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Regenerative estimator of the overflow probability in a tandem network

Irina Dyudenko, Evsey Morozov, Michele Pagano Institute of Applied Mathematical Research, Karelian Research Center and Petrozavodsk University

• Dept. of Information Engineering, University of Pisa, Italy

Rennes – RESIM 2008 September 2008

Outlines

2

☞ Network scenario ➳ Quality of Service (QoS) ➳ Bandwidth allocation techniques ☞ Effective Bandwidth (EB) ➳ Definition and physical meaning ➳ Scaled Cumulant Generating Function (SCGF) ☞ Effective Bandwidth estimators ➳ Standard estimator: the batch-mean approach ➳ Refined estimator for input traffic with a regenerative structure ☞ Performance Comparison ➳ Tandem network with 2 queues and deterministic service rate ➳ Variance of the estimator ☞ Conclusions

Regenerative estimator of the overflow probability in a tandem network

RESIM 2008 – September 2008

Michele Pagano

Network scenario

3

☞ Evolution of network services and architectures ➳ Quality of Service (QoS) requirements ➳ Packet loss (usually approximated through the overflow probability) as a typical QoS guarantee ➳ Traffic Engineering ☞ Necessity of dynamic control mechanisms ➳ Bandwidth allocation techniques ➳ Admission Control ☞ The notion of effective bandwidth has emerged as a powerful metric to quantify the amount of

resources needed by connections in order to guarantee a required QoS level

☞ Theorethical background: Large Deviation Theory ➳ For a wide class of buffered systems the overflow probability decreases exponentially fast as buffer

size increases

➳ The rate-function allows to calculate the required exponent ☞ Open issue: estimation of the effective bandwidth from traffic measurements

Regenerative estimator of the overflow probability in a tandem network

RESIM 2008 – September 2008

Michele Pagano

Workload process in a single server queue

4

C Xn Wn ☞ Stationarity condition: EXn < C ☞ Workload process: Wn = sup

u≤n

u

• i=1

Xi − C · u

• ☞ Under mild assumptions, stationary workload Wn ⇒ W exists and satisfies a Large Deviation

Principle (LDP)

P (W > b) ≈ e−δ b

Regenerative estimator of the overflow probability in a tandem network

RESIM 2008 – September 2008

Michele Pagano

Large Deviation Principle for the workload process

5

lim

x→∞

1 x log P(W > x) = −δ

where

☞ the exponent is defined as δ = δ(C) = sup{θ > 0 : Λ(θ) < Cθ} ☞ the quantity Λ(θ) = lim

n→∞

1 n log Eeθ Pn

i=1 Xi

∆

= lim

n→∞ Λn(θ)

is known as Limiting Scaled Cumulant Generating Function (LSCGF) In case of an i.i.d. sequence Xn, the previous limit leads to

Λ(θ) = log EeθX

Regenerative estimator of the overflow probability in a tandem network

RESIM 2008 – September 2008

Michele Pagano

Effective Bandwidth (EB)

6

☞ In real networks, it is reasonable to assume that the buffer size b is fixed, while the capacity C can be

varied

☞ Goal: determine the minimum service capacity CΓ which guarantees an overflow probability ≤ Γ CΓ = min(C : e−δ(C)b ≤ Γ) ☞ Hence, we obtain δ(CΓ) = θ∗ = −log(Γ) b

and, since

δ = δ(C) = sup{θ > 0 : Λ(θ) < Cθ}

we can express C as a function of θ∗ (provided that Λ(θ) is known):

CΓ = C(Γ, b) = Λ(θ∗) θ∗ ☞ Given the QoS parameter Γ and the buffer size b, the value CΓ = C(Γ, b) is called the effective

bandwidth of the incoming traffic

Regenerative estimator of the overflow probability in a tandem network

RESIM 2008 – September 2008

Michele Pagano

EB Estimation: batch-mean approach

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☞ If the samples Xi were i.i.d., then we could estimate the LSCGF for θ = θ∗ by means of the sample

mean estimator

ˆ Λn(θ∗) = log 1 n

n

• i=1

eθ∗Xi → log E

• eθ∗X

☞ Aggregation into blocks of size B X1 X1 X2 X2 X3 X3 XB XB XB+1 XB+1 XB+2 XB+2 X2B X2B X(k−1)B+1 X(k−1)B+1 XkB XkB ˆ X1 =

B

• i=1

Xi ˆ X2 =

2B

• i=B+1

Xi ˆ Xk =

kB

• i=(k−1)B+1

Xi ☞ If B is large enough, the dependence between blocks should be negligible, and ˆ Xi constitute

(approximately) an i.i.d. sequence

Regenerative estimator of the overflow probability in a tandem network

RESIM 2008 – September 2008

Michele Pagano

EB Estimation: batch-mean approach

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X1 X2 X3 XB XB+1XB+2 X2B X(k−1)B+1 XkB ˆ X1 =

