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CS 241 Spring 2007 System Programming 1 Queuing Framework for Process Management Evaluation Lecture 20 Klara Nahrstedt

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2 CS241 Administrative Read Stallings Chapter 9 No Quizzes this week, the next quiz will be on Monday 3/12 on SMP5

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3 Content of This Lecture Goals: Introduction to Principles for Reasoning about Process Management/Scheduling Things covered in this lecture Introduction to Queuing Theory

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4 Process States Finite State Diagram

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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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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21 Queueing Model Random Arrivals and the Poisson Distribution Elements of model

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22 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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23 Discussion What assumption have you made about the bus?

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24 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 7 8

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25 Hamburger Problem 7 Hamburgers arrive on average every time unit 8 Hamburgers are processed by Joe on average every unit 1) How long is a hamburger waiting to be eaten? (Do they get cold?) Ans = 7/8 time units 2) How many hamburgers are waiting in queue to be serviced? Ans = 49/8 Queue 7 8

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

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27 Queuing Theory ARRIVAL RATE ARRIVAL RATE SERVICE RATE Input Queue Server Single Server Queue

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28 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 1 t Av λ 0.5

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29 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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30 Useful Facts From Queuing Theory W q = mean time a customer spends in the queue = arrival rate L q = W q 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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31 Analysis of Single Server Queue Server Utilization: Time in System: Time in Queue: Number in Queue (Little):

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32 Example: How busy is the server? λ =2 μ=3 = 2/3 or 66% Busy

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33 How long is an eater in the system? λ =2 μ=3 = 1/(3-2)= 1

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34 How long is someone in the queue? λ =2 μ=3

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35 How many people in queue? λ =2 μ=3

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36 How many people in queue? λ =2 μ=2

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37 Interesting Fact If Arrival Rate = Service Rate, the queue length become infinitely large the longer you run the model.

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38 Until Now We Looked at Single Server, Single Queue Theory ARRIVAL RATE ARRIVAL RATE SERVICE RATE Input Queue Server

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39 Poisson Arrivals Sum ARRIVAL RATE 1 SERVICE RATE Input Queue Server ARRIVAL RATE 2 = 1+ 2 = 1+ 2

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40 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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41 Example Arrival 1 jobs/sec from Start Arrival 2 jobs/sec from Event queue Service 4 jobs/sec –Utilization=ρ=λ/μ=1+2/4=.75 –Time in system=1/(μ-λ)=1 –Time in queue=ρ/(μ-λ)=.75 –Length of queue=ρ*ρ/(1-ρ)=2.25

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42 As long as it’s a Poisson Distribution... ARRIVAL RATE ARRIVAL RATE SERVICE RATE 1 Input Queue Server Server SERVICE RATE 2 Combined = 1+ 2

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43 Question: MacDonalds Problem μ μ μ μ μ μ λ λ λ λ λ λ A) Separate Queues per Server B) Same Queue for Servers If W is waiting time for system A, and W is waiting time for system B, what is W/W B ? Integer answer; W A > W B ? If W A is waiting time for system A, and W B is waiting time for system B, what is W A /W B ? Integer answer; W A > W B ?

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44 Queuing Diagram for Processes Start Event Queue Exit Time Slice ARRIVAL RATE ARRIVAL RATE SERVICE RATE SERVICE RATE 1 ARRIVAL RATE 1

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45 Queuing Diagram for Processes Ready Queue I/O request in Device Queue Time Slice Expired Fork a Child Wait for an InterruptInterruptOccurs CreateChild CPU I/O UpdateAccounting Create Job

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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 Process Next Event on Queue Put new events on queue

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47 Summary Simulation Models of Scheduling Using Queuing Theory Average response time and variance important Simulation of Scheduling SS: Ch 9[405-422,428-435]

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