Day 3 Agenda for Today Formulate simple problem statement - - PowerPoint PPT Presentation

Day 3

Agenda for Today

Formulate simple problem statement Revisit the workload characterization problem. Present detailed (step by step) derivation of the workload.

Workload Modeling

• Workload modeling is used to generate synthetic

workloads based on real-life job execution observations.

• The goal is typically to be able to create workloads that

can be used in performance evaluation studies

Workload Submission

Workload description (may include QoS)

Workload Generation

J1 J1 Jn

Analysis SLA

Consolidate result

Negotiation

Problem Formulation

• Given 𝑄 number of available processors and m jobs,

𝒦 = 𝐾1, 𝐾2, ⋯ , 𝐾𝑛 , waiting in a queue to be processed

• Problem - allocate 𝑞𝑘 processors to job 𝐾𝑗 such that the overall

execution time is minimized

𝑛𝑗𝑜𝑛𝑗𝑡𝑓 𝑈

𝑘 𝑞𝑘 , 𝑛 𝑘=1

Subject to 𝑞𝑘 ≤ 𝑄, 𝑞𝑘∈ *1,2, … , 𝑞𝑛𝑏𝑦+

𝑛 𝑘=1

execution will starts only when 𝑞 = 𝑞𝑛𝑏𝑦 allocated – 𝑈

𝑘 𝑜 be the execution time function of job j,

– 𝑞𝑛𝑏𝑦 the maximum parallelism job j can have, – 𝑞𝑘 is the unknown processor allocation to job j, and

Problem Formulation

• Assume that we have four jobs 𝒦 = J1, J2, J3, J4 .
• For simplicity assume that each job request 3 processors

and the service demand of each job is 20 time units.

• Suppose we have 𝑄 = 7 available homogenous processors to

be assigned to the 4 jobs.

• The assignment of the processor to the jobs must minimize

the overall completion time of the jobs.

• Assume only space sharing allocation

Allocation Possibility

• One possible assignment is
• J1 = 3,
• J2 = 3
• J3 = 1
• J4 = 0

20 Time J1, J2 com completed sta tarted J1, J2 J3, J4 40 J3, J4 Initialization say 3 time units

• The total execution time is 43 units.
• Can we do better?

Job Description

• We have a set of jobs to be executed

𝒦 = 𝐾1, 𝐾2, ⋯ , 𝐾𝑜

• Number of tasks per job

– Each job 𝐾𝑗 has a set of tasks

𝐾𝑗 = 𝑈

1, 𝑈2, ⋯ , 𝑈 𝑛

– Note that a task represents a part of the work that must be done serially by a single processor – Interdependence among job tasks is important to consider

• A job is said to be “small” (or “large”) if it consists of

a small (or large) number of tasks.

• Tasks with a large service demand may introduce

large queuing delays for queued jobs.

Job Description

• A job is said to be “small” (or “large”) if it consists of a

small (or large) number of tasks.

• Based on he analysis of real workload logs used in

production

– the percentage of small jobs, with a small number of tasks, is higher than large jobs, with a large number of tasks. – For this reason, we examine the following distribution for the number of tasks per job.

• Tasks with a large service demand may introduce large

queuing delays for queued jobs.

Large jobs

Small jobs

Job Description

• A job is completely described by the following

parameters:

– Cumulative job service demand (W) – The arrival time – The number of task

• A job with one task is called a sequential job

and a job with multiple tasks is called parallel jobs.

Workload Modeling

• We will consider the following

workloads

– WK1 consists of curves with relatively good speedup. – WK2 consists of curves with not as good speedup as W1. – WK3 consists of curves with poor speedup. – WK4 contains jobs with all three speedup types, each appearing with approximately equal frequency.

Workload Generation

• Sevcik proposed the following model to represent

the execution time function of a job that can run on 𝑞 processors:

𝑈 𝑞 = 𝜚(𝑞) 𝑋 𝑞 + 𝛽 + 𝛾 ∙ 𝑞

• It has been shown that a wide range of

representative applications can be modeled by utilizing the above execution time function.

Workload Generation

• The execution time function captures both the scale

up and the overhead

𝑈 𝑞 = 𝜚(𝑞) 𝑋 𝑞

𝑡𝑑𝑏𝑚𝑓𝑣𝑞

+ 𝛽 + 𝛾 ∙ 𝑞

𝑝𝑤𝑓𝑠ℎ𝑓𝑏𝑒

• The runtime function allows to create different

workloads by choosing different values for the parameters: φ, W, β, and α.

Workload Generation

• There is certain level of load imbalance when

running a job on multiple processors.

