## 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 =