FINDING THE OPTIMAL QUANTUM SIZE Revisiting the M/G/1 Round-Robin Queue

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FINDING THE OPTIMAL QUANTUM SIZERevisiting the
M/G/1 Round-Robin Queue

• VARUN GUPTA
• Carnegie Mellon University

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M/G/1/RR
Incomplete jobs
Poisson arrivals
Jobs served for q units at a time
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M/G/1/RR
Incomplete jobs
Poisson arrivals
Jobs served for q units at a time
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M/G/1/RR
Incomplete jobs
Completed jobs
Poisson arrivals
Jobs served for q units at a time

• arrival rate ?
• job sizes i.i.d. S
• load

Squared coefficient of variability (SCV) of job
sizes C2 ? 0
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M/G/1/RR
q ? 0
q ?
M/G/1/FCFS
M/G/1/PS
? Preemptions cause overhead (e.g. OS
scheduling)
? Variable job sizes cause long delays
preemption overheads
job size variability
small q
large q
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GOAL Optimal operating q for M/G/1/RR with
overheads and high C2
effect of preemption overheads
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

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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

No preemption overheads
Effect of preemption overheads
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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 ETPS ? ETFCFS
• Monotonic in C2
• Increases from ETPS ? ETPS(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

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M/G/1/RR bounds

• Assumption job sizes ? 0,q,,Kq

THEOREM
Lower bound is TIGHT ES 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
• Minimize ETRR upper bound from Theorem

Common case
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Mean response time
h0
service quantum (q)
Job size distribution H2 (balanced means) ES
1, C2 19, ? 0.8
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Mean response time
1
h0
service quantum (q)
Job size distribution H2 (balanced means) ES
1, C2 19, ? 0.8
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Mean response time
2.5
1
h0
service quantum (q)
Job size distribution H2 (balanced means) ES
1, C2 19, ? 0.8
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Mean response time
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2.5
1
h0
service quantum (q)
Job size distribution H2 (balanced means) ES
1, C2 19, ? 0.8
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7.5
Mean response time
5
2.5
1
h0
service quantum (q)
Job size distribution H2 (balanced means) ES
1, C2 19, ? 0.8
19
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7.5
Mean response time
5
2.5
1
h0
service quantum (q)
Job size distribution H2 (balanced means) ES
1, C2 19, ? 0.8
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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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Conclusion/Contributions

• Simple approximation and bounds for M/G/1/RR
• Optimal quantum size for handling highly
  variable job sizes under preemption overheads

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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 APDTb

f10
f20
D2
Sufficient condition for upper bound DT ?
APDTb for all P
Sufficient condition for lower bound DT ?
APDTb for all P
D
D
D
D1
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Optimizing q

• Preemption overhead h
• q qh, ES ? ES (1h/q), ? ? ES

Heavy traffic
Small overhead