Performance Analysis of Non-stationary Peer-assisted VoD ...garetto/slides/infocom2-slides.pdf · Performance Analysis of Non-stationary Peer-assisted VoD Systems Delia Ciullo1, Valentina

Performance Analysis of Non-stationary Peer-assistedVoD Systems

Delia Ciullo1, Valentina Martina1, Michele Garetto2, Emilio Leonardi1,Gianluca Torrisi3

1Politecnico di Torino

2Universita di Torino

3CNR - Instituto per le Applicazioni di Calcolo

March 26-th, 2012

E. Leonardi Performance of P2P-VOD systems

Introduction

In peer-assisted Video-on-Demand (VoD) systems:

users browse a catalog of available videos and asynchronously issuerequests to watch a given content;

content is divided into chunks that can be retrieved either from

the serversother peers currently retrieving the same video (leechers)peers storing the whole video (seeds);

chunks must be retrieved by peers almost in sequence to guaranteesmall play-out delays;

a minimum average download rate equal to the video playback ratemust be sustained to guarantee service continuity; the system(exploiting servers bandwidth when needed) is able to steadily meetthis constraint.


Introduction








Introduction








Introduction




the servers

other peers currently retrieving the same video (leechers)peers storing the whole video (seeds);




Introduction




the serversother peers currently retrieving the same video (leechers)

peers storing the whole video (seeds);




Introduction








Introduction








Introduction








Assumptions

Video is downloaded by each user at constant rate d , greater or equalto the playback rate dv ;

upload available bandwidth Ui of peer i is a random variable with aassigned distribution (Ui are i.i.d.);

users contribute their upload bandwidth to the video distribution aslong as they are in the system;

the arrival process of requests (and users) for a video is a (possiblynon-homogeneous) Poisson process with intensity λ(t);

user’s sojourn time is described by an arbitrary random variable Twith finite mean T and complementary cumulative distributionfunction GT (x).


Assumptions







Assumptions







Assumptions







Assumptions







Assumptions







Preliminary Observations

The number of active users N(t) follows a Poisson distribution withmean

N(t) =

∫ ∞

0λ(t − x)GT (x)dx ;

τd = L/d is the time needed to download the whole video, andT d =

∫ τd0 GT (x)dx is the average time spent by peers downloading

the video, taking into account premature abandonments;

Nd(t) is the number of downloading users with meanNd(t) =

∫ τd0 λ(t − x)GT (x)dx , and Nseed(t) the number of seeds

with mean Nseed(t) = N(t)− Nd(t);

we define the average system load as:

γ =dT d

U T.




N(t) =

∫ ∞









γ =dT d

U T.




N(t) =

∫ ∞









γ =dT d

U T.




N(t) =

∫ ∞









γ =dT d

U T.


Goal

Our goal is:

to characterize the bandwidth requested from the servers S (and itsaverage S);

we develop an approximate efficient and accurate fluid model tocompute S ;our approach is able to capture several stochastic effects related topeer churn, upload bandwidth heterogeneity, non-stationary trafficconditions;our methodology can be exploited to design efficient peer-assisted VoDsystems and optimal resource allocation strategies.

In [1] we obtain rigorous bounds for the sequential delivery scheme andasymptotic results as the number of users increases.

[1] D.Ciullo, V.Martina, M.Garetto, E.Leonardi, G.L.Torrisi, “Stochastic Analysis ofSelf-Sustainability in Peer-Assisted VoD Systems”, Session TS05, Thursday, h.8.30-10.00


Goal

Our goal is:






Goal

Our goal is:


we develop an approximate efficient and accurate fluid model tocompute S ;

our approach is able to capture several stochastic effects related topeer churn, upload bandwidth heterogeneity, non-stationary trafficconditions;our methodology can be exploited to design efficient peer-assisted VoDsystems and optimal resource allocation strategies.




Goal

Our goal is:


we develop an approximate efficient and accurate fluid model tocompute S ;our approach is able to capture several stochastic effects related topeer churn, upload bandwidth heterogeneity, non-stationary trafficconditions;

our methodology can be exploited to design efficient peer-assisted VoDsystems and optimal resource allocation strategies.




Goal

Our goal is:






Goal

Our goal is:






A simple Lower Bound

A simple universal lower bound to S(t) for any chunk distribution schemeis

S(t) ≤ max{0, dNd(t)− U N(t)}.

Intuition: The additional server bandwidth is given by users requested

bandwidth minus their total upload bandwidth.

Note that this trivial lower bound was already shown in: C. Huang, J. Li, and K. W.Ross, Can Internet Video-on-Demand Be Profitable? in ACM SIGCOMM, 2007.


Analysis

Let Sd be the aggregate bandwidth requested by the downloading users.The aggregate upload bandwidth offered by the seeds is

Sseed =

Nseed∑i=1

Ui .

The bandwidth requested from the servers is:

S , max{0,Sd − Sseed}

where Sd is the bandwidth demanded by downloading peers.


Analysis


Sseed =

Nseed∑i=1

Ui .





Analysis


Sseed =

Nseed∑i=1

Ui .





