CS457 - System Performance Evaluation - Winter 2010

Public Service Announcements

1. Mid-term Examination: February 23rd, 16.30 at 18.00 in PHY145.
2. Assignment 3.

Lecture 19 - A Final Simulation Example

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 stop and start at widely spaced times
3. Batch jobs: which go on for very long times.

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

• Typical scheduling algorithm: try to provide some service to each
• For example, round-robin scheduling
  • N requests go into N queues
  • When one request has used up its current atom
    • put it back into its queue, and
    • put in the request in the next queue.
  • Abstraction
    • service time for each request is a large number of atoms of time
    • number of atoms a request gets per second depends on the number of requests being serviced
    • therefore number of seconds required is
      serv-time [atoms] / avail-atoms [atoms/sec] = serv-time [atoms] / ( (1/N) * unit [atoms/sec] )

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 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 = E((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 {x_1, x_2, x_3, ...} where ... may or may not terminate
   • P(X = x_i) = p_i
   • \sum_i p_i = 1
   • F(x) = \sum_(x_i<=x) p_i
   • Mean: E(X) = \sum_i x_i p_i
   • Variance: \sum_i (x_i - E(X) )^2 p_i

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.

• Random number generators normally produce random floating point numbers uniformally distributed between 0 and 1. (Technically, in the range [0.0..1.0).)
• We need to know how to transform this output into the distribution we want.

Random variables with arbitrary distributions

A random variable is uniquely defined by its CDF, F(x).

• 0 <= F(x) <= 1
• Suppose we want to express this random variable as a transformation Y = g(X) of a uniformly distributed random variable.
• P(Y < y) = F_y(y) = P(X < g^-1(y) ) = F_x( g^-1(y) ).
• Choose g(x) = F_x(x). Then P(Y < y) = 1/y, or f_y(y) = 1/(length of interval). Y is uniformly distributed.

Examples of Discrete Distributions

1. Bernoulli
2. Discrete uniform
3. Geometric Discrete
4. Binomial

Poisson

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