OREGON TECH CIV475 Lindgren1 CIV 475 Traffic Engineering Mannering / Kilareski Chapter 5 Queuing Theory

OREGON TECH CIV475 Lindgren2 Queue zA ‘queue’ is simply a line zThere were 16 cars in line at the toll booth zThe toll booth queue was 16 cars

OREGON TECH CIV475 Lindgren3 Queuing Theory zQueuing theory is a broad field of study of situations that involve lines or queues yretail stores ymanufacturing plants ytransportation xtraffic lights xtoll booths xstop signs xetc.

OREGON TECH CIV475 Lindgren4 Queuing Theory - acronyms zFIFO - a family of models that us the principle of “first in first out” zLIFO - “last in first out” za/d/N notation (aka Kendall notation) ya - arrival type ( either D- deterministic, or M- mechanistic ) yd - departure type ( either D- deterministic, or M- mechanistic ) yN - number of “channels”

OREGON TECH CIV475 Lindgren5 D/D/1 FIFO zEntrance gate to National Park zDeterministic arrivals and departures, one fee booth, first in first out zAt the opening of the booth (8:00am), there is no queue, cars arrive at a rate of 480veh/hr for 20 minutes and then changes to 120veh/hr zThe fee booth attendant spends 15seconds with each car zWhat is the longest queue? When does it occur? zWhen will the queue dissipate? zWhat is the total time of delay by all vehicles? zWhat is the average delay, longest delay? zWhat delay is experienced by the 200th car to arrive?

OREGON TECH CIV475 Lindgren6 More than D/D/1 zWhile D/D/1 queuing is easy to understand and graphical solutions are available, it may not be the best model to use in traffic situations since arrivals are not Deterministic ( as you will see by collecting data on some real traffic streams ) zDerivation of Stochastic (Mechanistic) queuing equations is beyond the scope of this course, but the equations are listed in the text book zread up on M/D/1, M/M/1 and M/M/N

OREGON TECH CIV475 Lindgren7 Queuing at traffic lights

OREGON TECH CIV475 Lindgren8 Graph of Flow vs. time Red is shown as darker gray, green is lighter gray Constant Arrival Flow

OREGON TECH CIV475 Lindgren9 zDuring the red interval for the approach, vehicles cannot depart from the intersection and consequently, a queue of vehicles is formed. zWhen the signal changes to green, the vehicles depart at the saturation flow rate until the standing queue is cleared. zOnce the queue is cleared, the departure flow rate is equal to the arrival flow rate. zDeparture flow rates are shown in the next figure

OREGON TECH CIV475 Lindgren10 Departure Flow Diagram

OREGON TECH CIV475 Lindgren11 zSketch a graph showing how the queue length changes with time during a red- green period for one movement of an intersection

OREGON TECH CIV475 Lindgren12 zDuring the red interval, the queue of vehicles waiting at the intersection begins to increase. zThe queue reaches its maximum length at the end of the red interval zWhen the signal changes to green, the queue begins to clear as vehicles depart from the intersection at the saturation flow rate

OREGON TECH CIV475 Lindgren13 zThere is another graph that allows us to glean even more information from our model. zImagine a plot where the x-axis is time and the y-axis contains the vehicle numbers according to the order of their arrival. zVehicle one would be the first vehicle to arrive during the red interval and would be the lowest vehicle on the y-axis. zIf you were to plot the arrival and departure (service) times for each vehicle, you would get a triangle

OREGON TECH CIV475 Lindgren14

OREGON TECH CIV475 Lindgren15 zFor a given time, the difference between the arrival pattern and the service pattern is the queue length. zFor a given vehicle, the difference between the service pattern and the arrival pattern is the vehicle delay. zIn addition, the area of the triangle is equivalent to the total delay for all of the vehicles.

OREGON TECH CIV475 Lindgren16 Assignment zText problems: 5.4, 5.9, 5.11, 5.14, 5.22 zCollect traffic arrival data (30-1minute increments) on a moderately congested road that is away from the influence of traffic signals, plot a histogram (figure 5.5) of your data, on the same histogram show a Poisson distribution model of the same # of arrivals, determine if your data follows a Poisson distribution zCollect 30 minutes of headway data, plot as per (figure 5.6), on the same graph, plot the exponential (Poisson) probabilities, determine if your data follows a Poisson distribution