mixed manual/semi-automated traffic: a macroscopic analysis

Transportation Research Part C 11 (2003) 439–462www.elsevier.com/locate/trc

Mixed manual/semi-automated traffic:a macroscopic analysis q

Arnab Bose a,*, Petros Ioannou b

a Real-Time Innovations, Inc., 155A Moffett Park Drive, Suite 111, Sunnyvale, CA 94089, USAb Department of Electrical Engineering-Systems, EEB200B, Center for Advanced Transportation Technologies,

University of Southern California, Los Angeles, CA 90089-2562, USA

Received 7 August 2000; accepted 22 April 2002

Abstract

The use of advanced technologies and intelligence in vehicles and infrastructure could make the current

highway transportation system much more efficient. Semi-automated vehicles with the capability of au-

tomatically following a vehicle in front as long as it is in the same lane and in the vicinity of the forward

looking ranging sensor are expected to be deployed in the near future. Their penetration into the current

manual traffic will give rise to mixed manual/semi-automated traffic. In this paper, we analyze the fun-

damental flow–density curve for mixed traffic using flow–density curves for 100% manual and 100% semi-automated traffic. Assuming that semi-automated vehicles use a time headway smaller than today�s manual

traffic average due to the use of sensors and actuators, we have shown using the flow–density diagram that

the traffic flow rate will increase in mixed traffic. We have also shown that the flow–density curve for mixed

traffic is restricted between the flow–density curves for 100% manual and 100% semi-automated traffic. We

have presented in a graphical way that the presence of semi-automated vehicles in mixed traffic propagates

a shock wave faster than in manual traffic. We have demonstrated that the presence of semi-automated

vehicles does not change the total travel time of vehicles in mixed traffic. Though we observed that with

50% semi-automated vehicles a vehicle travels 10.6% more distance than a vehicle in manual traffic for thesame time horizon and starting at approximately the same position, this increase is marginal and is within

the modeling error. Lastly, we have shown that when shock waves on the highway produce stop-and-go

qThis work is supported by the California Department of Transportation through PATH of the University of

California. The contents of this paper reflect the views of the authors who are responsible for the facts and accuracy of

the data presented herein. The contents do not necessarily reflect the official views or policies of the State of California

or the Federal Highway Administration. This paper does not constitute a standard, specification or regulation.* Corresponding author.

E-mail addresses: [email protected] (A. Bose), [email protected] (P. Ioannou).

0968-090X/$ - see front matter � 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/j.trc.2002.04.001

mail to: [email protected]

440 A. Bose, P. Ioannou / Transportation Research Part C 11 (2003) 439–462

traffic, the average delay experienced by vehicles at standstill is lower in mixed traffic than in manual traffic,

while the average number of vehicles at standstill remains unchanged.

� 2003 Elsevier Ltd. All rights reserved.

Keywords: Intelligent cruise control (ICC) vehicles; Semi-automated vehicles; Mixed traffic; Flow–density curve;

Macroscopic behavior; Shock waves; Space–time diagrams; Queuing theory

1. Introduction

During the past decade, researchers have focussed on improving traffic flow conditions with thehelp of automation. Several concepts that have been proposed include automation in the driver–vehicle system or the infrastructure or both. While fully automated vehicles on dedicated high-ways are seen as a far in the future objective, the use of partial or semi-automated vehicles oncurrent highways with manually driven vehicles is deemed as a near-term goal.

Semi-automated vehicles are those that have the capability of automatically following a vehiclein front as long as it is in the same lane and within the range of the forward looking ranging sensor(Bose and Ioannou, 1998). The vehicles use a longitudinal controller such as the intelligent cruisecontrol (ICC) system that comprises of throttle and brake subsystems (Ioannou and Xu, 1994,p. 345). The external input variables used by the ICC vehicle are the relative speed and the relativedistance between itself and the leading vehicle (if any), in addition to its own speed obtained usingdifferent sensors (Walker and Harris, 1993, p. 105).

The use of sensors and actuators makes it possible for semi-automated vehicles to have a lowerreaction time than manually driven vehicles. As a result, a semi-automated vehicle uses a smallertime headway and intervehicle spacing than current manual traffic average and reacts almostinstantaneously to a speed differential in comparison to a manual vehicle. Intervehicle spacing isthe distance between the same points in successive vehicles. Time headway is the time taken tocover the distance between the rear of the front vehicle to the front of the following vehicle.Human factor considerations demand that the response of an ICC vehicle be smooth for pas-senger comfort. Hence a semi-automated vehicle acts as a filter that smoothes out traffic flowdisturbances (Bose and Ioannou, 1999; Bose and Ioannou, in press). The principle question iswhat will be the effect of the gradual penetration of semi-automated vehicles among manual oneson the mixed traffic flow, especially during the presence of disturbances.

