CS457 - System Performance Evaluation - Winter 2010

Public Service Announcements

The weeks ahead
- class on assignment 3
- possible tutorial
Issues out of which on which exam questions could be based.
- anything in the notes with a question mark
- exercises for the reader

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

the parts of your program that populate the event set

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

lightly loaded single queue/single server
two types of events
Common sequences
- ADADADAD
- AA
- DD

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

Simulation Detritis

Generalize Server

several outputs rather than just one
- outputs are linked to
  - other servers, called targets, or
  - departure from the system
output chosen depends on
- request type
- chance (a polite way of saying, `I don't know.')
- possibly other aspects of the system state, such as
  - load of possible targets
- varies from instance to instance

Simplify Server

Every server has only one queue

Eliminate Scheduler

The generalized/simplified server described above can do scheduling.
Exercise for the reader. How?

Consequences

For the most part servers differ only by one table

Server Configuration Table
	Target 0	Target 1	...	Target M
Type 0	p00	p01		p0M
Type 1	p10	p11		p1M
...
Type N	pN0	pN1		pNM

You would then create a system by filling in one such table for each server
A more flexible alternative gives the server a function pointer or specially-named method
- It uses such a configuration table, and
- is able to take other aspects of system state into account.

Analytic Queueing Theory

Concepts We Know

Parameters

With some change in notation.

\lambda - arrival rate
1/\lambda - mean interrival time
1/\mu - mean service time
\mu - service rate

Carefully distinguish between

Throughput - X - metric - jobs done per second
Service rate - \mu - parameter - jobs that can be done per second when the server is processing

Stable system

Throughput, X = \lambda, arrival rate

for the steady state phase of processing
analysing data
- Wait until after the starting transient phase is finished
- Stop before the ending transient phase begins

Utilization

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, ...

E.g. state of a system at time t. Why?

Markov process

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

We could say something like, "The state of the system incorporates the past completely."
Discrete event simulation implicitly assumes that the system is Markovian.

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.)

In a Markov process with exponentially distributed transitions the mean transition rate is the only parameter that can vary with state.
That is, if in state S(j) then the transition rate is \lambda_j

Another useful property of the exponential distribution

Y1 - arrival time - exponential - \lambda
Y2 - service time - exponential - \mu
Ym - when the first of an arrival or end of service occurs - Ym = min(Y1, Y2)
P(Ym < y) = P(Y1 < y AND Y2 < y) = F1(y) * F2(y) = exp(-\lambda y) * exp( -\mu y) = exp( -(\lambda + \mu) * y )
Ym is exponentially distributed with mean 1 / (\lambda + \mu)
Why does this matter?

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

S(j-1), a death occurs.
Death rate \mu_j.
S(j+1), a birth occurs
Birth rate \lambda_j.
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

P(exactly one birth ) = (\lambda \Delta t) * exp(-\lambda \Delta t) = \lambda \Delta t + terms that are quadratic or higher order in \Delta t
- Write `terms that are quadratic or higher order in \Delta t' as o( (\Delta t)^2 )
- lim_x->0 o( x^2 ) / x = 0
P(exactly one death ) = \mu \Delta t + o( (\Delta t)^2 )
P(exactly zero births ) = exp(-\lambda \Delta t) = 1 - \lambda \Delta t + o( (\Delta t)^2 )
P(exactly one birth ) = 1 - \mu \Delta t + o( (\Delta t)^2 )
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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