# Excess Distribution of Renewal Process

## 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.
Interpretation: Fraction of time that = Fraction of “on” time for
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

Example: ~ exp
=
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.