B

• i=1

Xi ˆ X2 =

2B

• i=B+1

Xi ˆ Xk =

kB

• i=(k−1)B+1

Xi ☞ By using n = kB observations, we obtain a modified expression of the LSCGF Λ(θ∗, B) = lim

n→∞

1 n log Eeθ∗ Pk

i=1 ˆ

Xi = lim n→∞

k n log Eeθ∗ ˆ

X = 1

B log Eeθ∗ ˆ

X

☞ Batch-mean estimator for the LSCGF ˆ Λn(θ∗, B) = 1 B log 1 k

k

• i=1

eθ∗ ˆ

Xi = 1

B log B n

n/B

• i=1

eθ∗ ˆ

Xi

☞ Batch-mean estimator for the EB ˆ C(Γ, b) = ˆ Λn(θ∗, B) θ∗

Regenerative estimator of the overflow probability in a tandem network

RESIM 2008 – September 2008

Michele Pagano

EB Estimation: regenerative approach

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β0 = 0 α1 = β1 − β0 β1 β1 − 1 ˆ X1 =

β1−1

• i=β0

Xi α2 = β2 − β1 β2 β2 − 1 ˆ X1 =

β2−1

• i=β1

Xi αk = βk − βk−1 βk−1 ˆ Xk =

βk−1

• i=βk−1

Xi βk βk − 1 ☞ Assumption: regenerative structure of the input with regeneration points βn ➳ randomization of the block size with Eα < ∞ ➳ RVs ˆ Xn are i.i.d. ☞ Alternative definition of the EB estimator (which uses k regenerative blocks, corresponding to βk

• bservations)

ˆ Λk(θ∗) = k βk log 1 k

k

• i=1

eθ∗ ˆ

Xi

Regenerative estimator of the overflow probability in a tandem network

RESIM 2008 – September 2008

Michele Pagano

Heuristic justification

10

☞ Considering a large number of time slots n including k regeneration cycles, i.e. n = βk + (n − βk)

where n − βk

n → 0 ☞ Roughly speaking, we have: 1 n log Eeθ∗ Pn

i=1 Xi ≈ 1

βk log Eeθ∗ Pk

i=1 ˆ

Xi = k

βk log Eeθ∗ ˆ

X

☞ From renewal theory k βk → 1 Eα ☞ By the Strong Law of Large Numbers 1 k

k

• i=1

eθ∗ ˆ

Xi → Eeθ∗ ˆ X

☞ Hence ˆ Λk(θ∗) = k βk log 1 k

k

• i=1

eθ∗ ˆ

Xi →

1 Eα log Eeθ∗ ˆ

X

Regenerative estimator of the overflow probability in a tandem network

RESIM 2008 – September 2008

Michele Pagano

Lower bound

11

☞ Notation: ➳ k(n)

∆

= max(k : βk ≤ n) ➳ Eeθ∗ ˆ

X ∆

= a < ∞ ☞ By using conditional expectation and taking into account the independence between k(n) and the

regenerative blocks

1 n log Eeθ∗ Pn

i=1 Xi ≥ 1

n log E

• Eeθ∗ Pk(n)

i=1

ˆ Xi|k(n)

• = 1

n log Eak(n) ☞ By Jensen’s inequality log Eak(n) ≥ E log ak(n) = Ek(n) log a ☞ From the elementary renewal theorem Ek(n) n → 1 Eα ☞ Hence we obtain the desired lower bound: lim inf

n→∞

1 n log Eeθ∗ Pn

i=1 Xi ≥ log a

Eα

∆

= 1 Eα log Eeθ∗ ˆ

X

Regenerative estimator of the overflow probability in a tandem network

RESIM 2008 – September 2008

Michele Pagano

Simulation Scenario

12

C1 {tk} τk = tk − tk−1 b1 ν1 ν2 C2 C2 C2 b2 ☞ The first station is fed by a renewal input ☞ Minimal construction of regenerations for the input to the second station    β0 = 0 βn+1 = mink

• tk > βn : ν1(t−

k ) = 0

• n ≥ 0

☞ Since the queue-size is bounded, if P (τ > 1/C1) > 0 then the renewal process β = (βn) is

positive recurrent, that is it has a finite mean Eα < ∞

☞ The target parameter to be estimated is the (constant) service rate C2 for a given QoS level (overflow

probability ≤ Γ)

Regenerative estimator of the overflow probability in a tandem network

RESIM 2008 – September 2008

Michele Pagano

Regeneration Cycle

13 Packet arrivals and departure at the first queue

ν1(t) = 0 ν1(t) = 0 ν1(t) > 0 ν1(t) > 0 βk βk+1 αk = βk+1 − βk αk = βk+1 − βk

Regenerative estimator of the overflow probability in a tandem network

RESIM 2008 – September 2008

Michele Pagano

Simulation set–up

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C1 λ b1 ν1 ν2 C2 C2 C2 b2 ☞ The input traffic is Poisson(λ), i.e. τ ∈ E (λ) ☞ We compare two different estimatators ➳ Regenerative approach ( ˆ CREG