In this paper we focus our attention to the macroscopic analysis of mixed traffic flow. Twotopics addressed are fundamental flow–density diagrams and shock waves. A fundamental flow–density diagram defines the steady-state relation between the traffic flow rate and the trafficdensity (Lighthill and Whitham, 1955, p. 317). It is dependent on factors under which the trafficflow rate and the traffic density are observed like the length of time interval over which data isaggregated (Leutzbach, 1972). We outline flow–density diagrams for 100%manual and 100% semi-automated traffic. A linear follow-the-leader human driver model is used to model the dynamics ofmanually driven vehicles in a single lane of a highway (Drew, 1968; Chandler et al., 1958; Chabaniand Papageorgiou, 1997). The model assumes that the human driver observes only the vehicle infront and sets its vehicle acceleration depending on the relative speed and the relative distance. Arecent study shows that the model described in Pipes (1953, p. 271) closely approximates manual

A. Bose, P. Ioannou / Transportation Research Part C 11 (2003) 439–462 441

vehicle responses involving different drivers on a single lane on a highway (Bose and Ioannou, inpress). A manual traffic flow–density curve for a single lane on a highway is constructed using thishuman driver model. It has a stationary point that corresponds to maximum manual traffic flowrate. A flow–density curve for 100% semi-automated vehicles for a single highway lane is con-structed using the ICC design given in Ioannou and Xu (1994, p. 345). The highest point of thecurve corresponds to maximum semi-automated traffic flow rate. We show using these flow–density curves that for any given traffic density, mixed traffic flow rate is greater than manual trafficflow rate. We also show that the mixed traffic flow–density curve is restricted in the region en-capsulated by these curves. As the percentage of semi-automated vehicles increases, this curveconverges to the flow–density curve for 100% semi-automated traffic.

The effect of semi-automated vehicles among manual ones during the presence of traffic flowdisturbances such as shock waves is shown using space–time diagrams and demonstrated usingsimulations. Shock waves are discontinuous waves that occur when traffic on a section of a road isdenser in front and less dense behind. We present in a graphical way that the presence of semi-automated vehicles in mixed traffic propagates a shock wave faster than in manual traffic. Wedemonstrate that a vehicle in mixed traffic with 50% semi-automated vehicles travels 10.6% moredistance than a vehicle in manual traffic for the same time horizon and starting at approximatelythe same position. A linear follow-the-leader human driver model, namely Pipes model (Pipes,1953, p. 271; Chandler et al., 1958, p. 165) has been validated to closely model current manualdriving (Bose and Ioannou, in press). Hence it is used to model the dynamics of manual vehiclesduring the simulations. A three-dimensional representation of space–time diagrams is used toshow that the traffic flow rate and the traffic density in mixed traffic are greater than that inmanual traffic.

Lastly, we show that when shock waves produce stop-and-go traffic on the highway, the av-erage delay experienced by vehicles is lower in mixed traffic than in manual traffic, while theaverage number of vehicles at standstill remains unchanged.

This paper is organized as follows: Section 2 introduces the traffic flow variables and analyzesmixed traffic flow using flow–density diagrams for 100% manual, 100% semi-automated andmixed traffic. Section 3 deals with shock waves and the effect of semi-automated vehicles on them.Section 4 compares the average delay experienced by vehicles during stop-and-go conditions onthe highway in manual and mixed traffic. A summary of the main results is provided in theconcluding Section 5.

2. Fundamental flow–density diagram

Traffic flow rate q and traffic density k are average measures of traffic flow characteristics. Theprecise definition of q and k and the means of measuring them are explained in Lighthill andWhitham (1955, p. 317). In this section we consider the flow–density diagrams for 100% manualtraffic, 100% semi-automated traffic and analyze the flow–density curve for mixed manual/semi-automated traffic for a single lane on a highway. A fundamental flow–density diagram defines thesteady-state relation between the traffic flow rate and the traffic density (Lighthill and Whitham,1955, p. 317). It is dependent on factors under which the traffic flow rate and the traffic density areobserved like the length of time interval over which data is aggregated (Leutzbach, 1972).


2.1. Manual traffic

A linear car-following model is used to model the motion of a manually driven vehicle on asingle lane on a highway. The car-following theory assumes that traffic stream is a superpositionof vehicle pairs where each vehicle follows the vehicle ahead according to some specific stimulus–response equation (Drew, 1968; Pipes, 1953; Chabani and Papageorgiou, 1997). Using Fig. 1 wehave the acceleration of the ðnþ 1Þ-th vehicle given by Leutzbach (1972)

€xxnþ1ðt þ sÞ ¼ k½ _xxnðtÞ � _xxnþ1ðtÞ� ð1Þ

where s is the reaction time lag, k the sensitivity factor and k ¼ k0_xxmnþ1

ðtþsÞ½xnðtÞ�xnþ1ðtÞ�l

.

This is referred to as the ðm; lÞ model. For m ¼ 0, l ¼ 1:5 we obtain

€xxnþ1ðt þ sÞ ¼ k0½ _xxnðtÞ � _xxnþ1ðtÞ�½xnðtÞ � xnþ1ðtÞ�1:5

or,

_vvnþ1ðt þ sÞ ¼ k0_ssnþ1

s1:5nþ1

ð2Þ

where vnþ1 the speed of ðnþ 1Þ-th vehicle, snþ1 ¼ xnðtÞ � xnþ1ðtÞ the intervehicle spacing of theðnþ 1Þ-th vehicle.

The intervehicle spacing is defined as the distance between same points in successive vehicles.To analyze the macroscopic effect we neglect the reaction time lag s. In Leutzbach (1972) it is

shown that the time lag does not affect the equation that describes the macroscopic traffic flowbehavior. Dropping the subscripts in (2) we get

_vv ¼ k0_sss1:5

ð3Þ

It can be shown that the reciprocal of the average intervehicle spacing is the traffic density(Cassidy, 2002, p. 151). Therefore, using k ¼ 1

s where k represents the traffic density, we can ex-press (3) as

_vv ¼ �k0k1:5_kkk2

ð4Þ

Fig. 1. Vehicle following in a single lane.


