[PPT] - Window Flow Control Systems with Random Service Alireza Shekaramiz PowerPoint Presentation

SLIDE 1

Window Flow Control Systems with Random Service

Alireza Shekaramiz Joint work with Prof. J¨

rg Liebeherr and Prof. Almut Burchard

April 6, 2016

1 / 20

SLIDE 2

Content

1

Introduction

2

Related work

3

State-of-the-art

4

Results: Stochastic service analysis of feedback systems

5

Results: Variable bit rate server with feedback system

6

Results: Markov-modulated On-Off server with feedback system

7

Numerical results

2 / 20

SLIDE 3

Feedback system

Feedback system:

Network departures arrivals ≤ window throttle delay

For the analysis we use network calculus methodology Network calculus has analyzed feedback systems under deterministic assumptions Open problem in network calculus Analysis of feedback systems with probabilistic assumptions

3 / 20

SLIDE 4

Related work

Performance bonds for flow control protocols1

Deterministic analysis
Min-plus algebra
Window flow control model

A min,+ system theory for constrained traffic regulation and dynamic service guarantees2

Deterministic analysis
Min-plus algebra
Window flow control model

TCP is max-plus linear3

Deterministic service process
Max-plus algebra
TCP Tahoe and TCP Reno
1R. Agrawal et al. “Performance bonds for flow control protocols”.

In: IEEE/ACM Transactions on Networking 7.3 (1999), pp. 310–323.

2C.-S Chang et al. “A min,+ system theory for constrained traffic regulation and dynamic service guarantees”.

In: IEEE/ACM Transactions on Networking 10.6 (2002), pp. 805–817.

3F. Baccelli and D. Hong. “TCP is max-plus linear and what it tells us on its throughput”.

In: ACM SIGCOMM 30.4 (2000), pp. 219–230. 4 / 20

SLIDE 5

Related work

TCP congestion avoidance4

Deterministic analysis
Min-plus algebra
Window flow control model
TCP Vegas and Fast TCP

Window flow control in stochastic network calculus5

Stochastic analysis
Min-plus algebra
Window flow control model
4M. Chen et al. “TCP congestion avoidance: A network calculus interpretation and performance

improvements”. In: IEEE INFOCOM. vol. 2. 2005, pp. 914–925.

5M. Beck and J. Schmitt. “Window flow control in stochastic network calculus - The general service case”.

In: ACM VALUETOOLS. Jan. 2016. 5 / 20

SLIDE 6

Bivariate network calculus

(f ∧ g) (s, t) = min{f (s, t), g(s, t)} (f ⊗ g) (s, t) = min

s≤τ≤t{f (s, τ) + g(τ, t)}

(f ⊗ g)(s, t) = (g ⊗ f )(s, t) (∧,⊗) operations form a non-commutative dioid over non-negative non-decreasing bivariate functions discrete-time domain (t = 0, 1, 2, ...) Sub-additive closure: f ∗ δ ∧ f ∧ f (2) ∧ f (3) ∧ . . . =

∞

n=0

f (n) where f (n+1) = f (n) ⊗ f for n ≥ 1, f (0) = δ, and f (1) = f

6 / 20

SLIDE 7

Moment-generating function network calculus6

Moment-generating function of a random variable X: MX(θ) = E

eθX

Moment-generating function of operations ⊗ and ⊘: Mf ⊗g(−θ, s, t) ≤

t

τ=s

Mf (−θ, s, τ)Mg(−θ, τ, t) Mf ⊘g(θ, s, t) ≤

s

τ=0

Mf (θ, τ, t)Mg(−θ, τ, s) For Pr

S(s, t) ≤ Sε(s, t)
≤ ε, statistical service bound

Sε(s, t) = max

θ>0

1 θ

log ε − log MS(−θ, s, t)
6M. Fidler. “An end-to-end probabilistic network calculus with moment generating functions”.

In: IEEE

IWQoS. 2006, pp. 261–270.

7 / 20

SLIDE 8

State-of-the-art: Window flow control

Network departures arrivals ≤ window throttle

A A0 S

δ+w min

D Swin

D0

A′ = min

A, D′

δ+w(s, t) =

w

s ≥ t , ∞ s < t D′ = D ⊗ δ+w = D + w A′ − D = min {A, D + w} − D ≤ D + w − D = w

8 / 20

SLIDE 9

State-of-the-art: Window flow control

Delay element represent feedback delay: δd(s, t) = δ(s, t − d) Equivalent feedback service: Swin =

S ⊗ δd ⊗ δ+w∗ ⊗ S

A A0 S

δ+w min

D Swin D0

≡

A D

δd

9 / 20

SLIDE 10

Results: Exact result

A A0 S

δ+w min

D Swin

δd

D0 Feedback system with w > 0, d ≥ 0 and with an additive service process S(s, t) =

t−1

k=s

ck ck’s are arbitrary sequence of non-negative random variables If feedback delay is one (d = 1), Swin(s, t) =

t−1

k=s

min {ck, w}

10 / 20

SLIDE 11

Results: Upper and lower bounds

For the equivalent service process Swin of a general feedback system with window size w > 0, and feedback delay d ≥ 0, we have Upper and lower bounds: S′

win(s, t) < Swin(s, t) < min

S(s, t), t−s

d

w

S′

win(s, t) is the equivalent service process of the feedback system

with window size w′ = w/d and feedback delay d′ = 1 The lower bound corresponds to the exact result

