Queueing Theory, Highway Safety and Accident Analysis, Service Rates and Traffic Studies
1. Vehicl es arrive at an entrance of a recreational park. There is a single gate (at which all the
vehicles must stop), where a park attendant distributes a free brochure. The park opens at
8:00 AM., at which time vehicles arrive at the rate of 480 veh/hr. After 20 minutes the arrival
rate declines to 120 veh/hr, and it continuous at that level for the remainder of the day. If the
time required to distribute the brochure is 15 seconds, and is assuming D/D/1 queuing,
describe the operational characteristics of the queue.
2. However, let the average arrival rate be 180 veh/hr and Poisson distribute (exponential time
between arrival) over the entire opening time (8:00 AM) until closing at dusk. Compute the
average length of queue 9in vehicle), average waiting time in the queue, and average waiting
time spent in the system, assuming M/D/1 queuing.
3. Assume that the park attendant in No. 1 and 2 takes an average of 15 seconds to distribute
brochures but the distribution time varies depending on whether park patrons have
questions relating to the park operating policies. Given an average arrival rate of 180 veh/hr
as in No. 2, compute the average length of queue (in vehicles), average waiting time in the
queue, and average time spent in the system assuming M/M/1 queuing.
4. A ramp meter operates during the morning peak period. Ramp meter cycles vary with time
as shown in the table below. The metering scheme allows one vehicle per cycle to pass the
signal. The table gives the number of vehicles demanding service on the ramp during the
time intervals, the cumulative demand for the ramp for the morning peak and the ramp
meter cycle for each interval.
Time Period
15 min
volume
Cumulative
Volu me
Meter
Cycle (s)
6:30 – 6:45
75
75
6
6:45 – 7:00
100
175
10
7:00 – 7:15
125
300
12
7:15 – 7:30
110
410
12
7:30 – 7:45
80
490
10
7:45 – 8:00
65
555
6
A. What are the service rates for meter cycle 12s?
B. What are the service rates for meter cycle 10s?
C. What are the service rates for meter cycle 6s?
D. Determine the time the queue on the ramp begins and ends.
E. Determine the longest queue.
F. Determine the total delay.