Analysis(2)

We define Sd(k) , (Sd(t) | Nd(t) = k)

Theorem

Sd(k) satisfies the following recursive equation:

Sd(k) =

{d k = 1d + max{0, Sd(k − 1)− Uk} k > 1


Analysis(2)

We define Sd(k) , (Sd(t) | Nd(t) = k)

Theorem

Sd(k) satisfies the following recursive equation:

Sd(k) =

{d k = 1d + max{0, Sd(k − 1)− Uk} k > 1


Analysis(3)

We characterize the distribution of the server bandwidth using asecond-order approximation;

we approximate the distribution of the quantity Sd(k − 1)− Uk (foreach k ≥ 2) with a normal distribution matching the first twomoments of this quantity;

we apply standard formulas of the truncated normal distribution toderive the first two moments of Sd(k) as a function of the first twomoments of Sd(k − 1);

a similar approximation is subsequently applied to take into accountthe effect of the seeds.


Analysis(3)






Analysis(3)






Analysis(3)






Analysis(3)






Swarm size effect

d = dv = 1, T = T d = τd

0.1

1

10

100

1 10 100 1000

Aver

age

serv

er b

and

wid

th

Average number of users

approx - U = 0.9sim - U = 0.9

sim lower bound - U = 0.9


Swarm size effect

d = dv = 1, T = T d = τd

0.1

1

10

100

1 10 100 1000

Aver

age

serv

er b

and

wid

th


approx - U = 1.2sim - U = 1.2

sim lower bound - U = 1.2


Download rate impact

U = 1.2, dv = 1, T = T d = τd

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

2.25

2.5

2.75

3

3.25

3.5

1 1.2 1.4 1.6 1.8 2 2.2

download rate, d

simapprox

sim lower bound


Extension to non sequential download

A common approach in P2P-VoD is to allow users to receive alsoout-of-sequence chunks of the video within a limited sliding windowof data starting from the point currently played;

for simplicity, instead of considering an actual sliding window, wedivide the video into a fixed number W of non-overlapping segmentsof size LW = L/W ;

we assume that, within a segment, chunk based out-of-sequencedistribution can be exploited, and we extend our approximate modelto deal with partially non-sequential chunk delivery.

















Impact of non-sequential delivery

0

1

2

3

4

5

6

0 10 20 30 40 50 60 70 80 90 100

Av

erag

e se

rver

ban

dw

idth


sim - sequentialsim - W = 32sim - W = 16

sim - W = 8sim - W = 4sim - W = 2

sim - lower bound 0

1

2

3

4

5

6

0 10 20 30 40 50 60 70 80 90 100


approx - sequentialapprox - W = 32approx - W = 16approx - W = 8approx - W = 4approx - W = 2approx - W = 1


Impact of non stationarity

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

12 18 24 6 12 18 24 6 0

200

400

600

800

1000

1200v

ideo

req

ues

t ra

te, λ

(t)

nu

mb

er o

f d

ow

nlo

ader

s /

seed

s

λ

downloadersseeds

0

10

20

30

40

50

60

70

12 18 24 6 12 18 24 6

aver

age

serv

er b

and

wid

th

time of day (hours)

sim - traceapprox

approx - λ = 1


Conclusions

We have proposed a computationally-efficient methodology toestimate the server bandwidth requested in non-stationary P2P-VoDsystems;

we have discovered several interesting properties:

the server bandwidth can be minimized by a proper selection of thedownload rate;the server bandwidth increases with the variation coefficient of the peerupload bandwidth;the gain achievable by non-sequential schemes over the simplesequential scheme depends critically on the size of the sliding windowand the number of downloading users;non-stationary systems are affected by a misalignment problembetween downloaders and seeds.


Conclusions



the server bandwidth can be minimized by a proper selection of thedownload rate;

the server bandwidth increases with the variation coefficient of the peerupload bandwidth;the gain achievable by non-sequential schemes over the simplesequential scheme depends critically on the size of the sliding windowand the number of downloading users;non-stationary systems are affected by a misalignment problembetween downloaders and seeds.


Conclusions



the server bandwidth can be minimized by a proper selection of thedownload rate;the server bandwidth increases with the variation coefficient of the peerupload bandwidth;

the gain achievable by non-sequential schemes over the simplesequential scheme depends critically on the size of the sliding windowand the number of downloading users;non-stationary systems are affected by a misalignment problembetween downloaders and seeds.


Conclusions



the server bandwidth can be minimized by a proper selection of thedownload rate;the server bandwidth increases with the variation coefficient of the peerupload bandwidth;the gain achievable by non-sequential schemes over the simplesequential scheme depends critically on the size of the sliding windowand the number of downloading users;

non-stationary systems are affected by a misalignment problembetween downloaders and seeds.


Conclusions



the server bandwidth can be minimized by a proper selection of thedownload rate;the server bandwidth increases with the variation coefficient of the peerupload bandwidth;the gain achievable by non-sequential schemes over the simplesequential scheme depends critically on the size of the sliding windowand the number of downloading users;non-stationary systems are affected by a misalignment problembetween downloaders and seeds.


Thank you!


Documents

Performance Analysis of Non-stationary Peer-assisted VoD ...garetto/slides/infocom2-slides.pdf · Performance Analysis of Non-stationary Peer-assisted VoD Systems Delia Ciullo1, Valentina