TRAFFIC ANALYSIS TECHNIQUES INTRODUCTION The analysis of traffic flow is the basis for design of transportation system operating strategies, traffic control systems, and certain physical features of transportation systems. Several issues are involved. These include the interaction of traffic events in space and time, the analysis if travel times under various conditions, and relationships among traffic flow states such as flow, speed, and density. A variety of mathematical techniques are used in the analyzing space-time relationships, graphical and mathematical techniques for analyzing traffic queues and delays, and analysis of transportation networks. Chapter 9 presents techniques for the statistical analysis of traffic data and models of traffic flow for highways.
SPACE-TIME RELATIONSHIP Many traffic analysis problems involve the analysis of the motion of vehicles or traffic states in space and time. Such problems may vary a great deal in complexity. When such motion takes place along a single linear facility such as roadway or railway, it can often be represented by a graph known as space- time diagram. When it occurs in complicated transportation networks, no such simple approach is possible and satisfactory solutions to space- time problems may not yet exist. Space- Time diagrams have distance (or, more accurately, location) as one axis and time as the other axis. It is a matter of convention as to which axis represents time and which distance, and there are no firm rules; in what follows, either time or distance may be represented by the horizontal axis, depending on the situation. The figure represents a Simple Space- Time Diagram. The curve A-A represents the position of a vehicle (or some other event) at all times between t1 and t2. Such a curve is known as a Trajectory. In cases in which distance is the vertical axis, the slope of the trajectory is the speed of the vehicle; in cases in which distance is the horizontal axis, the reciprocal of speed (sometimes known as rate of travel) is re presented by the slope. Space- time diagrams are used in several different ways. Where the problems involved are simple enough, they may be used to provide direct graphical solutions. In more complicated cases, they may be used as an aid in developing analytical solutions to control problems, simulation models and theoretical models of traffic flow.
DEVELOPMENT OF ANALYTICAL SOLUTIONS Most space-time problems are complicated enough that it is not efficient to use graphical methods for routine solutions. In this case, space-time diagrams are often used to derive analytical relationships, relationships, which may then be used to produce a large number of solutions quickly and accurately . An example of this type of use of space-time diagrams is the derivation of basic relationships used in the analysis of capacity of capacity for airport runways used for landings only. • • • • •
Runway capacity is largely a function of the air traffic rules used by the air traffic control system. FUNDAMENTAL RULE: When flying under certain minimum conditions, all aircraft at a given elevation must maintain certain minimum longitudinal distance separations. Minimum distance separation is 3 nautical miles. COMMON APPROACH PATH – PATH – path from the entry gate to the runway threshold. CAPACITY OF A RUNWAY is the max number of aircraft it can handle per unit time (say, per hour). This, in turn, is the reciprocal of the average time separation between aircraft at the runway threshold. Case I Time separation at runway threshold vi
≤ v j
Where:
tij=time separation between the two aircraft δ=minimum distance separation v j =speed of the trailing aircraft
Case II Time separation at runway threshold vi > v j
Where:
v j = speed of the leading aircraft = length of the common approach path
DEVELOPMENT OF SIMULATION MODELS To solve even more complicated problems, space-time diagrams may be used to develop simulation models. Such models represent the behaviour of a system in a step-by-step manner. It is usually necessary to analyze both what is happening at the same location over time and what is happening at different locations at the same time. Development of a graphical solution is useful in determining the proper order of calculations to move from what is known to what is unknown, and can be used as a basis for developing a computer model of the behaviour of the system. • • • • • •
A simple example is that of the block signal control system for a rail line. BLOCK SIGNAL CONTROL SYSTEMS – intended to provide protection for trains against collisions and other hazards such as broken rails. BLOCKS – electrically insulated sections of track. OCCUPIED – whether a train is in particular block SIGNAL SYSTEM – used to warn or control trains approaching the occupied block. Most commonly involve a set of colored lights along the track or in the locomotive operator’s signals. ASPECTS – meaningful combinations of signal lights.
TWO-BLOCK, THREE-ASPECT SYSTEM It is the simplest system. Signal lights are either red, yellow, or green. A red signal means stop and proceed at some restricted speed (usually less than 7.5 km/h) prepared to stop behind the train ahead, a yellow signal means reduce speed and be prepared to stop at the next signal, and a green signal means to proceed at full speed. Because it would be dangerous to have a train entering an occupied block except after stopping, in this simplest system blocks must be at least as long as the maximum stopping distance of the trains.
Figure a
THREE-BLOCK, THREE-ASPECT SYSTEM It is in which an extra block is added. This is common on many subway lines. It permits a simple and reliable automatic train stop device to be activated at the second red signal behind a train, activating the brakes if the driver should fail to do so. The distance between following trains is increased, however, as can be seen from the Figure b, and capacity is reduced.
Figure b
THREE-BLOCK, FOUR-ASPECT SYSTEM It can be devised by using double lights. In this case red-red and green-green signals replace the red and green signals of the simpler systems, and there are two intermediate levels of restricted speed. GREEN-YELLOW signal - to approach the next signal at medium speed RED-YELLOW signal - prepare to stop Two blocks corresponds to the maximum stopping distance and capacity is increased.
Figure
Since speed control becomes more precise as the number of aspects increases, the size o f the blocks can be decreased. This, in turn, allows closer train spacing and greater line capacity. Hence there is a trade-off between the complexity of the control system and its effectiveness. Simulation of the system is useful in order to determine how a proposed signal system will affect the spacing of trains under a variety of assumptions as to their speed patterns.
NONTRAJECTORY SPACE-TIME DIAGRAMS Space-time diagrams may be used to display information about traffic (such as speed, flow rates, or densities) as well as vehicle trajectories. most common way of displaying traffic-state information in a space-time is by means of contour diagrams. In these, contours are used to isolate space-time regions with similar traffic-state values.
states The format
QUEUING ANALYSIS A second major issue in analysis of traffic systems is the analysis of times and delays. Normally travel time will be thought of as the time required to cover a fixed distance, and will be of course the reciprocal average speed at which that distance is covered.
travel
QUEUING SYSTEM SCHEMATIC DIAGRAM Delay is more subtle concept. It may be defined as the difference between the actual travel time on a given segment of a tran sportation system and some ideal travel time for the segment. This raises the obvious question of what ideal travel time should be used in measuring delay.
QUEUING THEORY FUNDAMENTALS The fundamentals of queuing theory may best be introduced by means of an example. Consider an airport runway being used for takeoffs only. Air traffic rules state that there can be only one aircraft on the runway at a time. Each aircraft spends about the same amount of time on the runway, so that a runway can process takeoffs at a fairly constant rate. In queuing theory jargon, the runway would be the server, and the maximum rate, at which it can process takeoffs, under given conditions, would be called its service rate. Arrivals of aircraft waiting to take off are not constant. The step function is called arrival function, A(t). Now suppose that each aircraft takes a fixed amount of time to clear the runway. Given this service time, it is possible to plot a departure function D*(t), which shows when each aircraft clears the runway. The departure function D(t) is usually of more interest that D*(t), since the takeoff run itself is a necessary part of the use of the runway, and should not be considered queuing delay. The difference between A(t) and D(t) represents aircraft which have arrived at the runway but have not begun(or completed) their takeoffs. These aircraft waiting to be served are said to be in queue, and the line of aircraft waiting to be served is called a Queue.
It is possible to determine a number of interesting things:
The number of customers in queue at any given time, or queue length. The delay to each customer, assuming the order of service is the same as the order of a rrival. The summation of delay to all customers. Given the density of the queue, the physical length of the queue, or the amount of space required for its storage.
In many applications it is appropriate to approximate the step functions A(t) and D(t), with smooth curves, and to ignore the finite service time if indeed, a finite service time even exists. In other cases, it is more convenient to approximate A(t) and D(t) with stepwise linear functions. Queuing diagrams have several important properties. It is very important that students understand these properties and be able to apply them to actual queuing problems. These are:
The slope of D(t) is the departure rate; the slope of A(t) is the arrived rate. The departure rate cannot e xceed the service rate or capacity of the server. It may be less. Cumulative departures can never exceed cumulative arrivals. D(t) can never be above A(t) in the queuing diagram. When a queue is present, the departure rate will equal the service rate. When no queue is present, the departure rate will equal the arrival rate. The queue first forms when the arrival rate first exceeds the service rate.
