(t)) is a Markov processwith transitions:

λ

µ(x+ y)

γy

x

y


Queueing model[Yang and De Veciana, 2004, Qiu and Srikant, 2004]

I Stability:I Since R(x, y) = µ(x+ y) scales with x, the system is always

stable (ergodic).

I Invariant distribution:I No product form. No explicit expression.

[Yang and De Veciana, 2004] performs numerical studies.

I Key issue: memoryless model, sensitive to the job sizedistribution.


Queueing modelWhat we would like to do:

I Build a tractable model for general file sizes (in particular,deterministic) based on the processor sharing feature.

Approaches:

I Use the insensitivity of PS networks.

I ...the queueing approach.

I Fluid models.

I ...large network scalings, asymptotic results, but more general.


Queueing modelWhat we would like to do:

I Build a tractable model for general file sizes (in particular,deterministic) based on the processor sharing feature.

Approaches:

I Use the insensitivity of PS networks.I ...the queueing approach.

I Fluid models.I ...large network scalings, asymptotic results, but more general.


Fixing the number of seeders

Consider the simpler problem, with fixed number of seeders y0:

λ

Leecher queue

x(t)

µ(x+ y0)

y0

qx,x+1 = λ,

qx,x−1 = µ(x+ y0), x > 0.

I Restrictive hypothesis? Not so much...I In practice the majority of leechers depart immediately.I It is a first step towards a slowly varying seeders model.

I Now is a standard birth-death process, easy to solve...I ...and PS, so insensitive to the job size.


Fixing the number of seeders

Consider the simpler problem, with fixed number of seeders y0:

λ

Leecher queue

x(t)

µ(x+ y0)

y0

qx,x+1 = λ,

qx,x−1 = µ(x+ y0), x > 0.

I Restrictive hypothesis? Not so much...I In practice the majority of leechers depart immediately.I It is a first step towards a slowly varying seeders model.

I Now is a standard birth-death process, easy to solve...I ...and PS, so insensitive to the job size.


Invariant distribution

Proposition

Denote ρ = λ/µ, the invariant distribution for the system is givenby:

π(n) = Kρn+y0

(n+ y0)!for n > 0,

with K =[∑∞

m=y0ρm

m!

]−1.

Remarks:

I The system is stable for any λ, µ and y0 (leecher contributionscales with congestion).

I π is a “shifted and rescaled” Poisson distribution.

I π’s pgf G(z) := Eπ[zX ], system averages, can be obtainedeasily.


Invariant distribution: two scenarios

0 20 40 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

k

π

0 20 40 600

0.05

0.1

0.15

0.2

k

π

0 20 40 600

0.01

0.02

0.03

0.04

0.05

0.06

0.07

k

π

ρ = 10, y0 = 20 ρ = 20, y0 = 20 ρ = 40, y0 = 20

←−ρ < y0

Seeders can cope with theload

M/G/1-like system

−→ρ > y0

Leecher contribution isessential

M/G/∞-like system


Invariant distribution: two scenarios

0 20 40 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

k

π

0 20 40 600

0.05

0.1

0.15

0.2

k

π

0 20 40 600

0.01

0.02

0.03

0.04

0.05

0.06

0.07

k

π

ρ = 10, y0 = 20 ρ = 20, y0 = 20 ρ = 40, y0 = 20

←−ρ < y0

Seeders can cope with theload

M/G/1-like system

−→ρ > y0

Leecher contribution isessential

M/G/∞-like system


Considering general file-sizesStoring remaining job sizes in the state

I Assume now jobs are iid with complementary CDF H(σ).

I State descriptor:

0 σ1 σ2 σ3 σ4

Φt =∑i

δσi(t)

I Φt is a measure-valued Markov process that stores theresidual service times σi(t) for each job in the system.

I Its equilibrium is well known!


Stationary distribution for M/G PS queues[Zachary, 2007]

In equilibrium a PS queues behaves as follows:

I Choose x, the number of customers with the invariantdistribution π of the exponential case.

I Given x = n, the residual service times are iid withdistribution:

H(σ) =

∫ ∞σ

H(s) ds.

called the residual lifetime distribution.

I This construction characterizes the point process Φt inequilibrium.

I We can compute its Laplace transform:

LΦ[f ] = E[e−

∫f(σ)Φ(dσ)

]= G

(∫ ∞0

e−f(σ) H(dσ)

)


Stationary distribution for M/G PS queues[Zachary, 2007]

In equilibrium a PS queues behaves as follows:

I Choose x, the number of customers with the invariantdistribution π of the exponential case.

I Given x = n, the residual service times are iid withdistribution:

H(σ) =

∫ ∞σ

H(s) ds.

called the residual lifetime distribution.

I This construction characterizes the point process Φt inequilibrium.

I We can compute its Laplace transform:

LΦ[f ] = E[e−

∫f(σ)Φ(dσ)

]= G

(∫ ∞0

e−f(σ) H(dσ)

)


Large network asymptotic

I Most P2P systems have a large number of peers.

I We would like to do asymptotic analysis.

I Choose L > 0 a scaling parameter (we will make L→∞)

I Consider a system with:

I Scaled arrival rates: λ 7→ Lλ.I Scaled number of seeders: y0 7→ Ly0.I Note that ρ = λ

µ 7→ Lρ.

I We distinguish the cases ρ < y0, ρ > y0.


The seeder sustained case (ρ < y0)

I In this case, the seeders can cope with the demand alone.

I In the scaled system, this contribution becomes dominant.

Theorem

If ρ < y0, the equilibrium distribution of the scaled system πLconverges in law to the equilibrium distribution of anM/G/1− PS queue with load ν = ρ

y0< 1.

Proof sketch:

I Write the pgf GL(z) of the scaled system distribution.

I Use large-deviations (Bahadur-Rao) for Poisson-likeprobability terms to solve the limit.

I The limit is the pgf of the geometric distribution.





Theorem


y0< 1.

Proof sketch:








Theorem


y0< 1.

Proof sketch:





The globally sustained case (ρ > y0)

I This case is more involved. The contribution of leechers isessential.

I As the system grows, so does the number of leechers onaverage.

I In order to obtain a non-trivial limit, we perform a law of largenumbers scaling of the state:

ΦL =1

L

xL∑i=1

δσLi,

where xL is the number of jobs in the scaled system.



We have the following law of large numbers result:

Theorem

If ρ > y0, the equilibrium distribution of the scaled system ΦL

converges in law to the deterministic measure (ρ− y0)H(dσ) onR+.

Proof sketch:

I Use the Laplace functional of the process and Taylorexpansions of GL(z).

I Prove the pointwise convergence of LΦL [f ] to that of thedeterministic measure.



I If we specialize Theorem 2 for deterministic job sizes (i.e. afile in P2P) we get:

I CCDF H(σ) = 1[0,1)(σ)I Therefore H(σ) = 1− σ ∀σ ∈ [0, 1],.

I The rescaled process then converges to a measure with totalmass (ρ− y0) uniformly distributed in [0, 1].

I Remark: the uniform distribution is a consequence ofdeterministic job sizes and the PS nature of the system.


A slowly varying model

I We now use the fixed-seeder case to analyze a slowly varyingseeder population.

I Model:

λ

Leecher queue

x(t)µ(x+ y)

α

1− αSeeder queue

y(t)γy

Fast time-scale Slow time-scale

Idea: For small α, γ, the system decouples in two timescales.




I Model:

λ

Leecher queue

x(t)µ(x+ y)

α

1− α

Seeder queue

y(t)γy






I Model:

λ

Leecher queue

x(t)µ(x+ y)

α

1− αSeeder queue

y(t)γy






I Model:

λ

Leecher queue

x(t)µ(x+ y)

α

1− αSeeder queue

y(t)γy




Time scale separationFast time-scale

λ

Leecher queue

x(t)µ(x+ y) λ

αµ(x+ y)

(1− α)µ(x+ y)

γy

x

y

I As α→ 0 the timescale for y becomes slower ⇒ constant y.

I Define ρx = λ/µ the load of the first queue.

I The invariant distribution for the number of leechers becomes:

π0(x | y) = K(y)ρx+yx

(x+ y)!


Time scale separationSlow time-scale

αλ

Seeder queue

y(t)γy λ

αµ(x+ y)

(1− α)µ(x+ y)

γy

x

y

I As α→ 0, x evolves faster, and the arrival rate to the seederqueue goes to λ.

