Frequency Scaling in Multilevel Queues IFIP Performance 2020 Maryam - - PowerPoint PPT Presentation

Frequency Scaling in Multilevel Queues

IFIP Performance 2020

Maryam Elahi1,3 Andrea Marin2 Sabina Rossi2 Carey Williamson3

1 Mount Royal University, Canada 2 Universit`

a Ca’Foscari Venezia, Italy

3 University of Calgary, Canada

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Talk overview

Introduction and Contribution The queueing model and its solution Case study Conclusion

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Introduction and Contribution

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Size based scheduling

• Goal: serve first the smallest jobs to improve the expected

response time

• Assumptions:
• Do we know the job size at the arrival epoch? (Shortest

Remaining Processing Time)

• Do we know the distribution of the job size? (Gittin’s policy)
• Is the job size distribution heavy tailed? (Least Attained

Service (LAS), Multilevel queues)

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How do multilevel queues work?

Example: two levels with Processor Sharing (PS) server

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Frequency scaling

• Goal: Energy saving
• Main idea: When there are few jobs in the system, we can

reduce the processor speed

• We increase the expected response time w.r.t. constant speed
• nly for the lucky jobs
• The service time is directly proportional to the service speed f
• The power consumption depends on the service speed as f α,

were 2 ≤ α ≤ 3

• Linear frequency scaling: The server speed is proportional

to the number of jobs in the system

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Related work (short list)

• Policies independent of the received service
• (George, Harrison): FCFS queues and frequency scaling
• (Wierman et al.): M/G/1/PS queues with frequency scaling
• Policies with known job size
• (Bansal et al.; Andrew et al.): Worst case analysis of SRPT

with frequency scaling

• (Andrew at al.; Elahi, Williamson): Unfairness in SRPT with

frequency scaling

• (Lassila; Aalto): LAS with sleeping servers

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Contribution

• We study a two-level queue with PS discipline and linear

speed scaling on the low-priority jobs (PS+IS)

• We give a numerical solution of the queueing system and

validate it with discrete event simulation

• We study the behaviour of the model with job size

distributions obtained by monitoring TCP flows of a data centre

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The queueing model and its solution

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Graphical representation

a X − a X Jobs with size < a Jobs with size ≥ a Infinite Server (IS) Processor Sharing (PS) Poisson(λ)

Note: The IS system works only when the PS is idle

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Job size distribution

• We use Generalized Hyperexponential (GH) distributions

fX(x) =

K

• k=1

pkµke−pkµk where K

k=1 pk = 1, pk ∈ R, µk ∈ R+, fX(x) > 0 for all

x ∈ R+

• GH distributions are dense in the domain of the distributions
• They can approximate any distribution arbitrary well

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Analysis of the queue: sketch

• We can see the system consisting of two queues:
• High priority one which is M/G/1/PS whose job sizes are

truncated at a

• Low priority one which works during the idle periods of the

PS which is a MB/G/∞ queue

• The arrival process is Poisson with intensity λ
• The batch size is the number of jobs that crossed the

threshold during a busy period of the PS level

• The generating function of the batch size distribution has

not an explicit form but has a characteristic equation (Kleinrock)

• The solution of the IS queue requires the distribution of the

batch size

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Computation of the batch size distribution

• We invert the generating function with the Lattice-Poisson

algorithm by Abate and Whitt

• The evaluation of the generating function is obtained with a

fixed point algorithm whose convergence is proved by resorting to Banach’s contraction mapping fixed point theorem

• The accuracy of the numerical procedure is validated in low

and heavy-load by comparing the first two moments of the distribution (which can be computed explicitly for GH distributions from the characteristic equation) with those

• btained by the numerical inversion

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Computation of the power consumption

• We resort to the literature for the power consumed by the PS

queue

• We provide a numerical solution for the IS queue and integer

values of the exponent α

• The power consumption is derived from the second (α = 2)

and third (α = 3) moments of the occupancy distribution in the IS queue

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Case study

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Dataset

• TCP flows monitored at the data centre of the Universit`

a Ca’ Foscari Venezia in November 2019

• Fitting with PH-Fit into an acyclic phase-type distribution
• Transformation of the acyclic phase-type distribution into a

GH

10-1 100 101 102 103 104 105 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1

(a) Probability density function in log-log scale.

10-1 100 101 102 103 104 105 106 107 108 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(b) Empirical and analytical cumulative density function in log-linear scale.

10-1 100 101 102 103 104 105 10-5 10-4 10-3 10-2 10-1 100

(c) Complementary cumulative density function in log-log scale.

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PS+IS vs. PS: Comparison of the expected response time

• PS queue has speed 1 and IS has speed f < 1

5 10 15 104 0.5 1 1.5 2 2.5 3 3.5 4

(a) Expected response time: ρPS = 0.85.

5 10 15 104 0.5 1 1.5 2 2.5 3 3.5 4

(b) Expected response time: ρPS = 0.92.

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PS+IS vs. PS: Comparison of the power consumption

0.5 1 1.5 2 2.5 3 3.5 4 104 0.7 0.75 0.8 0.85 0.9 0.95 1

(a) Power consumption: ρPS = 0.85 when 0 ≤ a ≤ 4 · 104.

0.5 1 1.5 2 2.5 3 3.5 4 104 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3

(b) Power consumption: ρPS = 0.92 when 0 ≤ a ≤ 4 · 104.

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PS+IS vs. PS: Slowdown

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0.5 1 1.5 2 2.5 3

(a) Slowdown of PS+IS conditioned to the job size x with ρPS = 0.85.

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0.5 1 1.5 2 2.5 3

(b) Slowdown of PS+IS conditioned to the job size x with ρPS = 0.92.

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PS+IS vs. PS: Comparison of the expected response times with same power consumption

5 10 15 104 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(a) Comparison of the expected response time with ρPS = 0.70 and f = 0.10.

5 10 15 104 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(b) Comparison of the expected response time with ρPS = 0.70 and f = 0.15.

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Simulation

• Simulation has been used to cross validate the numerical

results

• Simulation allows the investigation of other characteristics of

the system such as the distribution of the system speed

0.5 1 1.5 2 2.5 3

Speed

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Probability Speed of PS+IS with ;PS=0.70, a = 2 " 103 f = 0:05 f = 0:10 f = 0:15

(a) System speed: ρPS = 0.70.

0.5 1 1.5 2 2.5 3

Speed

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Probability Speed of PS+IS with ;PS=0.85, a = 2 " 103 f = 0:05 f = 0:10 f = 0:15

(b) System speed: ρPS = 0.85.

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Conclusion

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Conclusion

• We have introduced a two-level queueing system (PS+IS)

with linear speed scaling for the low-priority level

• A numerical solution procedure has been proposed and its

accuracy has been validated with discrete event simulation

• Experiments on real-world job size distributions have been

carried out

• We showed that the model-driven configuration of the PS+IS

system is crucial for obtaining the benefits of the speed scaling without compromising the slowdown of the system too much

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