3.29 Effect of traffic lights on
travel times A
vehicle travels a route from a to b, incurring a total travel
T = 6 +
where 6 = time to traverse
the distance (at
constant speed) from a to b
= delay incurred at traffic ight 1
= delay incurred at traffic light 2
As shown in Figure P3.29, the route is partitioned
into three 2-minute
travel-time segments. Each traffic light operates on a fixed cycle
of 1 minute green,
followed by I minute red. (We assume that no time is wasted in
accelerating, should one or two lights cause the vehicle to stop.)
Suppose that exactly at some prespecified time (say
12: 00 noon) we
examine the "phase" i, of each traffic
light (i = 1, 2). By definition,
i = time until
i next turns to green (0 i < 2)
For each light, we suppose that i,
is independently, uniformly distributed over [0, 2]. However,
is known, its value isfixedfor all time.
Throughout this problem, we assume that departure
times at a occur
independently of the phases of the traffic lights.
a. Find the mean and variance of the travel time
from a to b.
b. Find the probability density function of the
travel time from a to b.
c. Let k = number of traffic lights at which the
vehicle is delayed. Find
the z-transform of the probability mass function for k.
For parts (d) and (e), only [not for parts (f)-(h)],
C = event that for the most recent vehicle
traveling from a to b, the
vehicle was stopped only at traffic light 1
d. Find the conditional joint probability density
function for 1
e. Find the conditional probability density
function for the travel time
from a to b for the next vehicle to travel from a to b, given C.
(Assume that you know
nothing about when the next vehicle will leave a.)
For parts (f)-(h), suppose that vehicles leave a in
a Poisson manner with
= 1 vehicle per minute. Vehicles occupy zero space and, when
in a traffic light
queue, accelerate and decelerate instantaneously together.
f. Is the vehicle arrival process at b a
Poisson process? Why or
g. Determine the mean and variance of the queue
length (number of
vehicles) at traffic light I at the instant before the light turns
green. (This is a
primer for Chapter 4.)
h. A traffic engineer adjusts the phases of the
two traflac lights so that
= 0 (relative to 12: 00 noon). Suppose that at 12: 00 noon we are
information that no vehicles have left a during the last
8 minutes. Carefully
sketch and label the probability density function for the time of
arrival at b of the next
vehicle to arrive there.