I Define ρy = λα/γ the load of the seeder queue.

I The seeder queue becomes a standard M/G/∞.

π0(y) = e−ρy(ρy)

y

y!


Quasi-stationarity

Theorem (QS decomposition)

Let πα(x, y) be the invariant distribution for the system. Letα, γ → 0 such that:

λα

γ→ ρy

Then the invariant distribution of the system converges to:

π0(x, y) = π0(x | y)π0(y)

Proof sketch:

I Prove that the family of distributions {πα}α on N×N is tight.

I Take limit along subsequences to find the equilibriumdistributions in both timescales.

I Prove that the limit is unique.


Quasi-stationarityInsensitivity of the limit

Remarks:

I The above proof works in the Markovian setting (exponentialdistributions).

I Note however that both queues decouple to insensitiveprocessor sharing queues.

I Simulations validate that the limit is insensitive.

I Conjecture: The equilibrium distribution for the M/G systemalso decouples.


Agenda

Motivation




Ongoing work


Reversible descriptions of PS queues

I We now analyze a nice reversibility property of PS queues.

I It is well known that reversibility and insensitivity are deeplylinked [Zachary, 2007]...

I The following analysis incorporates the general file size intothe reversibility framework.

I For ease of exposition, we will focus on M/G/∞, but this isnot required.


Two state descriptorsThe residual service time

I Let {σi(t)} denote the residual service times of customerspresent in the system.

I Take (as before), the point mass measure:

Φt =∑i

δσi(t)

0 σ1(t) σ2(t) σ3(t) σ4(t)


Two state descriptorsThe residual service time dynamics

0 σ1 σ2 σ3 σ4

σ0

Drift towards 0: µdt

Mass arrives as λH(dσ)dt

I Φt is a measure valued Markov process[Robert, 2003, Dawson, 1993].

I We can characterize its infinitesimal generator.


Two state descriptorsThe residual service time dynamics

0 σ1 σ2 σ3 σ4σ0

Drift towards 0: µdt

Mass arrives as λH(dσ)dt

I Φt is a measure valued Markov process[Robert, 2003, Dawson, 1993].

I We can characterize its infinitesimal generator.


Two state descriptorsThe attained service time

I Let {ai(t)} denote the attained service times of customerspresent in the system.

I Take now the point mass measure:

Φt =∑i

δai(t)

0 a1(t) a2(t) a3(t) a4(t)

I Introduce the hazard-rate function:

η(σ) = −H′(σ)

H(σ)

“probability” that service ends immediately after σ units,given survival up to σ.


Two state descriptorsThe attained service time dynamics

0 a1 a2 a3 a4

Drift towards the right: µdt

Mass arrives at 0 as λdt Mass departs at rate µη(ai)

I Φt is also a measure valued Markov process, we cancharacterize its infinitesimal generator.


Two state descriptorsThe attained service time dynamics

0 a1 a2 a4

Drift towards the right: µdt

Mass arrives at 0 as λdt Mass departs at rate µη(ai)

I Φt is also a measure valued Markov process, we cancharacterize its infinitesimal generator.


Infinitesimal generatorsLaplace functionals

I We want to characterize the transition semigroup of theMarkov process.

I Consider the functionals:

Ff (Φ) = e−∫f(σ)Φ(dσ)

for a suitable family of functions f : R+ → R.

I The infinitesimal generator of the semigroup is characterizedby its actions on Ff (Φ), i.e.:

Q(Ff (Φ)) = limh→0

1

hE [Ff (Φt+dt)− Ff (Φt) | Φt = Φ]


Infinitesimal generatorsThe two dynamics

If we compute the above for both processes we get:

I Remaining service:

Q(Ff (Φ)) =

[λ

∫ ∞0

(e−f(σ) − 1

)H(dσ) + µ

∫ ∞0

f ′(σ)Φ(dσ)

]Ff (Φ)

I Attained service:

Q(Fg(Φ)) =

[λ(e−g(0) − 1

)− µ

∫ ∞0

g′(σ) − η(σ)(eg(σ) − 1

)Φ(dσ)

]Fg(Φ)



In this M/G/∞ scenario, the equilibrium distribution π is aPoisson point process:

I Total mass ρ = λ/µ.

