Lec 16, Ch16, pp.413-424: Intersection delay (Objectives)

Know the definitions of various delays taking place at Know the definitions of various delays taking place at signalized intersectionssignalized intersections

Be able to graph the relation between delay, waiting Be able to graph the relation between delay, waiting time, and queue lengthtime, and queue length

Become familiar with three delay scenariosBecome familiar with three delay scenarios Understand the derivation of Webster’s delay modelUnderstand the derivation of Webster’s delay model Understand the concept behind the modeling of Understand the concept behind the modeling of

overflow delayoverflow delay Know inconsistencies that exist between stochastic Know inconsistencies that exist between stochastic

and overflow delay modelsand overflow delay models

What we discuss today in class…

Definition of various delays and a typical time-space Definition of various delays and a typical time-space diagram for signalized intersectionsdiagram for signalized intersections

3 delay scenarios3 delay scenarios Webster’s delay modelWebster’s delay model Overflow delay model (v/c > 1.0)Overflow delay model (v/c > 1.0) Inconsistencies between stochastic and overflow delay Inconsistencies between stochastic and overflow delay

modelsmodels Introduction to the HCM delay modelIntroduction to the HCM delay model Theory vs. realityTheory vs. reality Sample delay computationsSample delay computations

Delays

Common MOEs:

• Delay

• Queuing

• No. of stops (or percent stops)

Stopped time delay: The time a vehicle is stopped while waiting to pass through the intersection

Approach delay: Includes stopped time, time lost for acceleration and deceleration from/to a stop

Travel time delay: the difference between the driver’s desired total time to traverse the intersection and the actual time required to traverse it.

Time-in-queue delay: the total time from a vehicle joining an intersection queue to its discharge across the stop-line or curb-line.

Time-space diagram to show approach delay

At saturation flow rate, s

Uniform arrival rate assumed, v Here we assume

queued vehicles are completely released during the green.

Note that W(i) is approach delay in this model.

Three delay scenarios

This is great.This is acceptable.

You have to do something with this signal.

A(t) = arrival function

D(t) = discharge function

UD = uniform delay

OD = overflow delay due to randomness (“random delay”). Overall v/c < 1.0

OD = overflow delay due to prolonged demand > supply (Overall v/c > 1.0)

Webster’s intersection delay model (Analytic model) for uniform delay

vs

vs

C

gCV

vs

vRt

sttRvV

C

gCR

c

cc

1

1

The area of the triangle is the total stopped delay, “Uniform Delay (UD)”.

vs

vs

C

gCheightbaseUDa

22 1

2

1))((

2

1

UDa

Total approach delay

To get average approach delay/vehicle, divide this by vC

sv

CgCUD

1

1

2

2

Webster’s intersection delay model (Analytic model) for random delay

UD = uniform delay

OD = overflow delay due to randomness (in reality “random delay”). Overall v/c < 1.0

Cgcvvc

cvv

cv

sv

CgCD

2312

22

65.0

/121

1

2

Adjustment term for overestimation (between 5% and 15%)

Analytical model for random delay

D = 0.90[UD + RD]

Webster’s optimal cycle length model

1

0

1

55.1

iisv

LC

C0 = optimal cycle length for minimum delay, sec

L = Total lost time per cycle, sec

Sum (v/s)i = Sum of v/s ratios for critical lanes

Delay is not so sensitive for a certain range of cycle length This is the reason why we can round up the cycle length to, say, a multiple of 5 seconds.

Modeling overflow delay when v/c>1.0

2

)(1

1

1

2

2

CgC

sv

CgCUDo

because c = s (g/C), (g/C)(v/c) = (v/s). And v/c = 1.0.

cvT

cTvTTODa 22

1 2

The aggregate overflow delay is:

Since the total vehicle discharged during T is cT,

12

cvT

OD

See the right column of p.418 for the characteristics of this model.

12

21

cvTT

OD

Inconsistencies between stochastic and overflow delay models

Cgcvvc

cvv

cv

sv

CgCD

2312

22

65.0

/121

1

2 1

2 cv

TOD

The stochastic model’s overflow delay is asymptotic to v/c = 1.0 and the overflow model’s delay is 0 at v/c =0. The real overflow delay is somewhere between these two models.

Comparison of various overflow delay model

Eq. 16-25

Eq. 16-26

Eq. 16-27

c

XXXX

XCg

CgCd

1611173

1

138.0

22

2

The HCM 1994 model looks like:

Theory vs. reality

Isolated intersections

Signalized arterials

HCM uses the Arrival Type factor to adjust the delay computed as an isolated intersection to reflect the platoon effect on delay.

Sample delay computations (p.421)

Sample computation A:

Approach volume v = 1000 vph

Saturation flow rate s = 2800 vphg (2 lanes?)

g/C = 0.55

Find average approach delay per vehicle

Sample computation B:

Chronic oversaturation

Two-hour period T = 2 hours

Approach volume v = 1100 vph

Saturation flow rate s = 2000 vphg (2 lanes?)

C = 120 sec

g/C = 0.52

Find the total average approach delay per vehicle for the 2 hour period and for the last 15 min

Sample computation C: Apply the HCM 1994 model to the condition described in Sample computation B. What is its implication?