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FINDING THE OPTIMAL QUANTUM SIZE Revisiting the M/G/1 Round-Robin Queue. VARUN GUPTA Carnegie Mellon University. M/G/1/RR. Incomplete jobs. Poisson arrivals. Jobs served for q units at a time. M/G/1/RR. Incomplete jobs. Poisson arrivals. Jobs served for q units at a time.

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finding the optimal quantum size revisiting the m g 1 round robin queue

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

VARUN GUPTA

Carnegie Mellon University

m g 1 rr
M/G/1/RR

Incomplete

jobs

Poisson

arrivals

Jobs served for q units at a time

m g 1 rr1
M/G/1/RR

Incomplete

jobs

Poisson

arrivals

Jobs served for q units at a time

m g 1 rr2
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

slide5

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

slide6

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
outline
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

approximate sensitivity analysis of m g 1 rr
Approximate sensitivity analysis of M/G/1/RR

Approximation assumption 1:

Service quantum ~ Exp(1/q)

Approximation assumption 2:

Job size distribution

approximate sensitivity analysis of m g 1 rr1

For high C2: E[TRR*] ≈E[TPS](1+q)

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)
outline1
Outline
  • Effect of q and C2 on mean response time
    • Approximate analysis
    • Bounds for M/G/1/RR
  • Choosing the optimal quantum size
m g 1 rr bounds
M/G/1/RR bounds

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}

outline2
Outline
  • Effect of q and C2 on mean response time
    • Approximate analysis
    • Bounds for M/G/1/RR
  • Choosing the optimal quantum size
optimizing q
Optimizing q

Preemption overhead = h

  • Minimize E[TRR] upper bound from Theorem:

Common case:

slide14

Mean response time

h=0

service quantum (q)

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

slide15

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

slide16

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

slide17

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

slide18

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

slide19

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

outline3
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

q

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