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Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol- Balter Carnegie Mellon Univ. Alan Scheller- Wolf Carnegie Mellon Univ. Uri Yechiali Tel Aviv Univ.
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Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

Dec 21, 2015

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Page 1: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

Fundamental Characteristics of Queues with Fluctuating Load

VARUN GUPTA

Joint with:

Mor Harchol-Balter

Carnegie Mellon Univ.

Alan Scheller-Wolf

Carnegie Mellon Univ.

Uri Yechiali

Tel Aviv Univ.

Page 2: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

2

Motivation

ClientsServer Farm

Requests

Page 3: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

3

Motivation

ClientsServer Farm

Requests

Page 4: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

4

Motivation

ClientsServer Farm

Requests

Page 5: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

5

Motivation

ClientsServer Farm

Requests

Page 6: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

6

Motivation

ClientsServer Farm

Requests

Page 7: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

7

Motivation

ClientsServer Farm

Requests

Page 8: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

8

Motivation

ClientsServer Farm

Requests

Page 9: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

9

Motivation

ClientsServer Farm

RequestsReal

World Fluctuating arrival

and service intensities

Page 10: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

10

A Simple Model

HL

exp(H)

exp(L)

HighLoad

LowLoad

Page 11: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

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• Poisson Arrivals• Exponential Job Size Distribution• H/H > L/L

• H>H possible, only need stability

A Simple Model

HighLoad

LowLoad

H,H

L,L

exp()

exp()

HH

LL

Page 12: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

12

The Markov ChainP

has

e

Number of jobs

L

H

H

H

0 1

0 1

2

2

L

L

H

H

L

L

. . .

. . .

Solving the Markov chain provides no behavioral insight

Page 13: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

13

HH

LL

• N = Number of jobs in the fluctuating load system

• Lets try approximating N using (simpler) non-fluctuating systems

Page 14: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

14

HH

LL

Method 1

Nmix

Page 15: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

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HH

LL

Q: Is Nmix ≈ N?

A: Only when 0

Method 1

Nmix

½

½

+

,

Page 16: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

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HH

LL

Method 2

Page 17: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

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avg(H,L)avg(H,L)

Method 2

≡ Navg

Q: Is Navg ≈ N?

A: When ,

Page 18: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

18

Example

H=1, H=0.99

L=1, L=0.01

E[Nmix] ≈ 49.5 E[Navg] = 1

0

Page 19: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

19

Observations

• Fluctuating system can be worse than non-fluctuating

0 and asymptotes can be very far apart

E[Nmix] > E[Navg]

E[Nmix] E[Navg]

Page 20: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

20

Questions

• Is fluctuation always bad?

• Is E[N] monotonic in ?

• Is there a simple closed form approximation for E[N] for intermediate ’s?

• How do queue lengths during High Load and Low Load phase compare? How do they compare with Navg?

More than 40 years of research has not

addressed such fundamental questions!

Page 21: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

21

Outline

Is E[Nmix] ≥ E[Navg], always?

Is E[N] monotonic in ?

Simple closed form approximation for E[N]

Application: Capacity Planning

Stochastic orderings for the number of jobs seen by an arrival into an H phase, L phase Not covered in this talk

Please read paper.

Page 22: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

22

Prior Work

Fluid/DiffusionApproximations

Transforms Matrix Analytic& Spectral Analysis

- P. Harrison- Adan and Kulkarni

Numerical ApproachesInvolves solution of cubic

- Clarke- Neuts- Yechiali and Naor

Involves solution of cubic

- Massey- Newell- Abate, Choudhary, Whitt

Limiting Behavior

But cubic equations have a close form solution…

?

Page 23: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

23

Good luck understanding this!

Page 24: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

24

Asymptotics for E[N] (H<H)

E[Navg]

E[Nmix]

E[N]

(switching rate)Highfluctuation

H=1, H=0.99

L=1, L=0.01

E[Nmix] > E[Navg]

Lowfluctuation

Page 25: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

25

Asymptotics for E[N] (H<H)

E[N]

E[Nmix]

E[Navg]

• Agrees with our example (H = L)

• Ross’s conjecture for systems with constant service rate:

“Fluctuation increases mean delay”

Q: Is this behavior possible?

A: Yes

E[N]

E[Navg]

E[Nmix]

Page 26: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

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Our Results

E[N]

(H-H) > (L-L)(H-H) = (L-L)(H-H) < (L-L)

• Define the slacks during L and H as• sL = L - L

• sH = H - H

E[N]

E[N]

Page 27: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

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Our Results

• Define the slacks during L and H as• sL = L - L

• sH = H - H

• Not load but slacks determine the response times!

sH > sLsH = sLsH < sL

KEY IDEA

E[N]

E[N]

E[N]

Page 28: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

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Outline

Is E[Nmix] ≥ E[Navg], always?