Integrating (4) we have
Z 0
v_vv ¼ �k0

Z kj

k

_kkk0:5

ð5Þ

where the limits are from the present point ðv; kÞ to the jam condition where all vehicles areat standstill, i.e. v ¼ 0 and k ¼ kj, where kj is the traffic density at jam condition. Evaluating (5)we get

v ¼ 2k0ðffiffiffiffikj

p�

ffiffiffik

pÞ ð6Þ

The proportionality constant k0 in (6) can be expressed in terms of speed and traffic density usingv ¼ vf when k ¼ 0, i.e. when the density is negligible, there is no vehicle-to-vehicle interaction andvehicles travel at mean free speed vf . The mean free speed is the vehicle speed that is not affectedby other vehicles and only subjected to constraints associated with the vehicle and road charac-teristics. That gives us

k0 ¼vf

2ffiffiffiffikj

p
Substituting for k0 in (6) we obtain the average speed of vehicles on a section of the road
v ¼ vf 1

�

ffiffiffiffikkj

s !ð7Þ

where vf is the mean free speed that corresponds to negligible traffic density when there is nointeraction among vehicles, k the traffic density on the section of the road, kj the jam density equalto 1=L where L is the average length of vehicles, i.e. the traffic density corresponding to jamconditions when the vehicles are stacked bumper-to-bumper and the speed is zero.

The traffic flow rate q at steady state measures the number of vehicles moving in a specifieddirection on the road per unit time and is given by

q ¼ kv ð8Þ
Substituting v from (7) we get an equation of the manual traffic flow–density relationship at steadystate given by
q ¼ kv ¼ kvf 1

�

ffiffiffiffikkj

s !� QðkÞ ð9Þ

which is also used to obtain the fundamental q–k diagram for a single lane on a highway as shownin Fig. 2.

The speed of waves carrying continuous changes of flow along the vehicle stream is the slope ofthe tangent to the q–k curve at a point and is given by

c ¼ dqdk

¼ dðkvÞdk

¼ vþ kdvdk

ð10Þ

The wave speed is smaller than the average speed of the vehicles when the latter decreases withincrease in traffic density, i.e. dv

dk 6 0. The equality holds when the density is very low and anyincrease does not affect the average speed of the vehicles. Such conditions exist in the vicinity of

flow(veh/hr)

qma

100% s.a.

qmm

100% man

kcm kca k j density(veh/km)

O

C

J

Fig. 2. Fundamental flow–density curves for 100% manual traffic and 100% semi-automated traffic.


the origin of the q–k diagram where the density is negligible and there is no vehicle-to-vehicleinteraction.

The q–k curve has a stationary point that corresponds to a critical density kcm that gives themaximum traffic flow rate qmm or the capacity on the section of the road. The slope of the linejoining a point on the q–k curve to the origin gives the average speed of vehicles at that point. Eq.(9) is an example of a flow–density model derived from the ðm; lÞ model and is used to qualita-tively describe manual traffic flow characteristics. Different values of m and l are used to obtain agood fit to actual traffic flow data. For example, in Lighthill and Whitham (1955, p. 317) it ismentioned that the flow–density model derived with m ¼ 0:8 and l ¼ 2:8 gives a good fit to dataobserved on an expressway in Chicago. The flow–density model derived using m ¼ 1 and l ¼ 3 isshown to be qualitatively valid for traffic flow characteristics observed in 3-lane Boulevard Pe-ripherique de Paris (Chabani and Papageorgiou, 1997). Likewise, m ¼ 0 and l ¼ 1 is used toderive a flow–density model that gives a good fit to actual traffic flow data taken in the LincolnTunnel in New York (Edie, 1974).

The point ðqmm; kcmÞ has been empirically observed to be unstable, i.e. it leads to a breakdownin traffic flow. When traffic flow conditions exist at or near this point, the traffic flow and theaverage speed decreases as traffic density increases, and the operating point moves towards thejam density kj on the q–k curve. This observation is explained in Transportation Research Board(1985). As capacity is approached, flow tends to become unstable as the number of available gapsreduces. Traffic flow at capacity means that there are no usable gaps left. A disturbance in such acondition due to lane changing or vehicle merging is not �effectively damped� or �dissipated�.Mathematically, this corresponds to the unattenuated upstream propagation of density distur-bances. This leads to a breakdown in traffic flow and �formation of upstream queues�. As pointedin Transportation Research Board (1985), this is the reason why all facilities are designed tooperate at lower than maximum traffic flow conditions.

2.2. Semi-automated traffic

While the behavior of human drivers is random and at best we can develop a manual trafficflow–density model that is qualitatively valid, the responses of semi-automated vehicles are de-


terministic due to the use of computerized longitudinal controllers. In this section, we utilize thisadvantage along with certain assumptions to develop a deterministic semi-automated traffic flow–density model for a single lane on a highway that is used to obtain the fundamental q–k diagram.

A semi-automated vehicle uses a longitudinal controller to automatically follow a vehicle in thesame lane and within the range of its forward looking ranging sensor. Many such controllerdesigns exist in literature that use constant time headway (Ioannou and Xu, 1994, p. 345; Ioannouet al., 1992) and constant spacing (Hedrick et al., 1991, p. 3107) policies. In this paper we considerthe ICC design given in Ioannou and Xu (1994, p. 345) which uses a constant time headwaypolicy. The time headway is defined as the time taken to travel the distance between the rear endof the lead vehicle and the front end of the following vehicle.

