# CS457 - System Performance Evaluation - Winter 2010

## Public Service Announcements

# Lecture 21 - Binomial Distributions, Poisson Processes (pdf)

## Other Distributions

The exponential distribution pretty well takes care of arrival events in
open systems. What about

- arrival events in closed systems, aka think time?
- service times?
- others?

These are normally modelled based on distributions that are abstractions
of behaviour observed in logs or traces. Here are a few that turn up from
time to time.

### Bernoulli & BinomialDistributions

Systems that are easily categorized into two distinct homogeneous
classes.

#### Examples

Think times

- Easy versus hard problems
- Need to access manual versus no need to access manual

Service times

- Slow CPU versus fast CPU
- Threaded versus unthreaded
- Local versus remote

#### Bernoulli

Coin-flipping is the natural analogue, but it covers any binary choice
made at (biased!) random.

- P(heads) = p
- P(tails) = 1-p

How do you sample a Bernoulli distribution in practice

- Get a random number, u, uniformly distributed on [0,L)
- If u < Lp then heads, else tails.

#### Binomial

Make N binary choices at random, all identical.

- P(k heads in N tosses) = (N choose k) p^k (1 - p)^(N-k)

How do you sample a binomial distribution in practice

- Small N: sample a Bernoulli N times,
- linear in N for each sample

- Medium N: divide the range [0,L) into N+1 parts proportional to (N
choose k) and do binary search with u
- linear in N to set up
- logarithmic in N for each sample

- Large N: converge to another distribution
- Poisson if Np is constant as N -> infinity
- Gaussian if p is constant as N -> infinity

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