Lec 16, Ch16, pp.413-424: Intersection delay (Objectives) Know the definitions of various delays taking place at signalized intersections Be able to graph the relation between delay, waiting time, and queue length Become familiar with three delay scenarios Understand the derivation of Webster’s delay model Understand the concept behind the modeling of overflow delay Know inconsistencies that exist between stochastic and overflow delay models

What we discuss today in class… Definition of various delays and a typical time-space diagram for signalized intersections 3 delay scenarios Webster’s delay model Overflow delay model (v/c > 1.0) Inconsistencies between stochastic and overflow delay models Introduction to the HCM delay model Theory vs. reality Sample 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. Common MOEs: Delay Queuing No. of stops (or percent stops)

Time-space diagram to show approach delay 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. At saturation flow rate, s

Three delay scenarios This is acceptable. This is great. UD = uniform delay OD = overflow delay due to prolonged demand > supply (Overall v/c > 1.0) OD = overflow delay due to randomness (“random delay”). Overall v/c < 1.0 A(t) = arrival function D(t) = discharge function You have to do something with this signal.

Webster’s intersection delay model (Analytic model) for uniform delay UDa Total approach delay The area of the triangle is the total stopped delay, “Uniform Delay (UD)”. To get average approach delay/vehicle, divide this by vC

Webster’s intersection delay model (Analytic model) for random delay UD = uniform delay Analytical model for random delay Adjustment term for overestimation (between 5% and 15%) OD = overflow delay due to randomness (in reality “random delay”). Overall v/c < 1.0 D = 0.90[UD + RD]

Webster’s optimal cycle length model 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 because c = s (g/C), (g/C)(v/c) = (v/s). And v/c = 1.0. The aggregate overflow delay is: Since the total vehicle discharged during T is cT, See the right column of p.418 for the characteristics of this model.

Inconsistencies between stochastic and overflow delay models 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. 16-26 Eq. 16-27 The HCM 1994 model looks like: Eq. 16-25

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?