Congestion Studies

Lecture notes in Transportation Systems Engineering

July 28, 2011

Introduction

Challenges in transportation system

Transportation system consists of a group of activities as well as entities interacting with each other to achieve the goal of transporting people or goods from one place to another. Hence, the system has to meet the perceived social and economical needs of the users. As these needs change, the transportation system itself evolves and problems occur as it becomes inadequate to serve the public interest. One of the negative impacts of any transportation system is traffic congestion. Traffic congestion occurs wherever demand exceeds the capacity of the transportation system. This lecture gives an overview of how congestion is generated, how it can be measured or quantified, and also the various countermeasures to be taken in order to counteract congestion. Adequate performance measures are needed in order to quantify congestion in a transportation system. Quality of service measures indicates the degree of traveller satisfaction with system performance and this is covered under traveller perception. Several measures have been taken in order to counteract congestion. They are basically classified into supply and demand measures. An overview of all these aspects of congestion is dealt with in this lecture.

Generation of traffic congestion

The flow chart in Fig. 1 shows how traffic congestion is generated in a transportation system. With the evolution of society, economy and technology, the household characteristics as well as the transportation system gets affected. The change in transport system causes a change in transport behaviour and locational pattern of the system. The change in household characteristics, transport behaviour, locational pattern, and other growth effects result in the growth of traffic. But the change or improvement in road capacity is only as the result of change in the transportation system and hence finally a situation arises where the traffic demand is greater than the capacity of the roadway. This situation is called traffic congestion.

Figure 1: Generation of traffic congestion

Effects of congestion

Congestion has a large number of ill effects which include:

1. Loss of productive time,
2. Increase in the fuel consumption,
3. Increase in pollutants (because of both the additional fuel burned and more toxic gases produced while internal combustion engines are in idle or in stop-and-go traffic),
4. Increase in wear and tear of automobile engines,
5. High potential for traffic accidents,
6. Negative impact on people's psychological state, which may affect productivity at work and personal relationships, and
7. Slow and inefficient emergency response and delivery services.

The summation of all these effects yields a considerable loss for the society and the economy of an urban area

Traffic congestion

A system is said to be congested when the demand exceeds the capacity of the section. Traffic congestion can be defined in the following two ways:

1. Congestion is the travel time or delay in excess of that normally incurred under light or free flow traffic condition.
2. Unacceptable congestion is travel time or delay in excess of agreed norm which may vary by type of transport facility, travel mode, geographical location, and time of the day.

Fig. 2 shows the definition of congestion. The solid line represents the travel speed under free-flow conditions and the dotted line represents the actual travel speed. During congestion, the vehicles will be travelling at a speed less than their free flow speed. The shaded area in between these two lines represents the amount of congestion.

Figure 2: Definition of congestion

Traffic congestion may be of two types:

1. Recurrent Congestion: Recurrent congestion generally occurs at the same place, at the same time every weekday or weekend day.
2. Non-Recurrent congestion: Non-Recurrent congestion results from incidents such as accidents or roadway maintenance.

Measurement of congestion

Congestion has to be measured or quantified in order to suggest suitable counter measures and their evaluation. Congestion information can be used in a variety of policy, planning and operational situations. It may be used by public agencies in assessing facility or system adequacy, identifying problems, calibrating models, developing and assessing improvements, formulating programs and policies and priorities. It may be used by private sector in making locational or investment decisions. It may be used by general public and media in assessing traveller's satisfaction.

System performance measurement

Performance measure of a congested roadway can be done using the following four components:

1. Duration,
2. Extent,
3. Intensity, and
4. Reliability.

Duration

Duration of congestion is the amount of time the congestion affects the travel system. The peak hour has now extended to peak period in many corridors. Measures that can quantify congestion include:

• Amount of time during the day that the travel rate indicates congested travel on a system element or entire system.
• Amount of time during the day that traffic density measurement techniques (detectors, aerial surveillance, etc.) indicate congested travel.

