Effect of higher moments of job size distribution on the performance of
an M/G/k system
VARUN GUPTA
Joint work with:
Mor Harchol-Balter
Carnegie Mellon University
Jim Dai, Bert Zwart
Georgia Institute of Technology
2
Multi-server/resource sharing systems are the norm today
Multicore chips
Call centers
ServerFarms
3
M/G/k: the classical multi-server model
Poisson arrivals (rate )
J1Ji+1JiJ2Ji+2
GOAL : Analysis of mean delay (time spent in buffer)
4
M/G/k model assumptions and notation
• Poisson arrivals
• Service requirements (job sizes) are i.i.d.• S ≡ random variable for job sizes
• Define
• Define
Per server utilization or load:0 < < 1
Squared coefficient of variability (SCV) of job sizes:
C2 0
5
M/G/k mean delay analysis
• Lets take a step back: M/G/1
6
M/G/k mean delay analysis
• Lets take a step back: M/G/1
7
M/G/k mean delay analysis
• Lee and Longton (1959)
– Simple and closed-form– Involves only first two moments of S– Exact for k=1– Asymptotically exact in heavy traffic [Köllerström[74]]
• No exact analysis exists• All closed-form approximations involve only the
first two moments of S – Takahashi[77], Hokstad[78], Nozaki Ross[78], Boxma
Cohen Huffels[79], Whitt [93], Kimura[94]
8
But…
Q1: Are 2 moments of S enough to reasonably approximate E[Delay]?
• Does the third moment have no/negligible effect?
Q2: How inaccurate can a 2-moment approximation be?
Q3: Are 3 moments enough? 4 moments?
9
Outline
Q1: Are 2 moments of S enough to reasonably approximate E[Delay]?
• Does the third moment have no/negligible effect?
Q2: How inaccurate can a 2-moment approximation be?
Q3: Are 3 moments enough? 4 moments?
10
H2
• The H2 class has three degrees of freedom
• Can vary E[S3] while keeping first two moments constant
• Can numerically evaluate M/H2/k using the matrix analytic method
11
E[Delay] vs. E[S3]k=10, E[S]=1, C2=19, =0.9
0
1
2
3
4
5
6
7
0 1 2 3 4 5 6
E[Delay]
E[S3] X104
2-moment approx
E[Delay]M/H2/k
12
0
1
2
3
4
5
6
7
0 1 2 3 4 5 6X104
E[S3]
E[Delay]
E[Delay]M/H2/k
2-moment approx
E[S3] can have a huge impact on mean delay!
The mean delay decreases as E[S3] increases!
13
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20
x)
Intuition for the effect of E[S3]
x) = load due to jobs smaller than x E[S]=1 C2=19
x
14
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20
x)
E[S3]=600
Intuition for the effect of E[S3]
x) = load due to jobs smaller than x E[S]=1 C2=19
x
15
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20
x)
E[S3]=600
E[S3]=700
Intuition for the effect of E[S3]
x) = load due to jobs smaller than x E[S]=1 C2=19
x
16
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20
x)
E[S3]=600
E[S3]=700
E[S3]=1200
Intuition for the effect of E[S3]
x) = load due to jobs smaller than x E[S]=1 C2=19
x
17
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20
E[S3]=600
E[S3]=700
E[S3]=1200
E[S3]=15000
x) = load due to jobs smaller than x E[S]=1 C2=19
Intuition for the effect of E[S3]
x
x)
18
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20
As E[S3] increases (with fixed E[S] and E[S2]):• Load gets ‘concentrated’ on small jobs• Load due to ‘big’ jobs vanishes• Bigs become rarer, usually see small jobs only• Causes drop in E[Delay]M/H2/k
x
x)
Increasing E[S3]
Intuition for the effect of E[S3]
19
0
1
2
3
4
5
6
7
0 1 2 3 4 5 6X104
E[S3]
E[Delay]
E[Delay] vs. E[S3]k=10, E[S]=1, C2=19, =0.9
E[Delay]M/H2/k
2-moment approx
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Outline
Q1: Are 2 moments of S enough to reasonably approximate E[Delay]?
• Does the third moment have no/negligible effect?
Q2: How inaccurate can a 2-moment approximation be?
Q3: Are 3 moments enough? 4 moments?
