CS457 - System Performance Evaluation - Winter 2008

Questions and Comments

1. Midterm: Q5e
2. Assignment 3

Lecture 21 - A Final Simulation

Processor Sharing

Time-slicing model: pre-emptive multi-tasking

For example, three classes of jobs

1. Jobs with active I/O: long think times, very little processing
2. Interactive jobs without active I/O: substantial processing that will come to an end before long
3. Batch jobs

Single server, three queues, needs a scheduling algorithm (discipline)

• Typical: try to provide some service to each
• Each queue has an importance, w(r), which means that it gets ... of the processor (Exercise for the reader: Fill in ... in a reasonable way.)
• In the example below each queue is of equal importance.

Important state

• number of non-empty queues, m
• number of jobs currently in each queue, N(r), r=1...M

Initialization

• The usual
• All queues empty
• One request (job) of each type in event-set

Arrival

• increment N(r)
• add request to queue
• if at head of queue,
  • increment m
  • reschedule existing departure events
    • one more queue shares the processor
    • scheduled departure events will occur later
    • multiply time remaining (dep-time - clock) by m/(m-1)
    • What about dividing by zero?
  • start-service

Departure

• remove job from queue, decrement N(r)
• if N(r) == 0
  • decrement m
  • reschedule existing departure events
    • one less queue shares the processor
    • scheduled departure events will occur earlier
    • multiply time remaining (dep-time - clock) by m/(m+1)
    • What about multiplying by zero?
• else
  • start-service

Start-service

• work out departure time
  • clock + service-time * m
• schedule departure event

Something is unrealistic about this model. What is it?

• no rescheduling within a queue
• A3 takes that into account

Something is unintuitive about this model. What is it?

• I have been justifying this approach in terms of implementation simplicity and efficiency.
• But these qualities are starting to recede as things get more complex. For example, queue-dependent importance.
• This is always the case.

Random Numbers

Text: chapter 12 & 29

Random Variables

The mathematical foundation for random number is the random variable, which is a variable that has different values on different occasions.

1. A probability distribution determines what values a random variable gets, and how often
2. Discrete versus continuous random variables
3. We usually assume that the random variables we want are independent and identically distributed (iid or IID).
   • Why is this assumption reasonable?
   • Think about where we get the distributions from.
4. Most random variables we use have underlying distributions that are non-negative.
   • Why?
5. Random variables are normally written as capitals.

Random variable concepts

1. Cumulative distribution function (cdf or CDF):
   • F(x) = P(X <= x)
   • F(0) = 0
   • F(infinity) = 1
2. Probability density function (pdf or PDF):
   • Continuous distributions only
   • F(x) = \int_0^x f(x') dx'
   • Mean: \mu = E(X) = \int_0^\infinity x'f(x')dx'
   • Variance: \sigma^2 = X((X-E(X))^2) = \int_0^\infinity (x'-E(X))^2 f(x')dx'
3. Probability mass function (pmf or PMF):
   • Discrete distributions only: x takes a value fro a countable set {x1, x2, x3, ...} where ... may or may not terminate
   • P(X = xi) = pi
   • \sum_i pi = 1
   • F(x) = \sum_(xi<=x) pi
   • Mean: E(X) = \sum_i xi pi
   • Variance: \sum_i (xi - E(X) )^2 pi

In discrete event simulation we model quantities like interarrival time or service time as IID random variables

• As a result we need to generate random variables from random number generators, which are discussed in a future lecture.

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