2

)

➳ Batch-mean approach ( ˆ CBM

2 )

➠ The block size B is chosen according to U [500; 20000] ➠ For a given value of B, 1000 blocks are considered ➠ 200 estimates are generated for different values of B ☞ Then variances of both estimators are calculated based on 200 sample paths

Regenerative estimator of the overflow probability in a tandem network

RESIM 2008 – September 2008

Michele Pagano

Simulation Results - b2 = 60

15 Simulation Parameters: C1 = 0.35, λ = 0.2, b1 = 100, Γ = 0.0001

C1 = 0.35, λ = 0.2, b1 = 100, Γ = 0.0001 C1 = 0.35, λ = 0.2, b1 = 100, Γ = 0.0001

50 100 150 200 250 2 4 6 8 10 12 14 16 18 20 Variance Simulation Trial BatchMean Regeneration

Regenerative estimator of the overflow probability in a tandem network

RESIM 2008 – September 2008

Michele Pagano

Simulation Results - b2 = 100

16 Simulation Parameters: C1 = 0.35, λ = 0.2, b1 = 100, Γ = 0.0001

C1 = 0.35, λ = 0.2, b1 = 100, Γ = 0.0001 C1 = 0.35, λ = 0.2, b1 = 100, Γ = 0.0001

10 20 30 40 50 60 70 80 90 100 2 4 6 8 10 12 14 16 18 20 Variance Simulation Trial BatchMean Regeneration

Regenerative estimator of the overflow probability in a tandem network

RESIM 2008 – September 2008

Michele Pagano

Simulation Results - b2 = 1000

17 Simulation Parameters: C1 = 0.35, λ = 0.2, b1 = 100, Γ = 0.0001

C1 = 0.35, λ = 0.2, b1 = 100, Γ = 0.0001 C1 = 0.35, λ = 0.2, b1 = 100, Γ = 0.0001

3 3.5 4 4.5 5 5.5 6 2 4 6 8 10 12 14 16 18 20 Variance Simulation Trial BatchMean Regeneration

Regenerative estimator of the overflow probability in a tandem network

RESIM 2008 – September 2008

Michele Pagano

Simulation Results - b2 = 10000

18 Simulation Parameters: C1 = 0.35, λ = 0.2, b1 = 100, Γ = 0.0001

C1 = 0.35, λ = 0.2, b1 = 100, Γ = 0.0001 C1 = 0.35, λ = 0.2, b1 = 100, Γ = 0.0001

3.5 4 4.5 5 5.5 6 2 4 6 8 10 12 14 16 18 20 Variance Simulation Trial BatchMean Regeneration

Regenerative estimator of the overflow probability in a tandem network

RESIM 2008 – September 2008

Michele Pagano

Simulation Results - block size

19 Simulation Parameters: C1 = 0.7, λ = 0.5, b1 = 100, b2 = 500, Γ = 0.00001

C1 = 0.7, λ = 0.5, b1 = 100, b2 = 500, Γ = 0.00001 C1 = 0.7, λ = 0.5, b1 = 100, b2 = 500, Γ = 0.00001

0.5 0.51 0.52 0.53 0.54 0.55 0.56 0.57 0.58 0.59 50 100 150 200 250 300 350 400 450 500 Estimated EB Block Size BatchMean Regeneration

Regenerative estimator of the overflow probability in a tandem network

RESIM 2008 – September 2008

Michele Pagano

Simulation Results - block size

20 Simulation Parameters: C1 = 0.7, λ = 0.5, b1 = 100, b2 = 500, Γ = 0.00001

C1 = 0.7, λ = 0.5, b1 = 100, b2 = 500, Γ = 0.00001 C1 = 0.7, λ = 0.5, b1 = 100, b2 = 500, Γ = 0.00001

0.5 0.51 0.52 0.53 0.54 0.55 0.56 0.57 0.58 0.59 50 100 150 200 250 300 350 400 450 500 Estimated EB Block Size BatchMean Regeneration

Regenerative estimator of the overflow probability in a tandem network

RESIM 2008 – September 2008

Michele Pagano

Conclusions

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☞ This work deals with Effective Bandwidths of QoS demanding traffic flows ☞ In particular a refined version of the mean bach EB estimator is proposed ☞ The novelty of the contribution relies on the randomization of the block size estimator based on the

identification of suitable renewall cycles

☞ Experimental results ➳ The estimator has been tested in a simple tandem network topology ➳ Simulation results show that the new estimator significantly outperforms the traditional batch mean

approach in terms of estimation variance

☞ Work in progress ... ➳ Simulations will be extended to a wider class of network topologies as well as input processes ➳ Upper bound (to prove the convergence of the estimator)

Regenerative estimator of the overflow probability in a tandem network

RESIM 2008 – September 2008

Michele Pagano