Lab Assignment 1COP 4600: Operating Systems Principles

Dr. Sumi HelalProfessor

Computer & Information Science & Engineering Department

University of Florida, Gainesville, FL 32611helal@cise.ufl.edu

Lecture Overview

• Go over Lab Assignment 1, one more time.

• Queuing Theory 101

• Simulation 101

Assignment

• λ = arrival rate, follows an arrival process

• μ = service rate, follows a service process

• ρ = utilization = λ/μ

λ μ

• Simulate a single queue/single server system, with a FIFO queuing discipline

• Report on the performance of the system

• Compare with analytic models.

Queuing Theory 101

• Must already know: – Random Variables– Basics of Probability

• Today, we will study & learn: – Probabilistic Processes– Little Law – M/M/1 analytic models

Exponential Process

• Suitable for describing time between successive events (e.g., arrival, service).

Time between Events

Pro

bab

ilit

y

€

P T > t( ) = e−λt T is a continuous Random Number

€

1

λ

Example

• Assume average time between arrival (or average inter-arrival time) is 45 sec.– Question: what is the prob. that inter-arrival

time is > 60 sec.?– Answer:

€

1

λ=1

45= 0.0222

P T > 60( ) = e−0.0222×60

= 0.2636

2636.060 TP

€

1

λ

Example

Poisson Process

• Poisson is suitable for describing arrivals or occurrence of events.

• Describes prob. of n arrivals in any time interval.

• If arrival process follows Poisson distribution, then the random variable representing inter-arrival time must follow the Exponential distribution.

€

Pt n( ) =λ t( )

ne−λt

n!

Quiz

• To make sure you follow so far, answer the following question:

– Prove that the probability that inter-arrival times are greater than the average inter-arrival time (that is > 1/λ), is 0.37, for any exponential distribution.

Definitions

• W = Average job wait time in the queue

• L = Average queue length

• N = Throughput (number of jobs completed per unit time)

Little’s Law:

€

L = λ •W

• Proof: – Shaded area is identical (=9 in

example)

1 2 3

Time inSystem(W)

Job# (N)

123

1 2 3 4 5 6 7

# in System(L) 1

23

Time (T)

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lii=1

T

∑ = τ jj=1

N

∑

lii=1

T

∑NT

=τ jj=1

N

∑NT

L

N=W

T

L =N

T•W

L = λ •W

Analytic Solutions

• Utilizing Little Law

• Utilization:

• L:

• W:

€

ρ =λμ

€

ρ1− ρ

=λ

μ − λ

€

1

μ − λ

• Quiz to check if you understand the implication of ρ• Calculate L and W for

ρ=0.09 (system under-utilized)

• Calculate the same for ρ=0.90 (system highly utilized)

• Calculate the same for ρ=0.999 (system over-utilized)

Effect of ρ – A Reality that Must be considered in any Operating System Design

λμ−= 1W

Simulation 101

• You have two independent events• At end of processing an independent event,

you must re-generate it. • All future events generated should be put in

an event list.• Simulation loop simply finds the next event

that will take place sooner in the future; remove it & process it. And yes, advance the clock to that selected next event.

Simulation 101

• At each new iteration in the simulation loop you check for exist criterion.

• You most update your counters and statistics every time: – The Clock is changed– A new job enters the system– A job exits the system– When the simulation loop exits.

Simulation 101

• Generating exponentially distributed random variables:– Use inverse inverse transform sampling as

follows: • X is RV with standard Uniform distribution [0,1],

then follows the exponential distribution with average arrival rate .

€

−ln(X) /λ

€

λ