• 𝜚(𝑞) parameter in the equation

𝑈 𝑞 = 𝜚(𝑞) 𝑋 𝑞 + 𝛽 + 𝛾 ∙ 𝑞 – 𝜚(𝑞) represents the degree to which the work is not evenly spread across the p processors (i.e., load imbalance) – Real measurements conducted by Wu shows that its value is in the range of: 1.1 ≤ 𝜚(𝑞) ≤ 1.2 – Therefore, φ(p) can be considered equal to 1.0.

Workload Generation

Note that adding processors to a job reduces computation time but there is certain level of increases in completion time due to sequential execution

• 𝛽 parameters in the equation captures the above

𝑈 𝑞 = 𝜚(𝑞) 𝑋 𝑞 + 𝛽 + 𝛾 ∙ 𝑞 – 𝛽 represents the increase of the work per processor due to parallelization (i.e., overhead)

Workload Generation

• Note that adding processors to a job reduces

computation time but increases communication time

• 𝛾 parameter in the equation captures the above

issue

𝑈 𝑞 = 𝜚(𝑞) 𝑋 𝑞 + 𝛽 + 𝛾 ∙ 𝑞 – 𝛾 represents the communication and congestion delays that increase with the increase in the number of processors assigned to job. – What this says is that the more the number of processors given to a job the higher the communication cost and congestion delays

Workload Generation

• 𝑋 in the runtime function represents the total

service demand of the job

𝑈 𝑞 = 𝜚(𝑞) 𝑋 𝑞 + 𝛽 + 𝛾 ∙ 𝑞 – The mean value 𝑋 is 13.76 and the coefficient of variation must be greater than one (e.g., 3.5, 10.0).

Large jobs

Small jobs

average service demand = 1.3 and account for 7/8

• f the jobs in the system

average service demand = 101 and accounts for 1/8 jobs in the system

Workload Speed Up

• We can determine the speed up of the application
• n p processors as follows:

𝑇 𝑞 = 1 + 1 𝑞𝑛𝑏𝑦 2 + 1 𝑞𝑛𝑏𝑦 2

𝜈

1 𝑞 + 𝑞 𝑞𝑛𝑏𝑦 2 + 1 𝑞𝑛𝑏𝑦 2

𝜈

• Where

– 𝜈 ∈ ∞, 0.2,0.4 – 𝑞𝑛𝑏𝑦 is the maximum number of processors assigned to the workload (WK1, WK2, WK3).

Workload Speed Up

• We can determine the speed up for WK3 with

𝜈 = 0.2 and 𝑞𝑛𝑏𝑦=1,4,6,9, p=32

𝑇 𝑞 = 1 + 1 𝑞𝑛𝑏𝑦 2 + 1 𝑞𝑛𝑏𝑦 2

𝜈

1 𝑞 + 𝑞 𝑞𝑛𝑏𝑦 2 + 1 𝑞𝑛𝑏𝑦 2

𝜈

• We can substitutive the above

𝑇 𝑞 = 1 + 1 4 2 + 1 4 2

𝜈

1 32 + 32 4 2 + 1 4 2

𝜈

Workload Speed Up

• The results for the speedup curve 𝑞𝑛𝑏𝑦 = 4

Workload Speed Up

• The results for the speedup curve when

𝑞𝑛𝑏𝑦 = 16

Workload Speed Up

• The results for the speedup curve when

𝑞𝑛𝑏𝑦 = 64

Workload WK1

• Consists of curves with relatively

good speedup.

• They correspond to 𝜈 = +∞
• Example is matrix multiplication

application

• The number of possible tasks is

given by 𝑜 = 𝑋 𝛾

• β reflects the communication

and congestion delays that increase with the number of processors.

Task 1 Task 2 Task 3 Task n-1 Structure of matrix multiplication application Task n

Workload WK1

• The actual job service demand value is obtained

from a two-stage hyper-exponential distribution depending on the coefficient of variation of the service time.

𝑔 𝑥 = 𝑄 1 − 𝑓

𝑋 101 𝑚𝑏𝑠𝑕𝑓 𝑝𝑐𝑡

+ 1 − 𝑄 1 − 𝑓

𝑋 1.3 𝑡𝑛𝑏𝑚𝑚 𝑘𝑝𝑐𝑡

Workload WK1

• The actual job service demand value is obtained

from a two-stage hyper-exponential distribution depending on the coefficient of variation of the service time.