Is E[N] monotonic in ?

Simple closed form approximation for E[N]

Application: Capacity Planning

Stochastic orderings for the number of jobs seen by an arrival into an H phase, L phase Not covered in this talk

Please read paper.

Page 29: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

29

Outline

Is E[Nmix] ≥ E[Navg], always? No

Is E[N] monotonic in ?

Simple closed form approximation for E[N]

Application: Capacity Planning

Stochastic orderings for the number of jobs seen by an arrival into an H phase, L phase Not covered in this talk

Please read paper.

Page 30: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

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Notation

• NH: Number of jobs in system during H phase

• NL: Number of jobs in system during L phase

• N = (NH+NL)/2

H,H

L,L

exp()

exp()NH NL

Page 31: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

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Analysis of E[N]

First steps:

– Note that it suffices to look at switching points

– Express

• NL = f(NH)

• NH = g(NL)

– The problem reduces to finding Pr{NH=0} and Pr{NL=0}

H,H

L,L

NH NL

NL=f(g(NL))

fg

Page 32: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

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– Find the root of a cubic (the characteristic matrix polynomial in the Spectral Expansion method)

– Express E[N] in terms of

E[N] =

The simple way forward…

H,H

L,L

fg

A

A-A

H(L -L)0H+ L(H-H)0

L - (L -L)(H-H)

2 (A -A)+

Where 0L = 0

H = (A-A)

L(-1)(H-H)

(A-A)

H(-1)(L-L)

NH NL Difficult to even prove the monotonicity of E[N] wrt

using this!

Page 33: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

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Our approach (contd.)

• Express the first moment as

E[N] = f1()r+f0()(1-r)

– r is the root of a (different) cubic– r1 as 0 and r0 as

KEY IDEA

Page 34: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

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Monotonicity of E[N]

• E[N] = f1()r+f0()(1-r)

• r is monotonic in E[N] is monotonic in • The cubic for r has maximum power of as 2

1

0

r

Page 35: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

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Monotonicity of E[N]

• E[N] = f1()r+f0()(1-r)

• r is monotonic in E[N] is monotonic in • The cubic for r has maximum power of as 2

1

0

r

Need at least 3 roots for when r=c1

but has at most 2 roots

c1

Page 36: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

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Monotonicity of E[N]

• E[N] = f1()r+f0()(1-r)

• r is monotonic in E[N] is monotonic in • The cubic for r has maximum power of as 2

1

0

r

Need at least 2 positive roots for when r=c2

but for r>1 product of roots is negative

c2

Page 37: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

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Monotonicity of E[N]

• E[N] = f1()r+f0()(1-r)

• r is monotonic in E[N] is monotonic in • The cubic for r has maximum power of as 2

1

0

r

E[N] is monotonic in !

Page 38: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

38

Outline

Is E[Nmix] ≥ E[Navg], always? No

Is E[N] monotonic in ?

Simple closed form approximation for E[N]

Application: Capacity Planning

Stochastic orderings for the number of jobs seen by an arrival into an H phase, L phase Not covered in this talk

Please read paper.

Page 39: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

39

Outline

Is E[Nmix] ≥ E[Navg], always? No

Is E[N] monotonic in ? Yes

Simple closed form approximation for E[N]

Application: Capacity Planning

Stochastic orderings for the number of jobs seen by an arrival into an H phase, L phase Not covered in this talk

Please read paper.

Page 40: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

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Approximating E[N]

• Express the first moment as

E[N] = f1()r+f0()(1-r)

– r is the root of a (different) cubic– r1 as 0 and r0 as

• Approximate r by the root of a quadratic

KEY IDEA

KEY IDEA

Page 41: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

41

Approximating E[N]

1

3

5

7

9

1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+0110-5 10-4 10-3 10-2 10-1 100 101

3

5

7

9

E[N]

ExactApprox.

H=L=1, H=0.95, L=0.2

Page 42: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

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Approximating E[N]

1

3

5

7

9

1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+0110-5 10-4 10-3 10-2 10-1 100 101

3

5

7

9

E[N]

ExactApprox.

H=L=1, H=0.95, L=0.2

Page 43: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

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Approximating E[N]

2

6

10

14

18

1.E-02 1.E-01 1.E+00 1.E+0110-2 10-1 100 10

ExactApprox.

H=L=1, H=1.2, L=0.2

2

6

10

14

18

E[N]

Page 44: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

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Approximating E[N]

2

6

10

14

18

1.E-02 1.E-01 1.E+00 1.E+0110-2 10-1 100 10

ExactApprox.

