An expressway off-ramp consisting of a single lane leads directly to a tollbooth. The rate of arrival of vehicles at the expressway can be considered to be Poisson with a mean of 45 veh/hr, and the rate of service to vehicles can be assumed to be exponentially distributed with a mean of 1 min. (a) What is the average number of vehicles waiting to be served at the booth (that is, the number of vehicles in queue, not including the vehicle being served)? (b) What is the length of the ramp required to provide storage for all exiting vehicles 90% of the time? Assume the average length of a vehicle is 18 ft and that there is an average space of 10 ft between consecutive vehicles waiting to be served. (c) What is the average waiting time a driver waits before being served at the tollbooth (that

Question

is, the average waiting time in the queue)?

What will be an ideal response?

Answer 1

a) Expected queue length
Using Equation 6.71,
E(m) = q2/[Q(Q – q)] = (45)2/[60(60 – 45)] = 2.25 vehicles
b) Ramp length
1.00 – 0.90 = (q/Q)^(N + 1)
0.10 = (45/60)^(N + 1)
0.10 = (0.75)^N + 1
ln(0.10) = ln(0.75)(N + 1)
8 = N + 1
N = 7 vehicles
Ramp length = 7 veh (18 ft/veh) + 6 spaces (10 ft/space) = 186 feet
c) Average waiting time per vehicle
Using Equations 6.73 and 6.72,
E(v) = 1/(Q – q) = 1/(60 – 45) = 0.067 hr = 4 minutes wait time including
queue time and service time
E(w) = q/(Q(Q – q)) = 45/(60(60 – 45)) = 0.05 hr = 3 minutes wait time in the
queue