𝑔 𝑥 = 𝑄 1 − 𝑓

𝑋 101 + 1 − 𝑄

1 − 𝑓

𝑋 101

Where – 𝑄 = 0125 – The mean value of 𝑋 is 13.76

Workload WK1

• The service demand of WK1 can now be computed as

𝑔 𝑥 = 0.125 1 − 𝑓 −13.76

101

+ 1 − 0.125 1 − 𝑓

−13.76 1.3

Workload WK2

• They correspond to 𝜈 = 0.4,

which indicate that its speedup is not as good as WK1

• Example: n-body simulations of

stellar or planetary movements, in which the movement of each body is governed by the gravitational forces produced by the system as a whole

• The number of possible tasks is

given by

𝑜 = 𝑋 𝛾

Workload WK3

• They correspond to 𝜈 = 0.2,

which indicate that its speedup is poorer than WK1 and WK2

• Example: Mean value analysis
• kinds of applications exhibit this

property.

• The number of possible tasks is

given by

𝑜 = 𝑋 𝛾

Workload WK3

• They correspond to 𝜈 = 0.2,

which indicate that its speedup is poorer than WK1 and WK2

• Example: Mean value analysis
• kinds of applications exhibit this

property.

• The number of possible tasks is

given by

𝑜 = 𝑋 𝛾

The Arrival Process

• Jobs arrive to the system in stochastic manner

J1 J2 J3 J4

• An exponential distribution with the mean

inter-arrival time can be derived as follows:

𝑔 𝑦 = 𝜇 ∙ 𝑓−𝜇∙𝑦

• where

1 𝜇 = 𝐹 𝑈 1 𝑂 ∗ 𝑀𝑝𝑏𝑒

The Arrival Process

• We already know that E(W) = 13.76.

𝐹 𝑈 1 = 𝐹 𝑋 + 𝐹 𝛾 + 𝐹 𝛽

• By using the theorem of total expectation across the

three different values of 𝑞𝑛𝑏𝑦, we find E(𝛾) = 0.30

𝐹 𝛽 = 0.0 𝑗𝑔 𝜈 = +∞ 2.0 𝑗𝑔 𝜈 = 0.4 5.0 𝑗𝑔 𝜈 = 0.2 2.3 mixed speedup

Today Lab

• We have seen the cloudlet in CloudSim. It just

runs with fixed values. We want to change it.

• Task 1: change the constant value to make the

arrival process based on exponential distributions

• Task 2: Currently the execution time of the

cloudlet is fixed. Change it to be generated using 2-stage hyper-exponential distribution.

32

Thank you.

Questions, Comments, …?

Exponential Distribution

• The Exponential:

– 𝜇 = measures how many things happen per unit 𝑔 𝑦 = 𝜇 ∙ 𝑓−𝜇∙𝑦

0.5 1 1.5 2 2.5 1 2 3 4 5 6 7 probability density functions X

Lambda

0.5 1 2

Hyper-exponential distribution

• Hyper-exponential distributions is used in

service demand modeling

• The hyper-exponential distribution is obtained

by selecting from a mixture of several exponential distributions.

• The simplest variant has only two stages:

𝜇1 𝜇2 𝑄 1 − 𝑄 𝜇1 ≠ 𝜇2

𝑔 𝑦 = 𝑞𝑗 ∙ 𝜇𝑗 ∙ 𝑓𝜇𝑗𝑦

𝑜 𝑗=1

0 ≤ 𝑞𝑗 ≤ 1 0 ≤ 𝜇1, 𝜇2𝑗 ≤ 1

Coefficient of Variation

• Coefficient of variation is defined as follows

𝐷𝑤 = 𝜏 𝜈

• Where

– 𝜏 is the standard deviation – 𝜈 is the mean

Long-lived vs short-lived Jobs

• Long-lived processes
• short-lived processes

𝑄 𝑈 > 𝜐 ∝ 𝛽2, 𝛽 ≈ 1

• This means that most processes are short, but

a small number are very long.

Coefficient of Variation

• Coefficient of variation is defined as follows

𝐷𝑤 = 𝜏 𝜈

• Where

– 𝜏 is the standard deviation – 𝜈 is the mean

Performance Measurements

• Classical performance metrics:

– Response time, – throughput, – scalability, – resource/cost/energy, – efficiency, – Elasticity – Availability, – reliability, and – security – SLA violation

References

• Thyagaraj Thanalapati, Sivarama P. Dandamudi:

An Efficient Adaptive Scheduling Scheme for Distributed Memory Multicomputers. IEEE Trans. Parallel Distrib. Syst. 12(7): 758-768 (2001)

• A. Iosup and D.H.J. Epema, Grid Computing

Workloads, IEEE Internet Computing 15(2): 19-26 (2011)

• Feitelson DG. Workload modeling for computer

systems performance evaluation. Cambridge University Press; 2015.