The intervehicle spacing s at a speed v followed by a semi-automated vehicle using a constanttime headway ha is given by

s ¼ havþ L ð11Þ
where L is the length of the vehicle. For simplicity we assume that all semi-automated vehicleshave the same length L and use the same time headway ha.
The average intervehicle spacing is the reciprocal of traffic density. When traffic density is suchthat the intervehicle spacing is greater than or equal to that given by (11), then during this periodat steady state the traffic flow rate given by (8) increases linearly with the density of semi-auto-mated vehicles according to

q ¼ kvf

where vf is the mean free speed. This situation is indicated by the line OC in Fig. 2.As the traffic density increases and reaches the value

k ¼ kca ¼1

havf þ L

the maximum semi-automated traffic flow rate qma or capacity is reached. Thereafter, the relationbetween q and k can no longer change unless the speed vf or the time headway ha changes. It isreasonable to assume that the time headway ha is fixed at all speeds and further increase in koccurs due to changes in speed. In such a case we have

s ¼ havþ L < havf þ L ð12Þ
and
k ¼ 1

havþ L> kca ¼

1

havf þ Lð13Þ

Thus, the speed of the semi-automated vehicles decreases with increasing k according to

v ¼ 1

ha

1

k

�� L

�ð14Þ

The traffic flow rate is given by

q ¼ kv ¼ 1

hað1� kLÞ


Graphically, the slope of the line joining the origin to a point on the q–k curve gives the speed ofthe semi-automated vehicles at that point. It can be seen that before the critical density the speedremains constant at vf . After reaching the critical density kca, the speed falls as the traffic densityincreases (Fig. 2).

Therefore the steady state fundamental flow–density diagram for 100% semi-automated trafficis given by

q ¼ kvf k6 kca1hað1� kLÞ k > kca

�ð15Þ

Eq. (15) describes the average steady state traffic flow characteristics. They do not capture anyeffects due to individual vehicle responses.

In region OC of the q–k curve, the critical density is not yet reached. The average speed of thesemi-automated vehicles is equal to the mean free speed vf . This region corresponds to ‘‘loose’’vehicle following where some semi-automated vehicles travel without following any vehicle anduse an intervehicle spacing greater than that given by (11). The rest follow a lead vehicle and usean intervehicle spacing given by (11). After C the average speed of the semi-automated vehiclesalong with the traffic flow rate begin to decrease with increase in density. This region indicated bythe line CJ corresponds to ‘‘tight’’ vehicle following where all semi-automated vehicles follow alead vehicle and use an intervehicle spacing given by (12). Finally, at maximum density kj all semi-automated vehicles are at dead stop and stacked bumper-to-bumper.

Let us now consider the stability of the point ðqma; kcaÞ. We mentioned in the previous sub-section that the corresponding point ðqmm; kcmÞ in manual traffic has been empirically observed tobe unstable. Obviously, it is not possible to empirically determine the stability of ðqma; kcaÞ.However, it is expected that the point ðqma; kcaÞ will also be unstable, i.e. at this operating pointthere will be a breakdown in traffic flow. As in manual traffic, flow will tend to become unstable ascapacity is approached and the number of available gaps reduces. At critical density there will beno usable gaps left. Any disturbance generated at this condition is expected to lead to the for-mation of upstream queues and a breakdown in traffic flow, as observed in manual traffic. Fur-thermore, it has been shown that traffic flow in the region CJ where the intervehicle spacing isgiven by (13) is unstable in the sense that density disturbances propagate upstream unattenuated(Swaroop and Rajagopal, 1999, p. 329).

2.3. Mixed manual/semi-automated traffic

In the previous subsections, at one end we developed a flow–density model that qualita-tively describes manual traffic flow characteristics. At the other end, we developed a determin-istic flow–density model for semi-automated traffic. A combination of these two models isexpected to characterize mixed manual/semi-automated traffic flow, which we analyze in thissection.

We assume that at steady state conditions vehicles of the same class use identical headways andintervehicle spacings. For semi-automated vehicles this is equal to the designed value of the ICCcontroller. For manual vehicles, we take it to be equal to the average value observed in currentmanual traffic.

Fig. 3. Mixed manual/semi-automated traffic.


Consider the mixed traffic flow shown in Fig. 3. Manual vehicles are marked �M� and semi-automated vehicles are marked �A�. Assuming moderately dense traffic conditions, the averageintervehicle spacing at steady state conditions is given by

�ss ¼ psa þ ð1� pÞsm ð16Þ
where p is the market penetration of semi-automated vehicles in mixed traffic, sa ¼ havþ L theintervehicle spacing used by semi-automated vehicles at speed v with time headway ha; L is theaverage length of the vehicle, sm ¼ hmvþ L the average intervehicle spacing used by manual ve-hicles at speed v for a time headway hm which is the average of current manual traffic timeheadway distribution.
Remark 1. Throughout this paper, we assume that due to the use of sensors and actuators insemi-automated vehicles, ha < hm, i.e. sa < sm at a given speed and for the same average length ofmanual and semi-automated vehicles (Bose and Ioannou, 1998).

The total mixed traffic density is given by

kmix ¼1

�ss� f ðp; sa; smÞ ð17Þ

We assume that the driver of a manual vehicle behaves the same way whether following a semi-automated vehicle or a manual vehicle. Under this assumption, we can use the fundamental flow–density curve for 100% manual traffic to determine the characteristics of manual vehicles in mixedtraffic on a single lane on a highway. Likewise, the flow–density curve for 100% semi-automatedtraffic can be used for semi-automated vehicles in mixed traffic on a single lane on a highway, astheir vehicle dynamics remain unchanged when following a manually driven vehicle.

