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FINDING THE OPTIMAL QUANTUM SIZE
Revisiting the M/G/1 Round-Robin Queue
VARUN GUPTA
Carnegie Mellon University
2
M/G/1/RR
Jobs served for q
units at a time
Poisson
arrivals
Incomplete
jobs
3
M/G/1/RR
Jobs served for q
units at a time
Poisson
arrivals
Incomplete
jobs
4
M/G/1/RR
• arrival rate =
• job sizes i.i.d. ~ S
• load
•
Jobs served for q
units at a time
Completed
jobs
Incomplete
jobs
Poisson
arrivals
Squared coefficient of
variability (SCV) of job sizes:
C2 0
5
M/G/1/RR
M/G/1/PS M/G/1/FCFS
q q → 0 q =
Preemptions cause overhead
(e.g. OS scheduling) Variable job sizes cause long delays
small q large q
preemption
overheads job size
variability
6
GOAL
Optimal operating q for M/G/1/RR with overheads and high C2
SUBGOAL
Sensitivity Analysis: Effect of q and C2 on M/G/1/RR performance
(no switching overheads)
PRIOR WORK
• Lots of exact analysis: [Wolff70], [Sakata et al.71],
[Brown78]
No closed-form solutions/bounds
No simple expressions for interplay of q and C2
effect of preemption overheads
7
Outline
Effect of q and C2 on mean response time
Approximate analysis
Bounds for M/G/1/RR
Choosing the optimal quantum size
No preemption
overheads
Effect of
preemption
overheads
8
Approximate sensitivity analysis of
M/G/1/RR
Approximation assumption 1:
Service quantum ~ Exp(1/q)
Approximation assumption 2:
Job size distribution
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Approximate sensitivity analysis of
M/G/1/RR
• Monotonic in q
– Increases from E[TPS] → E[TFCFS]
• Monotonic in C2
– Increases from E[TPS] → E[TPS](1+q)
For high C2: E[TRR*] ≈ E[TPS](1+q)
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Outline
Effect of q and C2 on mean response time
Approximate analysis
Bounds for M/G/1/RR
Choosing the optimal quantum size
11
THEOREM:
M/G/1/RR bounds
Assumption: job sizes {0,q,…,Kq}
Lower bound is TIGHT:
E[S] = iq
Upper bound is TIGHT within (1+/K):
Job sizes {0,Kq}
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Outline
Effect of q and C2 on mean response time
Approximate analysis
Bounds for M/G/1/RR
Choosing the optimal quantum size
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Optimizing q
Preemption overhead = h
1.
2. Minimize E[TRR] upper bound from Theorem:
Common case:
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0
5
10
15
20
25
30
0 0.5 1 1.5 2 2.5service quantum (q)
Me
an
re
sp
on
se
tim
e
Job size distribution: H2 (balanced means) E[S] = 1, C2 = 19, = 0.8
h=0
15
0
5
10
15
20
25
30
0 0.5 1 1.5 2 2.5service quantum (q)
Me
an
re
sp
on
se
tim
e
Job size distribution: H2 (balanced means) E[S] = 1, C2 = 19, = 0.8
approximation q*
optimum q
1%
h=0
16
0
5
10
15
20
25
30
0 0.5 1 1.5 2 2.5service quantum (q)
Me
an
re
sp
on
se
tim
e
Job size distribution: H2 (balanced means) E[S] = 1, C2 = 19, = 0.8
1% 2.5%
h=0
approximation q*
optimum q
17
0
5
10
15
20
25
30
0 0.5 1 1.5 2 2.5service quantum (q)
Me
an
re
sp
on
se
tim
e
Job size distribution: H2 (balanced means) E[S] = 1, C2 = 19, = 0.8
1% 2.5%
5%
h=0
approximation q*
optimum q
18
0
5
10
15
20
25
30
0 0.5 1 1.5 2 2.5service quantum (q)
Me
an
re
sp
on
se
tim
e
Job size distribution: H2 (balanced means) E[S] = 1, C2 = 19, = 0.8
1% 2.5%
5%
7.5%
h=0
approximation q*
optimum q
19
0
5
10
15
20
25
30
0 0.5 1 1.5 2 2.5service quantum (q)
Me
an
re
sp
on
se
tim
e
Job size distribution: H2 (balanced means) E[S] = 1, C2 = 19, = 0.8
1% 2.5%
5%
7.5%
10%
h=0
approximation q*
optimum q
20
Outline
Effect of q and C2 on mean response time
Approximate analysis
Bounds for M/G/1/RR
Choosing the optimal quantum size
Conclusion/Contributions
• Simple approximation and bounds for
M/G/1/RR
• Optimal quantum size for handling highly
variable job sizes under preemption
overheads
q
22
Bounds – Proof outline
• Di = mean delay for ith quantum of service
• D = [D1 D2 ... DK]
• D is the fixed point of a monotone linear system:
DT = APDT+b
D1
D2
f1=0
f2=0
D
D’
D*
Sufficient condition
for lower bound:
D’T APD’T+b
for all P
Sufficient condition
for upper bound:
D*T APD*T+b
for all P
23
Optimizing q
• Preemption overhead = h
• q’ = q+h, E[S]’ → E[S] (1+h/q), ’ = E[S]’
•
Heavy traffic: Small overhead:
