Models of Traffic Flow 1

Introduction Macroscopic relationships and analyses are very valuable, but A considerable amount of traffic analysis occurs at the microscopic level In particular, we often are interested in the elapsed time between the arrival of successive vehicles (i.e., time headway)

Introduction The simplest approach to modeling vehicle arrivals is to assume a uniform spacing This results in a deterministic, uniform arrival pattern—in other words, there is a constant time headway between all vehicles However, this assumption is usually unrealistic, as vehicle arrivals typically follow a random process Thus, a model that represents a random arrival process is usually needed

Introduction First, to clarify what is meant by ‘random’: For a sequence of events to be considered truly random, two conditions must be met: Any point in time is as likely as any other for an event to occur (e.g., vehicle arrival) The occurrence of an event does not affect the probability of the occurrence of another event (e.g., the arrival of one vehicle at a point in time does not make the arrival of the next vehicle within a certain time period any more or less likely)

Introduction One such model that fits this description is the Poisson distribution The Poisson distribution: Is a discrete (as opposed to continuous) distribution Is commonly referred to as a ‘counting distribution’ Represents the count distribution of random events

Poisson Distribution P(n) = probability of having n vehicles arrive in time t λ = average vehicle arrival rate in vehicles per unit time t= duration of time interval over which vehicles are counted e= base of the natural logarithm 6

Example Application Given an average arrival rate of 360 veh/hr or 0.1 vehicles per second; with t=20 seconds; determine the probability that exactly 0, 1, 2, 3, and 4 vehicles will arrive. 7

Poisson Example Example: Consider a 1-hour traffic volume of 120 vehicles, during which the analyst is interested in obtaining the distribution of 1-minute volume counts 8

Poisson Example What is the probability of more than 6 cars arriving (in 1-min interval)?

Poisson Example What is the probability of between 1 and 3 cars arriving (in 1-min interval)?

Poisson distribution The assumption of Poisson distributed vehicle arrivals also implies a distribution of the time intervals between the arrivals of successive vehicles (i.e., time headway)

Negative Exponential Substituting into Poisson equation yields To demonstrate this, let the average arrival rate, , be in units of vehicles per second, so that Substituting into Poisson equation yields (Eq. 5.25)

Negative Exponential Note that the probability of having no vehicles arrive in a time interval of length t [i.e., P (0)] is equivalent to the probability of a vehicle headway, h, being greater than or equal to the time interval t.

Negative Exponential So from Eq. 5.25, (Eq. 5.26) Note: This distribution of vehicle headways is known as the negative exponential distribution.

Negative Exponential Example Assume vehicle arrivals are Poisson distributed with an hourly traffic flow of 360 veh/h. Determine the probability that the headway between successive vehicles will be less than 8 seconds. Determine the probability that the headway between successive vehicles will be between 8 and 11 seconds.

Negative Exponential Example By definition,

Negative Exponential Example

Negative Exponential For q = 360 veh/hr

Negative Exponential

Queuing Systems Queue – waiting line Queuing models – mathematical descriptions of queuing systems Examples – airplanes awaiting clearance for takeoff or landing, computer print jobs, patients scheduled for hospital’s operating rooms 20

Characteristics of Queuing Systems Arrival patterns – the way in which items or customers arrive to be served in a system (following a Poisson Distribution, Uniform Distribution, etc.) Service facility – single or multi-server Service pattern – the rate at which customers are serviced Queuing discipline – FIFO, LIFO 21

D/D/1 Queuing Models Deterministic arrivals Deterministic departures 1 service location (departure channel) Best examples maybe factory assembly lines 22

Example Vehicles arrive at a park which has one entry points (and all vehicles must stop). Park opens at 8am; vehicles arrive at a rate of 480 veh/hr. After 20 min the flow rate decreases to 120 veh/hr and continues at that rate for the remainder of the day. It takes 15 seconds to distribute the brochure. Describe the queuing model. 23

M/D/1 Queuing Model M stands for exponentially distributed times between arrivals of successive vehicles (Poisson arrivals) Traffic intensity term is used to define the ratio of average arrival to departure rates: 24

M/D/1 Equations When traffic intensity term < 1 and constant steady state average arrival and departure rates: 25

M/M/1 Queuing Models Exponentially distributed arrival and departure times and one departure channel When traffic intensity term < 1 26

M/M/N Queuing Models Exponentially distributed arrival and departure times and multiple departure channels (toll plazas for example) In this case, the restriction to apply these equations is that the utilization factor must be less than 1. 27

M/M/N Models 28