## Excess Distribution of Renewal Process

– Course notes of Stochastic Process, 2014 Fall

### 1. Definition

Excess of renewal process is defined as (time until next event)

In the example of average time waiting bus, we drived

Now we are going to derive for a random .** Interpretation**: You show up “at random”. What is the probability that you wait more than x for the next event?

### 2. Derivation of

As we want to determine the fraction of time that .

Let if , and let otherwise.

**: Fraction of time that = Fraction of “on” time for**

__Interpretation__Let be “on” time during cycle , then

- if
- otherwise

Note: the ON time and OFF time for each cycle are dependent. A longer ON time implies a shorter OFF time.

Then we have

=

=

=

Since is an alternating renewal process, fraction of “on” time is

This is sometimes called

**,**__equilibrium distribution__### 3. Example of different distribution of

**: ~ exp**

__Example__ =

As expected (by memoryless property), excess distribution is an exponential distribution.

**Example: Pareto Distribution**

is also called the tail distribution

Some properties: (mean only exist if

**Example: Deterministic**

– Assume if , otherwise .

=

=

This is the CCDF of a uniform distribution on .

Note: the above assume that , If , then the result is 0.