I Distributed on the real line as H(dσ).

In the above notation:

Φ ∼ π ⇔ Eπ[Ff (Φ)] = e−ρ∫

(e−f(σ)−1)H(dσ)

It is easy to check that Eπ[Q(Ff (Φ))] = Eπ[Q(Ff (Φ))] = 0.



In this M/G/∞ scenario, the equilibrium distribution π is aPoisson point process:

I Total mass ρ = λ/µ.

I Distributed on the real line as H(dσ).

In the above notation:

Φ ∼ π ⇔ Eπ[Ff (Φ)] = e−ρ∫

(e−f(σ)−1)H(dσ)

It is easy to check that Eπ[Q(Ff (Φ))] = Eπ[Q(Ff (Φ))] = 0.


Reversibility

Theorem

The operators Q and Q are adjoint with respect to the innerproduct defined by the invariant distribution, i.e.

Eπ [Q(Ff (Φ))Fg(Φ)] = Eπ

[Ff (Φ)Q(Fg(Φ))

]

I This means the two processes are reversed versions of eachother.

Proof sketch:

I Compute the inner products, conditioning on the number ofpoints and clever use of integration by parts and properties ofH, H and η.


Reversibility

Theorem

The operators Q and Q are adjoint with respect to the innerproduct defined by the invariant distribution, i.e.

Eπ [Q(Ff (Φ))Fg(Φ)] = Eπ

[Ff (Φ)Q(Fg(Φ))

]

I This means the two processes are reversed versions of eachother.

Proof sketch:

I Compute the inner products, conditioning on the number ofpoints and clever use of integration by parts and properties ofH, H and η.


Some open questions...

I The above generalizes nicely to:I General service rates µ = µ(Φ).I Parallel systems with state dependent service rates (with

balance conditions)

I Routing between stations is harder to accommodate in thisframework.

I ¿Can we find other processes which are reversed of oneanother for other service disciplines?

I A really difficult question!


Agenda

Motivation




Ongoing work


A fluid model for the P2P system

I We would like to extend the analysis to include the transientdynamics.

I And also to more general situations (i.e. varying number ofseeders).

I The queueing model becomes hard to analyze.

I We turn our attention to fluid models.


Fluid model for Processor Sharing systems

I Let F (t, σ) denote the number of jobs with remaining workgreater than σ.

I In particular F (t, 0) = x(t), the total number of jobs.

I Note that F (t, σ) is the complementary CDF of the measureΦt.

I We now treat F as a fluid variable.

0 0.2 0.4 0.6 0.8 1 1.20

5

10

15

σ

F(t

,σ)

Point process Φ

CCDF F(t,σ)

0 0.2 0.4 0.6 0.8 1 1.20

5

10

15

σ

F(t

,σ)


Fluid model for the Processor Sharing systemsThe PDE approach

I As before:I Jobs arrive at rate λ, with job distribution CCDF H(σ).I They are served at a common rate r (possibly dependent of

the state).

I Evolution equation:

F (t+ dt, σ) = λH(σ)dt︸︷︷︸arrivals

+ [F (t, σ + rdt)]︸︷︷︸served but stay >σ

I Rearranging terms, in the limit dt→ 0:

∂F

∂t= λH(σ) + r

∂F

∂σ

σ

λH(σ)dt

rdtx(t)

Departing customers F (t, σ)




the state).





∂F

∂t= λH(σ) + r

∂F

∂σ

σ

λH(σ)dt

rdtx(t)





the state).





∂F

∂t= λH(σ) + r

∂F

∂σ

σ

λH(σ)dt

rdtx(t)



The case of bandwidth sharing networks

I Jobs arrive into a networkwith predefined routes.

I Arrival rate λi on route i.

I General file sizes, CCDFH(σ) and mean 1/µ.

λ2

λ1

λ3

Link capacities cl

Remark: Service rates must be assigned so they do not exceed cl(link capacity): ∑

i∈lϕi 6 cl


The case of bandwidth sharing networksα−fair resource allocation

I Let xi denote the number of ongoing connection in route i.