H=L=1, H=1.2, L=0.2

2

6

10

14

18

E[N]

Page 45: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

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Outline

Is E[Nmix] ≥ E[Navg], always? No

Is E[N] monotonic in ? Yes

Simple closed form approximation for E[N]

Application: Capacity Planning

Stochastic orderings for the number of jobs seen by an arrival into an H phase, L phase Not covered in this talk

Please read paper.

Page 46: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

46

Outline

Is E[Nmix] ≥ E[Navg], always? No

Is E[N] monotonic in ? Yes

Simple closed form approximation for E[N]

Application: Capacity Planning

Stochastic orderings for the number of jobs seen by an arrival into an H phase, L phase Not covered in this talk

Please read paper.

Page 47: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

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Scenario

Application: Capacity Provisioning

HH

LL

2HH

2LL

Aim: To keep the mean response times same

Page 48: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

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Scenario

Application: Capacity Provisioning

HH

LL

2H2H

2L2L

Question: What is the effect of doubling the arrival and service rates on the mean response time?

Page 49: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

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What happens to the mean response time when , are doubled in the fluctuating load queue?

Halves

Remains almost the sameReduces by less than half

Reduces by more than halfA:

D:C:

B:

Page 50: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

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What happens to the mean response time when , are doubled in the fluctuating load queue?

Halves

Remains almost the sameReduces by less than half

Reduces by more than halfA:

D:C:

B:

Page 51: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

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What happens to the mean response time when , are doubled in the fluctuating load queue?

Halves

Remains almost the sameReduces by less than half

Reduces by more than halfA:

D:C:

B:

Look at slacks!

A: sH = sL

B: sH > sL

C: sH < sL

D: sH < 0, 0

reduces by half more than half less than half remains same

Page 52: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

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Our Contributions

• Give a simple characterization of the behavior of E[N] vs.

• Provide simple (and tight) quadratic approximations for E[N]

• Prove the first stochastic ordering results for the fluctuating load model (see paper)

Page 53: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

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Bon Appetit!

Page 54: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

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Direction for future research

• Analysis of higher moments of response time

• Analysis of bursty arrival process

• General phase type distributions for phase lengths

• Analysis of alternating traffic streams – look at the workload process instead of number of jobs in system

• Conjecture: NH increases stochastically as switching rates decrease

Page 55: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

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Comparison of NL vs. NH

Jackpot!

Honey, I think we chose the wrong time to go out!

Page 56: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

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Stochastic Ordering refresher

• Random variable X stochastically dominates (is stochastically larger than) Y if:

Pr{Xi} Pr{Yi}

for all i.

• If X stY then E[f(X)] E[f(Y)] for all increasing f– E[Xk] E[Yk] for all k 0.

Page 57: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

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Comparison of NL vs NH

• NL ≥st NM/M/1/L

• NH ≤st NM/M/1/H

• NH ≥st NL

• NH ≥st Navg

• NL st Navg

Page 58: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

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Why do slacks matter?

• Fact: The mean response time in an M/M/1 queue is (-)-1

– Higher slacks Lower mean response times

• What is the fraction of customers departing during H

H,H

L,L

exp()

exp()

when ?

H,H

L,L

exp()

exp()

when 0?

H

H + L

H

H + L?

Page 59: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

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Why do slacks matter?

when ?

H,H

L,L

exp()

exp()

when 0?

H

H + L

H

H + L?

• Fact: The mean response time in an M/M/1 queue is (-)-1

– Higher slacks Lower mean response times

• What is the fraction of customers departing during H

H,H

L,L

exp()

exp()

Page 60: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

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Why do slacks matter?

when ?

H,H

L,L

exp()

exp()

when 0?

A H?

• Fact: The mean response time in an M/M/1 queue is (-)-1

– Higher slacks Lower mean response times

• What is the fraction of customers departing during H

As switching rates decrease, larger fraction of customers experience lower mean response times when sH>sL

H,H

L,L

exp()

exp()

Page 61: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

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Q: What happens to E[N] when we double ’s and ’s?

A:System A: , ,

System B: 2, 2,

?

Page 62: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

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Q: What happens to E[N] when we double ’s and ’s?

A:System A: , ,

System B: 2, 2,

System C: 2, 2, 2

E[N] remains same in going from A to C

A) sL = sH : remains same

B) sL > sH : increases, but by less than twice

C) sL < sH : decreases

D) 0, H>1 : queue lengths become twice as switching rates halve, E[N] doubles

Page 63: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie.

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Example

H=1.9, H=0.99

L=0.1, L=0.01

E[Nmix] ≈ 0.6 E[Navg] = 1

0