Queuing Framework for Process Management Evaluation

1
Queuing Framework for Process Management
Evaluation

• Lecture 20
• Klara Nahrstedt

2
CS241 Administrative

• Read Stallings Chapter 9
• No Quizzes this week, the next quiz will be on
  Monday 3/12 on SMP5

3
Content of This Lecture

• Goals
• Introduction to Principles for Reasoning about
  Process Management/Scheduling
• Things covered in this lecture
• Introduction to Queuing Theory

4
Process States Finite State Diagram
5
Queuing Models and Simulation

• Problem How do we
• size the ready queue, size any queue?
• decide how many jobs should be accepted?
• analyze scheduling algorithms?
• Decide how long we should wait for a job?
• Goals
• Simple arithmetic (back of envelope
  calculation) to calculate system behavior
• Basis for more complex analysis
• Approach to study systems too complex to produce
  simple mathematical model

6
CPU Scheduler Example

• Which is better Round Robin or FIFO?
• How long does A take to go thru system?
• How big should be the ready queue?
• Mean, Mode, Variance?

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Queuing Diagram for Processes
Exit

Start

Ready Queue
CPU
Time Slice
Event
Event Queue
8
Queuing Diagram for Processes
Exit

Start

Ready Queue
CPU
Time Slice
Event
Event Queue
9
Queuing Diagram for Processes
Exit

Start

Ready Queue
CPU
Time Slice
Event
Event Queue
10
Queuing Diagram for Processes
Exit

Start

Ready Queue
CPU
Time Slice
Event
Event Queue
11
Queuing Diagram for Processes
Exit

Start

Ready Queue
CPU
Time Slice
Event
Event Queue
12
Queuing Diagram for Processes
Exit

Start

Ready Queue
CPU
Time Slice
Event
Event Queue
13
Queuing Diagram for Processes
Exit

Start

Ready Queue
CPU
Time Slice
Event
Event Queue
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Queuing Diagram for Processes
Exit

Start

Ready Queue
CPU
Time Slice
Event
Event Queue
15
Queuing Diagram for Processes
Exit

Start

Ready Queue
CPU
Time Slice
Event
Event Queue
16
Queuing Diagram for Processes
Exit

Start

Ready Queue
CPU
Time Slice
Event
Event Queue
17
Queuing Diagram for Processes
Exit

Start

Ready Queue
CPU
Time Slice
Event
Event Queue
18
Queuing Diagram for Processes
Exit

Start

Ready Queue
CPU
Time Slice
Event
Event Queue
19
Queuing Diagram for Processes
Exit

Start

Ready Queue
CPU
Time Slice
Event
Event Queue
20
Queuing Diagram for Processes
Exit

Start

Ready Queue
CPU
Time Slice
Event
Event Queue
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Queueing Model

• Random Arrivals and the Poisson Distribution
• Elements of model

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Discussion

• If a bus arrives at a bus stop every 15 minutes,
  how long do you have to wait at the bus stop
  assuming you start to wait at a random time?

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Discussion

• What assumption have you made about the bus?

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Hamburger Problem

• 7 Hamburgers arrive on average every time unit
• 8 Hamburgers are processed by Joe on average
  every unit
• 1) Av. time hamburger waiting to be eaten? (Do
  they get cold?) Ans ????
• 2) Av number of hamburgers waiting in queue to be
  eaten? Ans ????

Queue
8
7
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Hamburger Problem

• 7 Hamburgers arrive on average every time unit
• 8 Hamburgers are processed by Joe on average
  every unit
• How long is a hamburger waiting to be eaten? (Do
  they get cold?) Ans 7/8 time units
• How many hamburgers are waiting in queue to be
  serviced? Ans 49/8

Queue
8
7
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Random Events

• Poisson Distribution
• Each event independent of other events
• Mean event rate, SD is same as mean
• Exponential distribution

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Queuing Theory
Single Server Queue
ARRIVAL RATE ?
Server


Input Queue
SERVICE RATE?
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Queueing Theory (Point of Interest)

• Steady state
• Poisson arrival with ? constant arrival rate
  (customers per unit time) each arrival is
  independent.
• P( ??t ) 1- e?t

0
0.5
1
t
Av ?
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Analysis of Queueing Behavior

• Probability n customers arrive in time interval t
  is
• e?t ??t? n/ n!
• Assume random service times (also Poisson) ?
  constant service rate (customers per unit time)

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Useful Facts From Queuing Theory

• Wq mean time a customer spends in the queue
• ? arrival rate
• Lq ? Wq number of customers in queue
• W mean time a customer spends in the system
• L ? W ( Little's theorem) number of customers
  in the system
• In words average length of queue is arrival
• rate times average waiting time

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Analysis of Single Server Queue

• Server Utilization
• Time in System
• Time in Queue
• Number in Queue (Little)

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Example How busy is the server?
µ3
?2
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How long is an eater in the system?
µ3
?2
1/(3-2) 1
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How long is someone in the queue?
µ3
?2
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How many people in queue?
µ3
?2
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How many people in queue?
µ2
?2
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Interesting Fact

• If Arrival Rate Service Rate, the queue length
  become infinitely large the longer you run the
  model.

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Until Now We Looked at Single Server, Single
Queue Theory
ARRIVAL RATE ?
Server


Input Queue
SERVICE RATE?
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Poisson Arrivals Sum
ARRIVAL RATE ?1
Server


Input Queue
SERVICE RATE?
ARRIVAL RATE ?2
? ?1 ? 2
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Example

• Arrival 1 jobs/sec from Start
• Arrival 2 jobs/sec from Event queue
• Service 4 jobs/sec
• Utilization?
• Time in system?
• Time in queue ?
• Length of queue?

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Example

• Arrival 1 jobs/sec from Start
• Arrival 2 jobs/sec from Event queue
• Service 4 jobs/sec
• Utilization??/µ12/4.75
• Time in system1/(µ-?)1
• Time in queue?/(µ-?).75
• Length of queue??/(1-?)2.25

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As long as its a Poisson Distribution...
SERVICE RATE?1
ARRIVAL RATE ?
Combined ??1?2

Input Queue
SERVICE RATE?2
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Question MacDonalds Problem
µ
µ
?
?
µ
µ
?
?
µ
?
?
µ
A) Separate Queues per Server
B) Same Queue for Servers
If WA is waiting time for system A, and WB is
waiting time for system B, what is WA/WB?
Integer answer WA gt WB ?
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Queuing Diagram for Processes
Exit

Start

ARRIVAL RATE ?
SERVICE RATE ?
Time Slice
SERVICE RATE?1
ARRIVAL RATE ?1
Event Queue
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Queuing Diagram for Processes
CPU
Create Job
Ready Queue
I/O request in Device Queue
I/O
Time Slice Expired
Update Accounting
Fork a Child
Create Child
Wait for an Interrupt
Interrupt Occurs
46
Simulations instead of maths

• Complicated logic and conditions about events-
  Write program that simulates events

While not end
Advance Clock to Next Event
Print results
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Summary

• Simulation Models of Scheduling
• Using Queuing Theory
• Average response time and variance important
• Simulation of Scheduling
• SS Ch 9405-422,428-435