I Assign ϕi according to:

Problem (Cong. control [Kelly et al., 1998, Srikant, 2004])

maxϕ

∑i

xiUi

(ϕixi

)subject to: ∑

i∈lϕi 6 cl

I Taking Ui(z) = z1−α

1−α we have α−fairness.

I Can be decentralized, α = 1 corresponds to proportionalfairness.

I TCP Congestion control works like this.


The natural stability condition

I Define ρi = λi/µi the load of each route.

I The natural stability condition for the system is:∑i∈l

ρi < cl ∀l

I For exponentially distributed file sizes,[de Veciana et al., 1999, Bonald and Massoulie, 2001] it iswell known that the system is stable.

I Several works analyzed the case of general file sizes[Massoulie, 2007]. It was conjectured that the result extends[Walton and Mandjes, 2011].

I What can we say using the PDE model?


The natural stability condition

I Define ρi = λi/µi the load of each route.

I The natural stability condition for the system is:∑i∈l

ρi < cl ∀l

I For exponentially distributed file sizes,[de Veciana et al., 1999, Bonald and Massoulie, 2001] it iswell known that the system is stable.

I Several works analyzed the case of general file sizes[Massoulie, 2007]. It was conjectured that the result extends[Walton and Mandjes, 2011].

I What can we say using the PDE model?


Stability of bandwidth sharing networksThe general file size case

I Let Fi(t, σ) denote the CCDF of remaining jobs in route i.

I Dynamics:∂Fi∂t

= λiH(σ) +

(ϕixi

)∂Fi∂σ

I xi(t) := Fi(t, 0).

I {ϕi} solves the network problem for current number of jobs x.


Stability of bandwidth sharing networksThe general file size case

Theorem ([Paganini et al., 2012])

If job sizes H(σ) have finite p > 1 moment, then the abovedynamics are globally asymptotically stable and converge to F ≡ 0in finite time.

Proof sketch:

I Based on the Lyapunov functional:

L =∑i

1

ραi

∫ ∞0

[Fi(t, σ)]α+1wi(σ)dσ

where wi(σ) is a space-weight appropriately chosen to ensureL 6 0.

I This result enables to prove stability for the stochastic systemin very general conditions (cf. Lee and Williams)


Back to P2P...

I Let’s go back to the P2P system with fixed seeders:

λ

Leecher queue

x(t)

µ(x+ y0)

y0 Memoryless fluid model:

x = λ− µ(x+ y0)

PDE model:

∂F

∂t= λH(σ) + µ

(x+ y0

x

)∂F

∂σ.

I Remark: If H(σ) = e−σ, separation of variables gives theODE model.


Back to P2P...

I Let’s go back to the P2P system with fixed seeders:

λ

Leecher queue

x(t)

µ(x+ y0)

y0 Memoryless fluid model:

x = λ− µ(x+ y0)

PDE model:

∂F

∂t= λH(σ) + µ

(x+ y0

x

)∂F

∂σ.

I Remark: If H(σ) = e−σ, separation of variables gives theODE model.


Equilibrium analysis

I Setting ∂F/∂t = 0 and integrating in the positive real line wecan deduce the equilibrium of the system.

I In the globally sustained case (ρ > y0) we get:

x∗ = ρ− y0

F ∗(σ) = (ρ− y0)

∫ ∞σ

H(s) ds = (ρ− y0)H(σ),

I Remark: The equilibrium distribution predicted by the PDEmodel corresponds to the earlier deterministic measure-valuedlimit.


Stability of the PDE model

Theorem

The equilibrium of the PDE dynamics with deterministic job sizesand processor sharing discipline is locally asymptotically stable.

Idea: linearize the PDE around equilibrium and work in the Laplace(frequency) domain.


Stability of the PDE model

Proof sketch:

I The system can be shown to have the following feedbackstructure:

G1(s)

G2(s)

n x−

G1(s) =1− e−τs

τs(PDE transport)

G2(s) = y0/ρ (seeders fbk.)

τ = 1/r∗ (avg. download time).

I Key point: the loop has small gain ‖G1G2‖∞ < 1.


Using frequency-domain techniques

I Add a noise source for random arrivals: n(t) ≈ white noisewith power λ.