Duration of congestion is the sum of length of each analysis sub period for which the demand exceeds capacity. The maximum duration on any link indicates the amount of time before congestion is completely cleared from the corridor. Duration of congestion can be computed for a corridor using the following equation:

$$H=N \times T$$ (1)

where, $H$ is the duration of congestion (hour), $N$ is the number of analysis sub periods for which $v/c > 1$, and $T$ is the duration of analysis sub-period (hour)The duration of congestion for an area is given by:

$$H_i=\frac{T\frac{v_i}{c_i}(1-r)}{1-r(\frac{v_i}{c_i})}$$ (2)

where, $H_i$ is the duration of congestion for link $i$ (hour), $T$ is the duration of analysis period (hour), $r$ is the ratio of peak demand to peak demand rate, $v_i$ is the vehicle demand on link $i$ (veh/hour), and $c_i$ is the capacity of link $i$ (veh/hour).

Extent

Extent of congestion is described by estimating the number of people or vehicles affected by congestion and by the geographic distribution of congestion. These measures include:

1. Number or percentage of trips affected by congestion.
2. Number or percentage of person or vehicle meters affected by congestion.
3. Percentage of the system affected by congestion.

Performance measures of extent of congestion can be computed from sum of length of queuing on each segment. Segments in which queue overflows the capacity are also identified. To compute queue length, average density of vehicles in a queue need to be known. The default values suggested by HCM 2000 are given in Table 1.

Table 1: Queue density default values

Subsystem	Storage density(veh/km/lane)	Spacing(m)
Free-way	75	13.3
] Two lane highway	130	7.5
Urban street	130	7.5

Queue length can be found out using the equation:

$$Q_i=\frac{T(v-c)}{N \times ds}$$ (3)

where; $Q_i$ is the queue length (meter), $v$ is the segment demand (veh/hour), $c$ is the segment capacity (veh/hour), $N$ is the number of lanes, $ds$ is the storage density (veh/meter/lane), and $T$ is the duration of analysis period (hour). If $v < c$, $Q_i$=0 The equation for queue length is similar for both corridor and area-wide analysis.

Intensity

Intensity of congestion marks the severity of congestion. It is used to differentiate between levels of congestion on transport system and to define total amount of congestion. It is measured in terms of:

• Delay in person hours or vehicle hours;
• Average speed of roadway, corridor, or network;
• Delay per capita or per vehicle travelling in the corridor, or per person or per vehicle affected by congestion;
• Relative delay rate (relative rate of time lost for vehicles);

Intensity in terms of delay is given by,

$$D_{PH}=T_{PH}-T_{PH}^0$$ (4)

where, $D_{PH}$ is the person hours of delay, $T_{PH}$ is the person hours of travel under actual conditions, and $T_{PH}^0$ is the person hours of travel under free flow conditions. The $T_{PH}$ is given by:

$$T_{PH}=\frac{O_{AV} \times v \times l}{S}$$ (5)

where, $O_{AV}$ is the average vehicle occupancy, $v$ is the vehicle demand (veh), $l$ is the length of link (km), and $S$ is the mean speed of link (km/h). The $T_{PH}$ is given by:

$$T_{PH}^0=\frac{O_{AV} \times v \times l}{S_0}$$ (6)

where, $O_{AV}$ is the average vehicle occupancy, $v$ is the vehicle demand (veh), $l$ is the length of link (km), and $S_0$ is the free flow speed on the link

Numerical example

On a 2.8 km long link of road, it was found that the demand is 1000 Vehicles/hour mean speed of the link is 12 km/hr, and the free flow speed is 27 km/hr. Assuming that the average vehicle occupancy is 1.2 person/vehicle, calculate the congestion intensity in terms of total person hours of delay.

