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

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

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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

Incomplete

jobs

Poisson

arrivals

Jobs served for q units at a time

Incomplete

jobs

Poisson

arrivals

Jobs served for q units at a time

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

q

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

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

- 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

Approximation assumption 1:

Service quantum ~ Exp(1/q)

Approximation assumption 2:

Job size distribution

For high C2: E[TRR*] ≈E[TPS](1+q)

- Monotonic in q
- Increases from E[TPS] →E[TFCFS]

- Monotonic in C2
- Increases from E[TPS] →E[TPS](1+q)

- Effect of q and C2 on mean response time
- Approximate analysis
- Bounds for M/G/1/RR

- Choosing the optimal quantum size

Assumption: job sizes {0,q,…,Kq}

THEOREM:

Lower bound is TIGHT:

E[S] = iq

Upper bound is TIGHT within (1+/K):

Job sizes {0,Kq}

- Effect of q and C2 on mean response time
- Approximate analysis
- Bounds for M/G/1/RR

- Choosing the optimal quantum size

Preemption overhead = h

- Minimize E[TRR] upper bound from Theorem:

Common case:

Mean response time

h=0

service quantum (q)

Job size distribution: H2 (balanced means)E[S] = 1, C2 = 19, = 0.8

approximation q*

optimum q

Mean response time

1%

h=0

service quantum (q)

Job size distribution: H2 (balanced means)E[S] = 1, C2 = 19, = 0.8

approximation q*

optimum q

Mean response time

2.5%

1%

h=0

service quantum (q)

Job size distribution: H2 (balanced means)E[S] = 1, C2 = 19, = 0.8

approximation q*

optimum q

Mean response time

5%

2.5%

1%

h=0

service quantum (q)

Job size distribution: H2 (balanced means)E[S] = 1, C2 = 19, = 0.8

approximation q*

optimum q

7.5%

Mean response time

5%

2.5%

1%

h=0

service quantum (q)

Job size distribution: H2 (balanced means)E[S] = 1, C2 = 19, = 0.8

approximation q*

optimum q

10%

7.5%

Mean response time

5%

2.5%

1%

h=0

service quantum (q)

Job size distribution: H2 (balanced means)E[S] = 1, C2 = 19, = 0.8

- Effect of q and C2 on mean response time
- Approximate analysis
- Bounds for M/G/1/RR

- Choosing the optimal quantum size

q

- Simpleapproximation and bounds for M/G/1/RR
- Optimal quantum size for handling highly variable job sizes under preemption overheads