I We have the following feedback structure:

H1(s)n1

G1(s)

G2(s)

x−

I The noise is “filtered” by the input-output n 7→ x transferfunction ⇒ variability estimation.


Transient analysis

I Using the PDE we can perform transient analysis.

I Example: assume at some point in time, the input is turnedoff. How long takes to empty the system?

T =1

µ

∫ ∞0

F (0, σ)

F (0, σ) + y0dσ.

I In particular for deterministic file sizes:

Tb 61

µ

x0

x0 + y06

1

µ

Scalable, bound independent of x0!


Transient analysis



T =1

µ

∫ ∞0

F (0, σ)

F (0, σ) + y0dσ.


Tb 61

µ

x0

x0 + y06

1

µ



Transient analysis



T =1

µ

∫ ∞0

F (0, σ)

F (0, σ) + y0dσ.


Tb 61

µ

x0

x0 + y06

1

µ



Transient analysisSimulation results

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

20

40

60

80

100

120

x0/y

0

Tim

e (

min

)

Time to empty the system

ODE model

PDE model

Simulations


Endogenously generated seeders

I Let’s now incorporate seeder variability:

λ

Leecher queue

x(t)

µ(x+ y)

Seeder queue

y(t)

γy

Memoryless fluid model:[Qiu and Srikant, 2004]

x = λ− µ(x+ y)

y = µ(x+ y)− γy

PDE model:

∂F

∂t= λH(σ) + µ

(x+ y

x

)∂F

∂σ

y = − µ(x+ y)

x

∂F

∂σ

∣∣∣∣σ=0

− γy.


The case of variable seeders: stability

I The feedback loop is now:

G1(s)

G2(s)

n x− G1(s) =

1− e−τs

τs(PDE transport)

G2(s) =s+ µ

s+ γ(seeders fbk.)

τ = 1/r∗ (avg. download time).I The small-gain argument still holds for the lead-lag G2 in the

leecher sustained case (µ > γ).

I Variability is studied analogously. Remark: cannot be donewith the queueing model.


Variable seeders: simulation example.

I By numerically integrating the transfer functions we canpredict variability.

I Example:I λ = 1.8 arr/min.I 1/γ ≈ 18 min.I 1/µ ≈ 1h.

0 2 4 6 8 10 12 14 160

20

40

60

80

100

120

Time (h)

Popula

tion s

ize

Leechers (sim)

Leechers (PDE)

Seeders (sim)

Seeders (PDE)


Agenda

Motivation




Ongoing work


Heterogeneous uplink bandwidths

I Typically peers contribute different rates to the network.

I Multi-class model: for each class i with uplink µi

∂Fi∂t

= λiH(σ) + ri(x, y0)∂F

∂σ

I Key point: how are resources allocated?

I A candidate:

ri(x, y) = µi︸︷︷︸Leecher reciprocity

+µ0y0∑i xi︸︷︷︸

Seeders do PS


Heterogeneous uplink bandwidths

I Typically peers contribute different rates to the network.

I Multi-class model: for each class i with uplink µi

∂Fi∂t

= λiH(σ) + ri(x, y0)∂F

∂σ

I Key point: how are resources allocated?

I A candidate:

ri(x, y) = µi︸︷︷︸Leecher reciprocity

+µ0y0∑i xi︸︷︷︸

Seeders do PS


Heterogeneous uplink bandwidthsStability via monotone systems

I Stability results for this model are harder to obtain.

I A key point is to use the theory of monotone dynamicalsystems.

I In fact, the dynamics in F (t, σ) is order preserving (for thenatural partial order between functions).

I This can be a very powerful tool also to go from ODE modelsto PDE models (recall bandwidth-sharing).


From queueing to fluid limits

I How does the stochastic and fluid model relate?

I We have seen that the equilibrium of the PDE modelcoincides with the scaled equilibrium of the queueing system.

I Idea: derive a fluid limit by scaling the measure-valued stateΦt and prove that the limit satisfies the PDE.

I Key points: tightness, martingale decomposition usingQ(Ff (Φt)). Also based on ideas from[Gromoll et al., 2002, Gromoll and Williams, 2009]. Ongoingwork...


