Excess Distribution of Renewal Process
– Course notes of Stochastic Process, 2014 Fall
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
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
– Assume if , otherwise .
This is the CCDF of a uniform distribution on .
Note: the above assume that , If , then the result is 0.