Queuing Theory and Simulation: Lab Assignment 1 Overview

1. Lab Assignment 1COP 4600: Operating Systems Principles Dr. Sumi Helal Professor Computer & Information Science & Engineering Department University of Florida, Gainesville, FL 32611 helal@cise.ufl.edu

2. Lecture Overview • Go over Lab Assignment 1, one more time. • Queuing Theory 101 • Simulation 101

3. Assignment • Simulate a single queue/single server system, with a FIFO queuing discipline • Report on the performance of the system • Compare with analytic models. • λ = arrival rate, follows an arrival process • μ = service rate, follows a service process • ρ = utilization = λ/μ μ λ

4. 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

5. Exponential Process • Suitable for describing time between successive events (e.g., arrival, service). T is a continuous Random Number

6. 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:

7. Example

8. 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.

9. 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.

10. Definitions • W = Average job wait time in the queue • L = Average queue length • N = Throughput (number of jobs completed per unit time)

11. 3 3 Time in System (W) 2 2 1 1 1 2 3 Job# (N) Little’s Law: • Proof: • Shaded area is identical (=9 in example) # in System (L) 1 2 3 4 5 6 7 Time (T)

12. Analytic Solutions • Utilizing Little Law • Utilization: • L: • W: • 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)

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

14. 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.

15. 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.

16. 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 .