QUEUE DISCIPLINE So far, no explicit consideration has been given to the relationship between the order in which customers arrive demanding service and the order in which they are served. This relationship is referred to as the queue discipline. The most common queue discipline encountered in transportation applications is first-in, first-out , or FIFO, in which customers are served in the order in which they arrive. Other disciplines are possible. One which has gotten considerable attention in queuing theory literature, but is rare in transportation applications, is last-in, last-out , or LIFO, in which the most recent arrival is always served next. Of more importance in transportation applications are random service, in which there is no relationship between the order of arrivals and the order of service, and priority service, in which certain classes of customers are served ahead of others. Although queue length and total delay are the same for a ll queue disciplines, given the same arrival and departure functions, delays to individual customers are not the same in all queue disciplines. In particular, the delay to an individual customer can be determined by measuring the horizontal distance on the diagram between A(t) and D(t) only if the queue discipline is FIFO. QUEUING THEORY Queuing Theory is the mathematics of waiting lines. It is extremely useful in predicting and evaluating system performance. It has been used for operations research, manufacturing and system analysis. Application of Queuing Theory • •
Traffic Control Airport Traffic, Airline ticket sales
Characteristic of Queuing Systems • •
Costumer – refers to anything that arrives at a facility and requires service, e.g., people, machines, truck (cars). Server – refers to any resource that provides the requested service, e.g., repairpersons, retrieval machines, runways at airport.
Queuing Example: SYSTEM
CUSTOMERS
SERVER
Reception Desk Hospital Airport Road Network Grocery
People Patient Airplanes Cars Shoppers
Receptionist Nurses Runway Traffic Light Checkout
Components of a Queuing System:
Arrival Process Queue or Waiting Line
Service Exit
Characteristics: Arrival
Waiting Line
1. Size of the population 2. Behavior of Arrivals 3. Statistical Distribution of the Arrival
Arrival Process:
According to source According to Time According to Numbers
1. 2.
Service 1. 2.
Limited vs. Unlimited Queue Discipline
Service Design Statistical Distribution of Service
Queue Structure First-come-first-served (FCFS) Last-come-first-served (LCFS) Service-in-random-order (SIRO) Priority Service
Service System
SINGLE SERVICE SYSTEM
MULTIPLE, PARALLEL SERVER, SINGLE QUEUE MODEL
MULTIPLE, PARALLEL FACILITIES WITH MULTIPLE QUEUES MODEL
SERVICE FACILITIES IN A SERIES
STOCHASTIC QUEUING MODELS Queuing Models are known as “deterministic queuing models”. They are appropriate for many transportation applications because of the prevalence of distinct traffic peaks in which the capacities of servers are exceeded on a regular basis. Queuing models can be represented Kendall's notation: A/B/S/K/N/D where: A is the interarrival time distribution B is the service time distribution S is the number of servers K is the system capacity N is the calling population D is the service discipline assumed
using Some standard notations for distributions (A or B) are: M for a Markovian (poisson, exponential) distribution Eκ for an Erlang distribution with κ phases D for degenerate (or deterministic) distribution (constant) G for general distribution (arbitrary) PH for a phase-type distribution
M / D / 1 Queue Model An M/D/1 queue represents the queue length in a system having a single server, where arrivals are determined by a Poisson process and job service times are fixed (deterministic). Features of interest for stochastic queuing systems include the average queue length “Q” , the average waiting time “ w” , and the average total delay time spent in the system (queue waiting time plus service time) “ t” .
Average Queue Length
Average Waiting Time
Average Total Delay time spent
M / M / 1 Queue Model Another commonly encountered stochastic queuing model, in which both arrivals and service times are exponentially distributed and there is one channel. For the M / M / 1 model, the average queue length, waiting time, and total delay are gi ven by: Average Queue Length
Average Waiting Time
Average Total Delay time spent
TRANSPORTATION APPLICATIONS OF QUEUING THEORY
Queuing theory may be used for analyze a wide variety of transportation-related problems. For airports, queuing theory can be used to model the operation of runways, airspace on runway approaches, baggage handling systems, ticketing and check-in operations, security checkpoints, and departure lounges. For highways, queuing theory can be used to analyze the operation of toll booths: the effects of capacity changes due to geometrics; the effects of temporary capacity changes due to incidents (such as accidents); the effects of traffic signals and other control devices; and the effect of traffic disruptions caused by railroad g rade crossings, drawbridges, and the like. For mass transit systems, queuing theory can be used to analyze the use of ticket windows or ticket dispensing machines, the operation of fare gates, and platform capacities.
For railroads, queuing theory can be used to analyze yard operations. For water transportation, it can be used to analyze the operation of locks and port operations. For all types of freight transportation, queuing theory can be used to analyze loading and unloading operations. Any situation in which either demand or capacity may change over time may be suitable for analysis by queuing theory. Figures a through d illustrate special situations frequently encountered in transportation queuing analysis. Figure a illustrates the case in which a server opens after arrivals begin. An example of this situation is an airport departure lounge. Figure b illustrates the case which the arrival rate temporarily exceeds a constant service rate. This situation is typical of fixed bottlenecks in freeway systems, airport runways, and other cases in which there are definite traffic peaks and limited capacity. Figure c illustrates the case in which a service rate varies. Examples of this situation are freeway incidents, ramp meters, and toll booth operations in w hich the number of booths in service varies with time. The extreme example of carrying service rate is that in which a server is temporarily shut down, as illustrated by Figure d. Examples of this situation are railroad crossings, lift or swing bridges, and traffic signals
Figure a Server opens after arrivals begin.
Figure b Arrival rate temporarily exceeds a constant service rate.
Figure c Service rate varies.
Figure d Server temporarily shut down.
Example problem 1: Morning peak traffic upstream of a toll booth is given below. The toll plaza consists of three booths, each of which can handle an average of one vehicle every 6s. Using a queuing diagram, determine the maximum queue, the longest delay to an individual vehicle, and the total delay. Time period
10-min. Volume
Cumulative Volume
7:00 – 7:10
200
200
7:10-7:20
400
600
7:20-7:30
500
1,100
7:30-7:40
250
1,350
7:40-7:50
200
1,550
7:50-8:00
150
1,700
Service rate:
⁄
Comparing 10-min arrival rates with the service rate, queue begins at 7:10. Queing diagram:
Maximum queue: 300 vehicles at 7:30 (from diagram). Maximum delay to individual vehicle: 10 min (from diagram) Total delay: Break area between A(t) and D(t) into triangles and trapezoids: 1/2 (100)(10) + 1/2 (100 + 300)(10) + 1/2 (300 + 250)(10) + 1/2 (250 + 150)(10) + 1/2 (150)(10) = 500 + 2,000 + 2,750 + 2,000 + 750 = 8,000 veh-min or 133.3 veh-h Example problem 2: A ramp meter operates during morning peak period. Ramp meter cycles vary with time as shown in the table below. The metering scheme allows one vehicle per cycle to pass the signal. The table below gives the number of vehicle demanding service on the ramp during particular time intervals, the cumulative demand for the ramp for the morning peak, and the ramp meter cycle for each interval. Using a queuing diagram, determine the maximum queue, the maximum delay to any vehicle, and the total delay. Time period
15-min volume
Cumulative Volume
Meter cycle, s
6:30-6:45 6:45-7:00 7:00-7:15 7:15-7:30 7:30-7:45 7:45-8:00
75 100 125 110 80 65
75 175 300 410 490 555
6 10 12 12 10 6
Service rates:
( )() ) ( )( ) ( )( Comparing 15-min arrival rates with service rates, queue begins at 6:45.