Solution:

Person hours of delay is given as

$$\begin{eqnarray*} D_{PH}=T_{PH}-T_{PH}^0 \end{eqnarray*}$$

Person hours of travel under actual conditions,

$$\begin{eqnarray*} T_{PH}&=&\frac{O_{AV} \times v \times l}{S}\\ &=& \frac{1.2 \times 1000 \times 2.8}{12} \\ &=& 280 \mathrm{~person~hours} \end{eqnarray*}$$

Person hours of travel under free flow conditions,

$$\begin{eqnarray*} T_{PH}^0 &=& \frac{O_{AV} \times v \times l}{S_0} \\ &=& \fra... ...\times 1000 \times 2.8}{27} \\ &=& 124.4 \mathrm{~person~hours} \end{eqnarray*}$$

Therefore, person hours of delay,

$$\begin{eqnarray*} D_{PH}= &=& 280-124.4 \\ &=& 155.6 \mathrm{~person~hours}. \end{eqnarray*}$$

Relationship between duration, extent, and intensity of congestion

The relationship between duration,extent, and intensity of congestion can be show in a time-distance graph Fig. 3 The extent of congestion is seen on the x-axis, the duration on the y-axis. The intensity is shown in the shading. Based on the extent and duration the congestion can be classified into four types as shown in Fig.4

Figure 3: Intensity of congestion-relation between duration and distance

The variation in extent and duration of congestion indicates different problems requiring different solutions. Small delay and extent indicates limited problem, small delay for large extent indicates general congestion, great delay for small extent indicates critical links and great delay for large extent indicates critical system-wide problem.

Figure 4: Intensity of congestion-Relation between extent and duration of delay

The product of extent and duration indicates the intensity, or magnitude of the congestion problem.

Congestion countermeasures

Various measures to address congestion are discussed here. These include supply side, demand side, and pricing.

Supply measures:

Congestion countermeasures on the supply side add capacity to the system or make the system operate more efficiently. They focus on the transportation system. Supply measures include

1. Development of new or expanded infrastructure. This includes civil projects (new freeways, transit lines etc), road widening, bridge replacement, technology conversions(ITS),etc.
2. Small scale capacity and efficiency improvement. This includes signal system upgrade and coordination, freeway ramp metering, re-location of bus stops etc.

Demand measures:

Demand measures focuses on motorists and travelers and attempt to modify their trip making behaviour. Demand measures include:

1. Parking pricing: It discourages the use of private vehicles to specific areas, thereby reducing the demand on the transportation system.
2. Restrictions on vehicle ownership and use: It includes heavy import duties, separate licensing requirement, heavy annual fees, expensive fuel prices, etc. to restrain private vehicle acquisition and use.

Congestion pricing

Congestion pricing is a method of road user taxation, charging the users of congested roads according to the time spent or distance travelled on them. The principle behind congestion pricing is that those who cause congestion or use road in congested period should be charged, thus giving the road user the choice to make a journey or not.

Economic principle behind congestion pricing

Journey costs are made of private journey cost, congestion cost, environmental cost, and road maintenance cost. The benefit a road user obtains from the journey is the price he prepared to pay in order to make the journey. As the price gradually increases, a point will be reached when the trip maker considers it not worth performing or worth performing by other means. This is known as the critical price. At a cost less than this critical price, he enjoys a net benefit called as consumer surplus(es) and is given by:

$$s=x-y$$ (7)

where, $x$ is the amount the consumer is prepared to pay, and $y$ is the amount he actually pays. The basics of congestion pricing involves demand function, private cost function as well as marginal cost function. These are explained below.

Demand

Fig. 5 shows the general form of a demand curve. In the figure, area QOSP indicates the absolute utility to trip maker and the area SRP indicates the net benefit.

Figure 5: Demand Curve

Private cost

Total private cost of a trip, is given by:

$$c=a+\frac{b}{v}$$ (8)

where, $a$ is the component proportional to distance, $b$ is the component proportional to speed, and $v$ is the speed of the vehicle (km/h) which is given by:

$$v=d-eq$$ (9)

where, $q$ is the flow in veh/hour, $d$ and e are constants.

Marginal cost

Marginal cost is the additional cost of adding one extra vehicle to the traffic stream. It reduces speed and causes congestion and results in increase in cost of all journey. The total cost incurred by all vehicles in one hour($C_T$) is given by:

$$C_T=cq$$ (10)

Marginal cost is obtained by differentiating the total cost with respect to the flow($q$) as shown in the following equations.

$$\frac{d(cq)}{dq} = c+q\frac{dc}{dq}$$ (11)

$$\frac{dc}{dq} = \frac{dc}{dv}\times\frac{dv}{dq}$$ (12)

$$= (-b)/v^2 \times-e$$ (13)

$$= be/v^2$$ (14)

$$\frac{d(cq)}{dq} = c+q\frac{dc}{dq}$$ (15)

$$= a+\frac{b}{v}+\frac{d-v}{e}\times\frac{be}{v^2}$$ (16)

Note that c and q in the above derivation is obtained from Equations 8 and 9 respectively. Therefore the marginal cost is given as:

$$M=a+\frac{b}{v}+\frac{(d-v)b}{v^2}$$ (17)

Fig. 6 shows the variation of marginal cost per flow as well as private cost per flow.

Figure 6: Private cost/flow and cost and marginal curve

It is seen that the marginal cost will always be greater than the private cost, the increase representing the congestion cost.

Equilibrium condition and Optimum condition

Superimposing the demand curve on the private cost/flow and marginal cost/flow curves, the position as shown in Fig. 7 is obtained. The intersection of the demand curve and the private costs curve at point A represents the equilibrium condition, obtained when travel decisions are based on private costs only. The intersection of the demand curve and the marginal costs curve at point B represents the optimum condition. The net benefit under the two positions A and B are shown by the areas ACZ and $BYC_{Y}Z$ respectively. If the conditions are shifted from point A to B, the net benefit due to change will be given by area $CC_{y}YX$ minus AXB. If the area $CC_{y}YX$ is greater than arc AXB, the net benefit will be positive. The shifting of conditions from point A to B can be brought about by imposing a road pricing charge BY. Under this scheme, the private vehicles continuing to use the roads will on an average be worse off in the first place because BY will always exceed the individual increase in benefits XY.

Figure 7: Relation between material cost, private cost and demand curves.

Numerical example

Vehicles are moving on a road at the rate of 500 vehicle/hour, at a velocity of 15 km/hr. Find the equation for marginal cost.

Solution:

$$\begin{eqnarray*} c &=& \frac{a+b}{v} \\ &=& \frac{a+b}{15} \\ v &=& d-eq \\ ... ...frac{(d-v)b}{v^2} \\ &=& a+\frac{b}{15}+\frac{(d-15)b}{225} \\ \end{eqnarray*}$$

Uses of congestion pricing

1. Diverts travelers to other modes
2. Causes cancellation of non essential trips during peak hours
3. Collects sufficient fund for major upgrades of highways
4. Cross-subsidizes public transport modes

Requirements of a good pricing system

1. Charges should be closely related to the amount of use made of roads
2. Price should be variable at different times of day/week/year or for different classes of vehicles
3. It should be stable and ascertainable by road users before commencement of journey
4. Method should be simple for road users to understand and police to enforce
5. Should be accepted by public as fair to all
6. Payment in advance should be possible
7. Should be reliable
8. Should be free from fraud or evasion
9. Should be capable of being applied to the whole country

Conclusion

In this lecture, we have discussed about the causes and effects of congestion and how congestion can be defined. We also discussed how congestion can be quantified by various performance measures such as duration, extent and intensity. The measures to be taken in order to counteract congestion were also discussed. The principle and process of congestion pricing was also discussed.

Bibliography