Question

A left turn lane is to be designed at a signalized intersection to accommodate leftturning
vehicles during the peak period of traffic. If the design peak hour volume for the left
turn movement is 150 vehicles per hour and the red time interval per signal cycle for the left turn
movement is 60 seconds.

a) Compute the average number of left turn vehicle arrivals per red interval of each signal
cycle during the design peak hour.

b) Assume that left turn vehicles arrive the intersection in a random fashion (i.e., following
Poisson distribution). Compute and plot the probability mass function (PMF) for the
number of left turn vehicles arriving during the red interval (use of Excel spreadsheets is
preferred).

c) If the design of the left-turn lane requires accommodating (or storing) the left-turn queue
95% of time during the peak hour, recommend the minimum left-turn lane length in feet
(assuming 25 feet per vehicle, including space between vehicles).

Answer 1

Solution

Back-up Theory

If a random variable X ~ Poisson (λ), i.e., X has Poisson Distribution with mean λ then

probability mass function (pmf) of X is given by P(X = x) = e ^{– λ}.λ^x/(x!),where x = 0, 1, 2, ……. , ∞ ..................................(1)

Values of p(x) for various values of λ and x can be obtained by using Excel Function, Statistical, POISSON……………. (1a)

Now to work out the solution,

Part (a)

Given

Design peak hour volume for the left turn movement is 150 vehicles per hour (60 minutes) …… (2)

Red time interval per signal cycle for the left turn movement is 60 seconds (i.e., 1 minute) …….. (3)

(2) and (3) => Average number of left turn vehicle arrivals per red interval of each signal
cycle during the design peak hour = 150/60 = 2.5 Answer 1

Part (b)

Let X = number of left turn vehicles that arrive at the intersection during red interval of each signal
cycle during the design peak hour.

Given, that left turn vehicles arrive at the intersection in a random fashion (i.e., following
Poisson distribution) => X ~ Poisson (λ), where λ = mean = 2.5 [vide Answer 1]. Thus,

X ~ Poisson (2.5)

Vide (1). The pmf is: p(x) = e ^{– 2.5}.(2.5)^x/(x!) ………………………………… Answer 2

For all possible values of X, the pmf values are obtained, vide (1a) and tabulated below:

x	p(x)
0	0.082085
1	0.2052125
2	0.2565156
3	0.213763
4	0.1336019
5	0.0668009
6	0.0278337
7	0.0099406
8	0.0031064
9	0.0008629
10	0.0002157
> 10	6.163E-05

Answer 3

Part (c)

First, to find the number of left turn vehicles that are likely to be at the intersection 95% of the times, we use the above table and obtain the cumulative probability P(x) as shown in the table below:

x	p(x)	P(x)
0	0.082085	0.082085
1	0.2052125	0.2872975
2	0.2565156	0.54381312
3	0.213763	0.75757613
4	0.1336019	0.89117802
5	0.0668009	0.95797896
6	0.0278337	0.98581269
7	0.0099406	0.9957533
8	0.0031064	0.99885975
9	0.0008629	0.99972265
10	0.0002157	0.99993837
> 10	6.163E-05	1.00000

So, a maximum of 5 left turn vehicles are expected at the intersection during red interval of each signal cycle during the design peak hour.

Given, (assuming 25 feet per vehicle, including space between vehicles),

recommended minimum left-turn lane length in feet = 25 x 5 = 125 feet. Answer 4

DONE