Queuing diagram:
Maximum queue: 95 vehicles at 7:30 (from diagram). Maximum delay to individual vehicle: 15.5 min (from diagram) Total delay: Break area between A(t) and D(t) into triangles and trapezoids: 1/2 (10)(15) + 1/2 (60 + 10)(15) + 1/2 (60 + 95)(15) + 1/2 (95 + 85)(15) + 1/2 (85)(15) = 75 + 525+ 1,162.5 + 1,350 + 6 37.5 = 3,750 veh-min or 62.5 veh-h Example problem 3: Vehicles arrive at a stop sign at an average rate of 300 per hour. Average waiting time at the stop sign is 10 s per vehicle. If both arrivals and departures are exponentially distributed, what are the average queue length and the average delay per vehicle? Queue is an M/M/1 queue λ = 300 veh/h μ = (1veh/10s)(3600s/h) = 360veh/h ρ = 300/360 = 0.833 Q = (0.833)^2/(1-0.833) = 4.017 veh w = 300/360(360-300) = 0.014 veh = 0.83min Queue Density, Storage, and Spillback In the study of highway traffic, the problem is often one of locating queues and thus bottlenecks in the system. Since it is easy to estimate density by direct surveillance, observation of high densities is often used to spot bottlenecks. Actual measurement of density is less practical, however, because it usually involves taking aerial photographs and counting the number of vehicles in a section of roadway by hand. Where automatic data collection systems are available, a measure known as occupancy (the fraction of time vehicles are over the d etector) is often used as a substitute for density. A second reason that queue density is of interest is that it can be used to determine the physical length of the queue, and hence, the amount of space that must be provided to store it. As a general rule, it is desirable to avoid situations in which the queue from a bottleneck blocks traffic no bound for that bottleneck. In this case the queue is said to spill back into the upstream section.
NETWORK ANALYSIS Actual traffic systems usually consist of facilities in which there are several possible paths that can be followed between two points. Network - consist of a set of nodes connected by links. Nodes may be either origins or destination of trips or decision points. Points at which transportation facilities meet.
Links
elements in the network which have cost . are transportation facilities such as road, sidewalk, or railway segments.
Costs
Usually include travel time, travel distance, money costs or some sort of weighted sum of several cost elements.
There are two network analysis problems of particular interest in transportation. 1. Traffic Assignment Problem – which consists of predicting traffic volumes on each link in a network. (This problem will be discussed further in Chapter 13.) 2. Minimum cost path – determination of the minimum length of travel time (generally) from a certain origin to a certain destination. Most efficient solution is Dijkstra’s algorithm.
Example network
Minimum path algorithm, Step 1
Minimum path algorithm, Step 2
Minimum path algorithm, Step 3
Minimum path algorithm, Step 4
Minimum path algorithm, Step 5
Minimum path algorithm, Step 6
The minimum path from node A to node F is found by tracing the last-node portion of the label back from node F to node A. The path (in reverse order) is found to be F-C-B-E-D-A
-
-
-
Network cannot be represented by means of diagram unless to the extent if the information contained in the network diagram is stored in the form of two -dimensional array. Nodes are numbered and links are identified by the nodes they connect. Example:
Links may be either one-way or two-way. The two- way of the link-cost ij equals to that link ji.
Negative numbers are known as flag that can be identified by the computer as fictitious. These are also called as non - existing link. The number of node just labeled is used as subscript to identify column (or row) in the link-cost array. As in the manual version, existing tentative labels are replaced only if the new cost is less than the existing value.
SUMMARY Important traffic analysis techniques include analysis of space-time relationships, analysis of queuing, and network analysis. Spacetime diagrams may be used to develop direct graphical solutions for simple problems, and analytical solutions or simulation models for more complicated ones. Also, non trajectory space time displays may be used for such purposes as locating freeway queues. Que uing analysis is used to qualify delays. Deterministic queuing problems are solved by means of queuing diagrams; solutions to stochastic queuing problems involving various assumptions as to distributions of the arrivals and departures exist in the form of analytical formulas. Network models are used to analyze travel times where there is more than one possible path between the ultimate origin and destination of a trip. Efficient algorithms exist to solve for the minimum cost path where link costs are independent of flow s.