21
Outline
Q1: Are 2 moments of S enough to reasonably approximate E[Delay]?
• Does the third moment have no/negligible effect?
Q2: How inaccurate can a 2-moment approximation be?
Q3: Are 3 moments enough? 4 moments?
22
{G|C2} ≡ positive distributions with mean 1 and SCV C2
E[Delay]
G1
G2
GAP Error of2-moment approx
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Our Theorems• Upper bound
• Lower bound– <1-1/k
– 1-1/k
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E[D
elay
]
GAP
25
E[D
elay
]
0
D* has the smallest third
moment in {G|C2}
third moment as
0
26
E[D
elay
]
Conjecture
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Outline
Q1: Are 2 moments of S enough to reasonably approximate E[Delay]?
• Does the third moment have no/negligible effect?
Q2: How inaccurate can a 2-moment approximation be?
Q3: Are 3 moments enough? 4 moments?
28
Outline
Q1: Are 2 moments of S enough to reasonably approximate E[Delay]?
• Does the third moment have no/negligible effect?
Q2: How inaccurate can a 2-moment approximation be?
Q3: Are 3 moments enough? 4 moments?
29
What about higher moments?
{G|C2}
H2
H*3
• H*3 class has four
degrees of freedom
• Can vary E[S4] while keeping first three moments constant
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0
1
2
3
4
5
6
7
0 1 2 3 4 5 6X104
E[S3]
E[Delay]Increasing fourth moment
E[Delay] vs. E[S4]
k=10, E[S]=1, C2=19, =0.9
E[Delay]M/H2/k
2-moment approx
31
0
1
2
3
4
5
6
7
0 1 2 3 4 5 6X104
E[S3]
E[Delay]Increasing fourth moment
E[Delay]M/H2/k
2-moment approx
• Even E[S4] can have a significant impact on mean delay!• High E[S4] can nullify the effect of E[S3]!
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E[D
elay
]
The BIG picture
LB1=E[Delay]M/D/k
UB1,2=(C2+1)E[Delay]M/D/k
LB1,2,3
UB1,2,3,4
LB1,2,3,4,5
Odd/Even moments refine the Lower/Upper bounds on mean delay
33
Outline
Q1: Are 2 moments of S enough to reasonably approximate E[Delay]?
• Does the third moment have no/negligible effect?
Q2: How inaccurate can a 2-moment approximation be?
Q3: Are 3 moments enough? 4 moments?
34
Conclusions
• Shown that 2-moment approximations for M/G/k are insufficient
• Shown bounds on inaccuracy of 2-moment only approximations– (C2+1) inaccuracy factor
• Observed alternating effects of odd and even moments
35
Thank you!
36
Open Questions
• Proof (or counter-example) of conjectures on bounds
• Are there other attributes of service distribution that characterize it better than moments?– For example, mean and variability of small and big
jobs
• Where do real world service distributions sit with respect to these attributes?
37
{G|C2} ≡ positive distributions with mean 1 and SCV C2
H2
• The H2 class has three degrees of freedom (s, p, ps)
• Can vary E[S3] while holding first two moments constant
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Look at the moments of H2 … • Load due to big jobs vanishes as E[S3] increases
• When k>1, a big job does not block small jobs
• This reduces the effect of variability (C2) as third moment increases
39
Observations
• < 1-1/k UB/LB (C2+1)– No 2-moment approximation can be accurate in this
case
• [Kiefer Wolfowitz] [Scheller-Wolf]: When > 1-1/k, E[Delay] is finite iff C2 is finite.– Matches with the conjectured lower bound– Also popular as the “0 spare server” case
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Proof outline: Upper bound
• THEOREM:
• PROOF: Consider the following service distribution
• Intuition for conjecture: k>1 should mitigate the effect of variability; D* exposes it completely
• Note: D* has the smallest third moment in {G|C2}
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Proof outline: Lower bound
• THEOREM:– < 1-1/k
– 1-1/k
• PROOF: Consider the following sequence of service distributions in {G|C2} as 0
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What about higher moments?
{G|C2}
H2
H*3
H*3
• H*3 allows control over fourth
moment while holding first three moments fixed
• The fourth moment is minimized when p0=0 (H2 distribution)
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