Handout #1

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Poisson Distribution � dicrete

Probability mass function (pmf):

Cumulative distribution function (cdf):

Example:

A computer repair person is beeped each time there is a call for service, and the number of beeps is represented by a Poisson Distribution, with a mean of two beeps per hour.

What is the probability of getting beeped three times in the next hour?

What is the probability of getting two or more beeps in a one hour period?

Exponential Distribution � continuous

Probability Density Function (pdf):

Continuous Distribution Function (cdf):

Note:� the book is strange!� Usually 1/b is replaced by l.

Example:

Suppose a lamp is exponentially distributed with a failure rate of one in every 3000 hours.

What is the probability that the lamp will last longer than its mean life of 3000 hours?

Relationship of Poisson & Exponential

To develop queuing formulas we use (usually):

� Poisson to characterize the arrivals.

� Exponential to characterize their servicing.

Poisson has a mean of l, so the arrival rate is l.

Service depends on arrivals.� If arrival rate is l, then the interarrival rate is 1/l.

Poisson & Given Interval

P[n(T) = i] = (e^-lT (lT)ⁱ) / (i!), for T � 0 and n = 0, 1, 2, �

{Note that n(T) is the number of arrivals in T time units.}

If there are no arrivals in x time units, then i = 0.

P[n(T) = 0] = (e^-lT (lT)⁰) / (0!) = e^-lT

Then assume there's an arrival, then no arrival for x time units, then an arrival, etc.

So x is the interarrivals with distribution 1 � e^-lx, and probability density function le^-lx, which is the exponential pdf with b = 1/l.

Thus, the Poisson process has interarrival times that are exponentially distributed with mean 1/l.

Poisson Approximation to the Binomial Distribution

If x is binomially distributed:

Let l = np.� Then, p = l/n.

So now we get:

Which, through a little bit of mathematics, becomes:

Which is Poisson!