# Alternating Renewal Process

## Alternating Renewal Process

### 1. Definition

Consider a process with “on” periods and “off” periods
• Let ON time of cycle j
• Let OFF time of cycle j
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 =