fundamental characteristics of queues with fluctuating load varun gupta joint with: mor...
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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](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d635503460f94a459d6/html5/thumbnails/1.jpg)
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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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](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d635503460f94a459d6/html5/thumbnails/3.jpg)
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](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d635503460f94a459d6/html5/thumbnails/4.jpg)
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](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d635503460f94a459d6/html5/thumbnails/5.jpg)
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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](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d635503460f94a459d6/html5/thumbnails/6.jpg)
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](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d635503460f94a459d6/html5/thumbnails/7.jpg)
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](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d635503460f94a459d6/html5/thumbnails/8.jpg)
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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](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d635503460f94a459d6/html5/thumbnails/9.jpg)
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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](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d635503460f94a459d6/html5/thumbnails/10.jpg)
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A Simple Model
HL
exp(H)
exp(L)
HighLoad
LowLoad
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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
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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
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HH
LL
• N = Number of jobs in the fluctuating load system
• Lets try approximating N using (simpler) non-fluctuating systems
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HH
LL
Method 1
Nmix
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HH
LL
Q: Is Nmix ≈ N?
A: Only when 0
Method 1
Nmix
½
½
+
,
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HH
LL
Method 2
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avg(H,L)avg(H,L)
Method 2
≡ Navg
Q: Is Navg ≈ N?
A: When ,
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Example
H=1, H=0.99
L=1, L=0.01
E[Nmix] ≈ 49.5 E[Navg] = 1
0
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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]
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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](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d635503460f94a459d6/html5/thumbnails/21.jpg)
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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 22: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d635503460f94a459d6/html5/thumbnails/22.jpg)
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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…
?
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Good luck understanding this!
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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
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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]
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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]
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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](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d635503460f94a459d6/html5/thumbnails/28.jpg)
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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](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d635503460f94a459d6/html5/thumbnails/29.jpg)
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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](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d635503460f94a459d6/html5/thumbnails/30.jpg)
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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
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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](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d635503460f94a459d6/html5/thumbnails/32.jpg)
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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!
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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](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d635503460f94a459d6/html5/thumbnails/34.jpg)
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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
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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
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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
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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](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d635503460f94a459d6/html5/thumbnails/38.jpg)
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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](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d635503460f94a459d6/html5/thumbnails/39.jpg)
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](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d635503460f94a459d6/html5/thumbnails/40.jpg)
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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
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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 42: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d635503460f94a459d6/html5/thumbnails/42.jpg)
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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
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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](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d635503460f94a459d6/html5/thumbnails/44.jpg)
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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](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d635503460f94a459d6/html5/thumbnails/45.jpg)
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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](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d635503460f94a459d6/html5/thumbnails/46.jpg)
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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 47: Fundamental Characteristics of Queues with Fluctuating Load VARUN GUPTA Joint with: Mor Harchol-Balter Carnegie Mellon Univ. Alan Scheller-Wolf Carnegie](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d635503460f94a459d6/html5/thumbnails/47.jpg)
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Scenario
Application: Capacity Provisioning
HH
LL
2HH
2LL
Aim: To keep the mean response times same
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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?
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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:
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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:
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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
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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)
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Bon Appetit!
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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
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Comparison of NL vs. NH
Jackpot!
Honey, I think we chose the wrong time to go out!
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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.
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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
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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?
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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()
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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()
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Q: What happens to E[N] when we double ’s and ’s?
A:System A: , ,
System B: 2, 2,
?
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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
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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