11 / 20

SLIDE 12

Results: Equivalent service

Feedback system with window size w > 0 and delay d ≥ 0: A A0 S

δ+w min

D Swin

δd

D0 Swin(s, t) = ⌈ t−s

d ⌉

n=0
min

Cn(s,t)

n

i=1
S(τi−1, τi − d)
+ S(τn, t)
+ nw
where Cn(s, t) is given as

Cn(s, t) =

s = τo ≤ · · · ≤ τn ≤ t

∀i = 0, . . . , n τi − τi−1 ≥ d
12 / 20

SLIDE 13

Results: Feedback system with VBR

Variable Bit Rate (VBR) server S(s, t) =

t−1

k=s

ck where ck’s are independent and identically distributed random variables For a feedback system with VBR server with window size w > 0 and delay d ≥ 0: MSwin(−θ, s, t) ≤

Mc(−θ)d + de−θw⌊ t−s

d ⌋

Mc(θ) is the moment-generating function of ck, Mc(θ) = E

eθck
13 / 20

SLIDE 14

Results: Feedback system with MMOO

Markov-modulated On-Off (MMOO) server operates in two states: ON (state 1): The server transmits a constant amount of P > 0 units of traffic per time slot, ck = P OFF (state 0): The server does not transmit, ck = 0 The MMOO server offers an additive service process S(s, t) =

t−1

k=s

ck For a feedback system with MMOO server with window size w > 0 and delay d ≥ 0, if p01 + p10 < 1: MSwin(−θ, s, t) ≤

m+(−θ)d + de−θw⌊ t−s

d ⌋

m+(θ) is the larger eigenvalue of the matrix L(θ) =

p00

p01 p10 p11 1 eθP

14 / 20

SLIDE 15

Numerical results: Statistical service bounds

Sε

win(s, t) = max θ>0

1 θ

log ε − log MSwin(−θ, s, t)
VBR server with exponential ck

MMOO server with p00 = 0.2, p11 = 0.9, P = 1.125 Mb

Time (ms) 10 20 30 40 50 Service (Mb) 5 10 15 20 Sε

win

Sε

upper

Sε

lower and Sε win for d = 1 ms

d = 20 ms w = 10 Mb d = 10 ms w = 5 Mb d = 400 ms w = 200 Mb d = 100 ms w = 50 Mb d = 1 ms w = 500 Kb d = 5 ms w = 2.5 Mb

Time (ms) 10 20 30 40 50 Service (Mb) 5 10 15 20 Sε

win

Upper bound Lower bound and Sε

win for d = 1 ms

d = 400 ms w = 200 Mb d = 100 ms w = 50 Mb d = 20 ms w = 10 Mb d = 10 ms w = 5 Mb d = 5 ms w = 2.5 Mb d = 1 ms w = 500 Kb

Average rate = 1 Gbps, time unit = 1 ms, w/d = 500 Mbps, ε = 10−6

15 / 20

SLIDE 16

Numerical results: Effective capacity

γSwin(−θ) = lim

t→∞ − 1

θt log MSwin(−θ, 0, t) VBR server with exponential ck MMOO server with p00 = 0.2, p11 = 0.9, P = 1.125 Mb

θ (×10−5) 0.2 0.4 0.6 0.8 1 Effective capacity (Mbps) 100 200 300 400 500 350 400 450 500 Lower bounds for γSwin(−θ) Lower bound Upper bound

d = 1 ms w = 500 Kb d = 5 ms w = 2.5 Mb d = 20 ms w = 10 Mb d = 10 ms w = 5 Mb d = 100 ms w = 50 Mb

θ (×10−5) 0.2 0.4 0.6 0.8 1 Effective capacity (Mbps) 100 200 300 400 500 Lower bounds of γSwin(−θ) Lower bound Upper bound

d = 10 ms w = 5 Mb d = 1 ms w = 500 Kb d = 100 ms w = 50 Mb d = 20 ms w = 10 Mb d = 5 ms w = 2.5 Mb

Average rate = 1 Gbps, w/d = 500 Mbps

16 / 20

SLIDE 17

Numerical results: Backlog and delay bounds

A(s, t) =

t−1

k=s

ak with exponential ak and average rate λ Backlog bound Delay bound

Arrival rate λ (Mbps) 50 100 150 200 250 300 350 400 Backlog bound (Mb) 5 10 15 20 25 30 35 40 ε = 10−9 ε = 10−6 ε = 10−3

Sim. ε = 10−6

w/d = 500 Mbps w/d = 100 Mbps Arrival rate λ (Mbps) 50 100 150 200 250 300 350 400 Delay bound (ms) 50 100 150 200 250 ε = 10−9 ε = 10−6 ε = 10−3

Sim. ε = 10−6

w/d = 100 Mbps w/d = 500 Mbps

Exponential VBR, time unit = 1 ms, feedback delay d = 1 ms

17 / 20

SLIDE 18

Conclusions

A A0 S

δ+w min

D Swin

δd

D0 Results: Exact results Upper and lower service bounds Equivalent service of the feedback system Bounds for a feedback system with VBR server Bounds for a feedback system with MMOO server Backlog and delay bounds

18 / 20

SLIDE 19

Technical Report - July 2016

A. Shekaramiz, J. Liebeherr, and A. Burchard. Window Flow

Control Systems with Random Service. arXiv:1507.04631, July 2015.

19 / 20

SLIDE 20

Thank you Q & A

20 / 20