The steady state speed of mixed traffic is determined by the speed of the slowest moving vehicleclass (manual or semi-automated). Using the above assumption, we take the inverse of the averageintervehicle spacing for each vehicle class as their density, i.e. km ¼ 1

smand ka ¼ 1

sa. Then use (7) and

(14) to determine the average speeds at which the manual and the semi-automated vehicles cantravel. The lower of the two determines the speed at which all vehicles travel in mixed traffic. Notethat from (17) the density of each class is dependent on the other.

Lemma 1. The q–k curve for mixed manual/semi-automated traffic remains in the region between theq–k curves for 100% manual and 100% semi-automated traffic. Furthermore, the mixed traffic flowrate is greater than the manual traffic flow rate for the same traffic density.

Fig. 4. vm < va: The mixed traffic q–k operating point om lies between the 100% manual traffic operating point o2 andthe 100% semi-automated traffic operating point o01 corresponding to vmix ¼ vm.


Proof. Consider a mixed manual/semi-automated traffic with traffic densities of manually drivenvehicles and semi-automated vehicles equal to km and ka, respectively. From Fig. 4, since vm < va,the average steady state mixed traffic speed is vmix ¼ vm.

From (16), we have

�ss ¼ psa þ ð1� pÞsm ð18Þ

) 1

kmix

¼ pka

þ ð1� pÞkm

ð19Þ

) 1

kmixvmix

¼ pkavmix

þ ð1� pÞkmvmix

ð20Þ

) 1

qmix

¼ pqa

þ ð1� pÞqm

ð21Þ

Thus the throughput for mixed manual/semi-automated traffic is the weighted harmonic mean ofthe throughput of fully manual and fully semi-automated traffic. Since p > 0, it can be seen that atany given speed vmix, the mixed traffic throughput is between the manual traffic and the semi-automated traffic throughputs. In other words, the q–k curve for mixed manual/semi-automatedtraffic remains in the region between the q–k curves for 100% manual and 100% semi-automatedtraffic, as shown in Figs. 5 and 6. The new point in the fundamental diagram of mixed traffic isgiven by om corresponding to mixed traffic flow rate fmix. h

Fig. 5. vm > va: The mixed traffic q–k operating point om lies between the 100% semi-automated traffic operating point

o2 and the 100% manual traffic operating point o01 corresponding to vmix ¼ va.


Lemma 1 and its proof give an insight into the evolution of mixed traffic. To observe the be-havior of the mixed traffic q–k curve operating point as the percentage p of semi-automated ve-hicles increases, we consider two separate cases:

Case 1. Assume that as p increases, the average speed is equal to the speed of the manual vehiclesand ka 6 kca. Then from (7) as the percentage of semi-automated vehicles increases, i.e. p ! 1,km ¼ k ! 0 and v ! vf . From (17) kmix ! ka to give us

q ¼ limp!1

kmixv ¼ kavf ð22Þ

This is the same as (15) for ka 6 kca.

Case 2. Assume that as p increases, the average speed is equal to the speed of the semi-automatedvehicles. This means that ka > kca. Otherwise the average speed of semi-automated vehicles will bevf , the maximum for manual vehicles, which is not possible. Then from (14) as p ! 1, v ! 1

hað 1ka �

LÞ and from (17) kmix ! ka to give us

q ¼ limp!1

kmixv ¼1

hað1� kaLÞ ð23Þ

This is the same as (15) for ka > kca.

Thus we observe how the q–k curve for mixed traffic converges to the q–k curve for 100% semi-automated traffic as the percentage of semi-automated vehicles increases.

Fig. 6. vm ¼ va: The mixed traffic q–k operating point om lies between the 100% semi-automated traffic operating point

o2 and the 100% manual traffic operating point o1 corresponding to vmix ¼ vm ¼ va.


Now we draw a q–k curve for a given market penetration of semi-automated vehicles with thehelp of the analyses presented above. The initial points of the curve merge with the q–k curves for100% semi-automated and 100% manual traffic as seen in Fig. 7, denoting the region where allvehicles travel at mean free speed and there is no vehicle-to-vehicle interaction. Thereafter wetrace the bivariate relationship for mixed traffic using the average intervehicle spacings used by themanual and the semi-automated vehicles as outlined previously. Each unique p corresponds to aunique mixed traffic q–k curve.

Lastly, we investigate the mixed traffic critical density. Each curve has a maximum traffic flowpoint that corresponds to the critical density. Therefore, the mixed traffic critical density dependson the combination of the intervehicle spacings sa and sm used by the semi-automated and themanual vehicles, respectively. Furthermore, the operating point on the q–k curve at critical densitycorresponding to maximum mixed traffic flow is expected to be unstable and lead to a breakdownin mixed traffic flow. Since at critical density there are no available gaps left, a phenomenonsimilar to that observed in manual traffic is expected to occur in the event of a traffic disturbance.

3. Shock waves in mixed traffic

Shock waves are discontinuous waves that occur when traffic on a section of a road is denser infront and less dense behind. The waves on the less dense section travel faster than those in the

Fig. 7. Fundamental q–k diagrams for 100% semi-automated, 100% manual and p mixed traffic at steady state con-

ditions.


dense section ahead and catch up with them. Then the continuous waves coalesce into a dis-continuous wave or a �shock wave� (Lighthill and Whitham, 1955, p. 317). It can be shown thatshock waves travel at a speed given by

u ¼ DqDk

ð24Þ

where Dq and Dk are the traffic flow rate and traffic density differences, respectively, between thetwo sections. In the fundamental diagram for manual traffic, this is given by the slope of the chordjoining the two points that represent conditions ahead and behind the shock wave at a and b,respectively (Fig. 8).

The question that we pose is: how does the mixing of semi-automated and manual vehiclesaffect the shock waves? Since different degree of penetration of semi-automated vehicles corre-spond to different flow–density diagrams with different stationary maximum flow points, thisquestion cannot be easily addressed using mixed traffic flow–density q–k curves and thus, warrantsfurther investigation. We answer the question using space–time diagrams that represent vehicletrajectories on a single lane on a highway. This simple theory of traffic evolution will provide aninsight into the effect of semi-automated vehicles among manually driven ones on the propagationof shock waves. The theory is also used in Cassidy (2002, p. 151) and is a version of one originallydeveloped in Lighthill and Whitham (1955, p. 317). It is presented in a graphical way and assumesthe following: no overtaking, i.e. all vehicles follow another vehicle in a single lane and vehicles of

Fig. 8. Shock wave in manual traffic.


the same class (i.e., manual and semi-automated) exhibit identical headways, spacings and ve-locities within a given state.

Fig. 9(a) shows the space–time graph for five vehicles in manual traffic. At time instant td thereis a disturbance that causes the lead vehicle to slow down, denoted by a change in slope in thediagram. We assume that the vehicle decelerations occur instantaneously. After a time lag s, thefollowing vehicle reacts and decelerates to maintain the intervehicle spacing depending on aconstant time headway policy. Likewise, the rest of the manual vehicles respond as shown. Theshock wave that exists at the boundary of the two regions moves back at a speed given by theslope of the arrow in Fig. 9(a).

For mixed traffic shown in Fig. 9(b), the lines marked �s� denote semi-automated vehicleswhile the rest are manually driven ones. As before, the disturbance at td causes the first vehicleto slow down. The following semi-automated vehicle reacts after a time t, where t depends onthe sensors and the actuators of the semi-automated vehicle and t � s, i.e. much smaller thanthe time lag for human drivers in manual vehicles. This effect continues upstream with thesemi-automated vehicles reacting almost instantaneously to speed changes while the manualvehicles have a time lag s. As shown in Fig. 9(b), the resultant effect is that the average slopeof the shock wave increases and as a result it travels upstream at a higher speed than in Fig.9(a).

This phenomenon is illustrated in a three-dimensional representation of the space–time dia-gram. We construct an axis of cumulative number of vehicles at three different distance points andat three different time instances. The surface of the cumulative vehicle count Nðx; tÞ is staircase-shaped, with each step representing a vehicle trajectory. It follows that if the surfaces are takento be continuous, then the traffic flow rate at a given point can be obtained from the Nðx; tÞ–tdiagrams directly by using

qxðtÞ ¼oNðx; tÞ

otð25Þ

Fig. 9. Space–time graph showing traffic evolution and the propagation of shock waves in (a) manual traffic and (b)

mixed traffic.


and the traffic density at a given time can be obtained from the Nðx; tÞ–x diagrams using

ktðxÞ ¼ � oNðx; tÞox

ð26Þ

We use a negative sign in (26) because the cumulative vehicle count is considered by counting thenumber of vehicles that have not yet crossed the line at a time instance t in the space–time dia-grams. This is the number of vehicles to the left of t, which is opposite to the motion of the ve-hicles and hence the negative derivative.

It is important to note that the continuum approximation to a discrete flow is valid only in theregime of dense traffic. Writing (25) and (26) together, we obtain the continuity equation

oqox

þ okot

¼ 0 ð27Þ

The cumulative vehicle count curve Nðx2; tÞ for the point x2 in the Nðx; tÞ–t diagrams in Fig. 10 isobtained by shifting the curve Nðx1; tÞ horizontally to the right by the time it takes a vehicle totravel from x1 to x2. However, this is not the proper representation of the cumulative curve at x2due to the presence of the shock wave. Thus we perform the following transformations. We shift

Fig. 10. (a) Three-dimensional representation of manual traffic. (b) Three-dimensional representation of mixed traffic.



the cumulative curve Nðx3; tÞ for x3 (a point after the shock wave) horizontally to the right by thetime it takes the shock wave to travel from x3 to x2, and vertically upwards by the number ofvehicles that cross the shock wave interface. The number of vehicles that cross the shock waveinterface is given by the product of traffic density in the region after the shock wave and thedistance from x2 to x3. This curve is then combined with Nðx2; tÞ to obtain the correct cumulativecurve N 0ðx2; tÞ at x2. We use a similar procedure to construct the cumulative vehicle count curveNðx; t2Þ at time instant t2 in Nðx; tÞ–x diagrams in Fig. 10. The cumulative N -curves are the onesthat will be observed if vehicles counts are taken using measurement devices at positions x1, x2 andx3 at time instances t1, t2 and t3.

We observe from the cumulative curves in Fig. 10 that the traffic flow rate and the trafficdensity are greater in mixed traffic than in manual traffic. Furthermore, we observe from theNðx; tÞ–t diagrams that the traffic flow rate at a given point decreases more rapidly in mixed trafficthan in manual traffic. Likewise, from the Nðx; tÞ–x diagrams, we can conclude that the rate ofincrease in traffic density is higher in mixed traffic than in manual traffic. These are consequencesof the shock wave travelling faster in mixed traffic than in manual traffic.

To demonstrate this we perform the following simulations for manual and mixed traffic on asingle lane on a highway. Consider a stretch of road of length 2.5 km subdivided into five sec-tions of 500 m each. A constant traffic flow rate is assumed along the road. The manual vehicledynamics are modeled using the Pipes linear car-following model from Pipes (1953, p. 271) andthe semi-automated vehicle dynamics are modeled using the ICC design presented in Ioannou andXu (1994, p. 345). The Pipes linear car following and the ICC models have been validated us-ing actual vehicle following experiments in Ioannou and Xu (1994, p. 345) and Bose and Ioannou(in press). All vehicles follow a constant time headway policy. The time headways for the manualvehicles are generated according to a lognormal distribution given in Cohen (1964) while for semi-automated vehicles they are considered to be 1 s. It is important to note that the time headwaydefined in Cohen (1964) is the time taken to cover the distance that includes the vehicle length. Themaximum value for the manual vehicle time headway is taken as 4 s. This is done to make thestudy applicable to current manual traffic where seldom a vehicle in moderately dense trafficconditions uses a time headway greater than 4 s. Fig. 11(a) shows the time headway of vehicles in100% manual traffic on the road. Initially all vehicles are travelling at 15 m/s. We introduce adisturbance in the 3rd section that lasts for 10 s and results in all vehicles in that section to slowdown to 5 m/s. The traffic density and average speed of each section of the road for 50 s are shownin Fig. 11(b) and (c), respectively. We observe a shock wave developing in Section 3 that causes apile-up of vehicles.

Next, we assume 50% semi-automated vehicles in mixed traffic that are placed randomly amongmanually driven vehicles on the road. The time headways for the vehicles are shown in Fig. 12(a).A similar disturbance is introduced that results in all vehicles in Section 3 to slow down to 5 m/s.We observe in Fig. 12(b) and (c) that the shock wave in Section 3 moves back at a speedhigher than that in manual traffic. As a result the vehicle pile-up spills over in Section 2 (Fig.12(c)).

We observe that the presence of semi-automated vehicles do not affect the total travel timeduring traffic disturbances when we compare the space–time graphs of vehicles in manual andmixed traffic in Fig. 13. The 27th vehicle in mixed traffic starts at approximately the same positionas the 20th vehicle in manual traffic, and travels 10.6% (covering 365 m) more distance than the

(b)

0

10

20

30

40

50

1 1.5 2 2.5 3 3.5 4 4.5 5

5

10

15

time [s]

100% manual traffic

section

aver

age

velo

city

[m/s

]

0

10

20

30

40

50

11.5 2 2.5 3 3.5 4 4.5 5

0

10

20

30

40

time [s]

100% manual traffic

section

aver

age

dens

ity [v

eh/k

m]

(c)

0 10 20 30 40 50 600.5

1

1.5

2

2.5

3

3.5

4

car #

head

way

[s]

100% manual traffic

(a)

Fig. 11. Macroscopic behavior of vehicles in 100% manual traffic. (a) Time headways of vehicles. (b) Average speed

distribution of vehicles in five sections of the highway. (c) Traffic density distribution of vehicles in the five sections

of the highway.


latter (covering 330 m) in 50 s. However, this increase is marginal and is within the modelingerror.

0 100.5

1 1.5 2 2.5 3 3.5 4 4.5

6

8

10

12

14

16

50% s.a. veh

section

aver

age

velo

city

[m/s

]

(b)

Fig. 12. Macroscopic behavior of vehicles in mixed traffic where 50% are semi-automated vehicles. (a) Time headways

of vehicles. (b) Average speed distribution of vehicles in five sections of the highway. (c) Traffic density distribution

of vehicles in five sections of the highway.


0 5 10 15 20 25 30 35 40 45 50850

900

950

1000

1050

1100

1150

1200

1250

1300

20th veh

19th veh

18th veh

time [s]

dist

ance

cov

ered

[m]

100% manual traffic

0 5 10 15 20 25 30 35 40 45 50850

900

950

1000

1050

1100

1150

1200

1250

1300

27th veh26th veh

25th veh

time [s]

dist

ance

cov

ered

[m]

(a)

(b)

Fig. 13. Distance covered by vehicles starting at approximately the same place in (a) 100% manual traffic and (b) mixed

traffic with 50% semi-automated vehicles.


4. Stop-and-go traffic

In this section, we show that the average delay experienced by vehicles on the highway due tostop-and-go conditions is shorter in mixed traffic than in manual traffic. Consider Fig. 14 that

x

time

q1

q2

1

3

2

1-22-3

Fig. 14. Stop-and-go traffic.


shows two shock waves 1–2 and 2–3 that create a stop-and-go condition. The vehicles come to acomplete stop after the first shock wave 1–2 as shown in the shaded region 3. Thereafter reaching thesecond shock wave 2–3 they start moving again. As before, we consider vehicles in a single lane andassume instantaneous acceleration/deceleration of vehicles and vehicles of the same class use identicaltime headways and spacings at a given state. Consider random vehicle arrivals at the shock wave 1–2at a rate q1 per unit time, which is the traffic flow rate in region 1.We take the vehicle discharge rate tobe equal to q2 that represents the traffic flow rate in region 2. The stop-and-go situation can be viewedas a first-come-first-serve system as the first vehicle to stop is the first one to start moving. Thereforewe can model the system as an M/M/1 queue with Poisson arrivals and exponential service rate(Kleinrock, 1975). The theoretical Poisson distribution model is the most commonly used model forvehicular traffic (Cleveland and Capelle, 1964; Edie, 1954, p. 107). It gives a ‘‘satisfactory fit’’ withempirical traffic data, even at high traffic flow rates as shown in Edie (1954, p. 107).

We have the following arrival and service rates

k ¼ q1 and l ¼ q2

The utilization factor is given by

q ¼ kl¼ q1

q2ð28Þ

We assume that q2 > q1, i.e. q < 1, which is required for ergodicity (Kleinrock, 1975).

4.1. Manual traffic

For manual traffic we have the following:

q1 ¼ kmv1 ¼v1

hmv1 þ L

where q1 is the traffic flow rate in region 1, km ¼ 1hmv1þL is the traffic density, hm is the average time

headway in manual traffic, v1 is the average speed of vehicles in region 1 and L is the averagelength of vehicles.


q2 ¼ kmv2 ¼v2

hmv2 þ L

where q2 is the traffic flow rate in region 2 and v2 is the average speed of vehicles in region 2.

Remark 2. For ergodicity, we have

q2 > q1 ) v2 > v1 ð29Þ
The average delay experienced by a vehicle in region 3 due to the stop-and-go condition is givenby Kleinrock (1975)
T1 ¼1=l1� q

¼ 1

q2 � q1¼ ðhmv1 þ LÞðhmv2 þ LÞ

Lðv2 � v1Þð30Þ

4.2. Mixed traffic

For mixed traffic we have

�qq1 ¼ kmixv1 ¼v1

hmixv1 þ L

where �qq1 is the mixed traffic flow rate in region 1, hmix ¼ pha þ ð1� pÞhm is the average timeheadway in mixed traffic, ha is the time headway of semi-automated vehicles and p is the per-centage of semi-automated vehicles in mixed traffic.

�qq2 ¼ kmixv2 ¼v2

hmixv2 þ L

where �qq2 is the mixed traffic flow rate in region 2.So we have the average delay given by

T2 ¼1=l1� q

¼ 1

�qq2 � �qq1¼ ðhmixv1 þ LÞðhmixv2 þ LÞ


To show that the average delay experienced by vehicles at standstill in region 3 of Fig. 14 isshorter in mixed traffic than in manual traffic, we need to verify that

T1 � T2 > 0 8p p 2 ð0; 1Þ ð32Þ
Using (30) and (31) we obtain
T1 � T2 ¼ðhmv1 þ LÞðhmv2 þ LÞ

Lðv2 � v1Þ� ðhmixv1 þ LÞðhmixv2 þ LÞ


The denominator is always positive by (29). The numerator can be expressed as

h2mv1v2 þ hmLðv1 þ v2Þ � h2mixv1v2 � hmixLðv1 þ v2Þ ð34Þ

) ðh2m � h2mixÞv1v2 þ ðhm � hmixÞLðv1 þ v2Þ ð35Þ
which is always positive for all p as hm > hmix (since hm > ha and 0 < p < 1). Hence (32) is sat-isfied.


It is interesting to note that though the above phenomenon occurs, the average number of ve-hicles at standstill remains unchanged in manual and mixed traffic. It is given by Kleinrock (1975)

N ¼ q1� q

¼ v1v2 � v1

ð36Þ

which is independent of the average intervehicle spacings and hence the type of traffic.The above results agree with intuition. In Section 3 we observed that shock waves travel faster

in mixed traffic than in manual traffic. Thus the shaded region 3, where the vehicles are atstandstill, travels upstream faster in mixed traffic than in manual traffic. This explains (32).However, the relative speed between the two shock waves remains the same in both types oftraffic. Thus, the area or the number of vehicles they cover also remains unchanged, which isshown in (36). Furthermore, from the environmental perspective, lower average delay for vehiclesin mixed traffic during the presence of shock waves that produce stop-and-go traffic should reducefuel consumption and air pollution.

5. Conclusion

In this paper, we derive a q–k diagram for mixed traffic using q–k diagrams for 100% manualand 100% semi-automated vehicle traffic on a single lane on a highway, assuming that semi-au-tomated vehicles use a time headway smaller than current manual traffic average. We showgraphically that semi-automated vehicles propagate traffic disturbances such as shock waves fasterwithout affecting the total travel time. Furthermore, we use an M/M/1 queue to show that theaverage delay experienced by vehicles in mixed traffic is lower than those in manual traffic duringshock waves that produce stop-and-go traffic. Based on the macroscopic mixed traffic analysispresented in this paper, we conclude the following results:

• Mixed traffic flow rate is greater than manual traffic flow rate for the same traffic density.• Mixed traffic q–k curve remains in the region between the q–k curves for 100% manual and

100% semi-automated traffic.• Presence of semi-automated vehicles in mixed traffic increases the speed of propagation of

shock waves without affecting the total travel time.• Presence of semi-automated vehicles in mixed traffic increases the traffic flow rate and traffic

density.• The average delay experienced by vehicles during stop-and-go conditions is shorter in mixed

traffic than in manual traffic while the average number of vehicles at standstill during stop-and-go conditions remains the same in manual and mixed traffic.

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