IEOR 4106, Spring 2009, Professor Whitt
Introduction to Renewal Theory: Thursday, April 9
1. Speeding Ticket Revenue
A policeman spends his entire day on the lookout for speeders.
The policeman cruises
on average approximately 10 minutes before stopping a car for some offense. Of the cars he
stops, 90% of the drivers are given speeding tickets with an $80 fine. It takes the policeman
an average of 5 minutes to write such a ticket. The other 10% of the stops are for more serious
offenses, leading to an average fine of $300. These more serious charges take an average of 30
minutes to process. In the long run what is the rate of money brought in by fines?
———————————————————————
ANSWER
The average time between successive stops is
10 + (0
.
9
×
5) + (0
.
1
×
30) = 10 + 4
.
5 + 3
.
0 = 17
.
5
minutes
.
The average fine revenue per stop is
0
.
9
×
80 + 0
.
1
×
300 = 72 + 30 = 102
Hence, by the renewal reward theorem, the longrun average rate fine revenue is accrued is
E
[fine per stop]
E
[time per stop]
=
$102
17
.
5
minutes
= $5
.
28
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 Spring '08
 Whitt
 Probability theory, Ri, longrun average rate, renewal reward theorem, longrun average reward, renewal reward processes

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