Alternating Renewal Process
1. Definition
Consider a process with “on” periods and “off” periods
Then and must satisfy the following property in order for to be regenerative
The pair must be i.i.d; in particular, independent of for . That is
- are i.i.d
- are i.i.d
- However, and may be independent for the same j
Cycle time .
Then by the regenerative process, we have
the average “up” time =
2. Example
Cars pass a point on highway according to Poisson process with rate . of cars are speeding (>10 mph over the pretend speed limit). Assume time to issue a ticket~UNIF[10,14] minutes(one officer)
Question: What fraction of speeding cars pass when the officer is busy.
Answer: The answer to the question is equivalent to the fraction of time the officer is busy.
- Let = time spent waiting to give a ticker
- Let = time spent giving a ticker
Speeders arrive according to Poisson process with rate per min.
Fraction of time busy =