CS457 - System Performance Evaluation - Winter 2010

Public Service Announcements

  1. The weeks ahead
  2. Issues out of which on which exam questions could be based.

Lecture 24 - Entia non sunt multiplicanda praeter necessitatem

Random Number Detritis

The Truth about Random Number Generation

How good do they have to be?

Look carefully at how your simulation uses random numbers

What type of correlations are you likely to care about? For example,

Never send a man to do a boy's job.

Simulation Detritis

Generalize Server

Simplify Server

Eliminate Scheduler


Analytic Queueing Theory

Concepts We Know


With some change in notation.

  1. \lambda - arrival rate

    1/\lambda - mean interrival time

  2. 1/\mu - mean service time

    \mu - service rate

Carefully distinguish between

Stable system

Throughput, X = \lambda, arrival rate


U = \lambda / \mu

Stability requires \lambda <= \mu

Little's Law

XE(r) = E(n)

E(r) - mean response time

E(n) - mean number of jobs in the system

Note the change in notation

New Concepts

Stochastic process

A sequence of random variables indexed by time: S0, S1, S2, ...

Markov process

A stochastic process in which S(n+1) is independent of S(0), ..., S(n-1), but may depend on S(n).

In performance evaluation, when we talk of a Markov process we usually also mean that the next transition occurs at a time distributed by an exponential distribution. Why? (You should already know this.)

Another useful property of the exponential distribution

Birth-death process

A special process where only transitions to neighbouring states are possible That is, if we are in S(j) then the next state can be

  1. S(j-1), a death occurs.

    Death rate \mu_j.

  2. S(j+1), a birth occurs

    Birth rate \lambda_j.

  3. S(j), neither a birth nor a death occurs, or both a birth and death occur.

We now want to examine what happens, in a birth-death process, in the short time between t and t + \Delta t. Use the Poisson distribution

  1. P(exactly one birth ) = (\lambda \Delta t) * exp(-\lambda \Delta t) = \lambda \Delta t + terms that are quadratic or higher order in \Delta t
  2. P(exactly one death ) = \mu \Delta t + o( (\Delta t)^2 )
  3. P(exactly zero births ) = exp(-\lambda \Delta t) = 1 - \lambda \Delta t + o( (\Delta t)^2 )
  4. P(exactly one birth ) = 1 - \mu \Delta t + o( (\Delta t)^2 )
  5. P(more than one birth and/or death) = o( (\Delta t)^2 )

Exercise for the reader. Explain how it is that this is equivalent to only Arrival and Departure events affecting the system state.

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