Take home messages

I About P2P systems...I P2P systems give rise to interesting queueing networks.I General job size distributions have to be incorporated in the

analysis.I This can be done using the insensitivity of PS queues.

I About reversibility...I General PS systems have two alternative state descriptors.I They show a nice reversibility property between the associated

processes.

I About fluid limits...I PDE models are a powerful tool to analyze fluid PS networks.I We showed some successful applications to bandwidth sharing

and P2P.


Take home messages




processes.


and P2P.


Take home messages




processes.


and P2P.


Thank you!

¿?

[email protected]

References I

Bonald, T. and Massoulie, L. (2001).

Impact of Fairness on Internet Performance.

In ACM SIGMETRICS/Performance pp. 82–91,.

Clevenot, F. and Nain, P. (2004).

A simple model for the analysis of SQUIRREL.

In Proc. of IEEE Infocom, Hong Kong.

Dawson, D. A. (1993).

Measure valued Markov Processes notes.

Ecole D’Ete de Probabilites de Saint-Flour, France.

de Veciana, G., Lee, T.-J. and Konstantopoulos, T. (1999).

Stability and Performance Analysis of Networks Supporting Services withRate Control - Could the Internet Be Unstable?

In IEEE Infocom pp. 802–810,.


References II

Gromoll, H., Puha, A. and Williams, R. (2002).

The fluid limit of a heavily loaded processor sharing queue.

Annals of Applied Probability 12, 797–859.

Gromoll, H. and Williams, R. (2009).

Fluid limits for networks with bandwidth sharing and general documentsize distributions.


Hajek, B. and Zhu, J. (2011).

The missing piece syndrome in peer-to-peer communication.

Stochastic Systems 1, 246–273.

Kelly, F., Maulloo, A. and Tan, D. (1998).

Rate control in communication networks: shadow prices, proportionalfairness and stability.

Journal of the Operational Research Society 39, 237–252.


References III

Kesidis, G., Konstantopoulos, T. and Sousi, P. (2007).

Modeling file-sharing with BitTorrent-like incentives.

In Proc. of IEEE Int. Conf. on Acoustics, Speech and SignalProcessing, Honolulu, HI.

Leskela, L., Robert, P. and Simatos, F. (2010).

Interacting branching processes and linear file-sharing networks.

Adv. in Applied Probability 42, 834–854.

Massoulie, L. (2007).

Structural properties of proportional fairness: stability and insensitivity.


Massoulie, L. and Vojnovic, M. (2008).

Coupon replication systems.

IEEE/ ACM Transactions on Networking 16, 603–616.


References IV

Paganini, F., Tang, K., Ferragut, A. and Andrew, L. (2012).

Stability of networks under general file size distribution and alpha fairbandwidth allocation.

IEEE Trans. on Automatic Control 57, 579–591.

Qiu, D. and Srikant, R. (2004).

Modeling and performance analysis of BitTorrent-like peer-to-peernetworks.

ACM SIGCOMM Computer Communication Review 34, 367–378.

Robert, P. (2003).

Stochastic Networks and Queues.

Stochastic Modelling and Applied Probability Series, Springer-Verlag, NewYork.

Simatos, F., Robert, P. and Guillemin, F. (2008).

Analysis of a Queueing System for Modeling a File Sharing Principle.

In Proc. of ACM Sigmetrics, Annapolis, MD.


References V

Srikant, R. (2004).

The Mathematics of Internet Congestion Control.

Birkhauser, Boston, MA.

Walton, N. and Mandjes, M. (2011).

A stability conjecture on bandwidth sharing networks.

Queueing Systems 68, 237–250.

Yang, X. and De Veciana, G. (2004).

Service capacity of peer-to-peer networks.

In Proc. of IEEE Infocom, Hong Kong.

Zachary, S. (2007).

A note on insensitivity in stochastic networks.

Journal of Applied Probability 44, 238–248.

Zhu, J. and Hajek, B. (2012).

Stability of a peer-to-peer communication system.

IEEE Trans. on Information Theory 58, 4693–4713.


Documents

Processor sharing queues motivated by P2P networks · Processor sharing queues motivated by P2P networks Andres Ferragut Universidad ORT Uruguay Collaborators: F. Paganini, M. Jonckheere