# Two-Lane Rural Roads

This applet exe,plifies the macroscopic theory proposed in this paper for predicting the operation on two-lane, two-way roads. In this theory, the interaction between fast and slow vehicles obeys Newell's kinematic wave theory of moving bottlenecks. Calibration is not required as all parameters are fully observable. Closed-form expressions for the capacity, average speed, percent time spent following and overtaking rates are proposed, which can be viewed in the applet below.

### Problem formulation

Consider a long two-lane, two-way highway where no-passing zones are neglectable. Each lane obeys a triangular fundamental diagram defined by its free-flow speed u, wave speed -w and capacity Q; see Fig 1a. For one direction of travel the input demand corresponds to state A in the figure, where a small (time-mean) proportion of SVs, r, travels at a free-flow speed v < u. Variables on the opposite direction are denoted by a prime (eg, q_{A¢}, r¢, etc).

Figure 1: Problem formulation in the (a) fundamental diagram and in the (b) time - space diagram.

We are interested in describing congested situations where platoons form due to the presence of SVs; ie, when SVs hold back a queue upstream while free-flow conditions are observed downstream (until the next platoon). Let D be the long-term average free-flow state observed downstream of a SV when it holds back a queue, and let U be the state of this queue. Notice that this definition implies that the overtaking process will be described in its mean value over time, including periods where passing is impossible and periods where passing takes place at some maximum rate. Notice too that q_{D} will be endogenously defined in the model, es explained next.

The flow corresponding to a given traffic state S is denoted Q_{S} , its speed v_{S} and its density K_{S}. To simplify the exposition we use adimensional flows q_{S}=Q_{S}/Q and density in units of pace k_{S}=K_{S}/Q; note that q_{S}=k_{S}v_{S} still holds. For clarity in notation, the only exception to this notation rule is q_{U*} (see Fig. 1a), which will be denoted c, ie

(1)

Notice that using adimensional flows together with (1) will enable us to express q_{D} in terms of a single exogenous parameter, c, which is fully observable.

According to moving bottleneck theory [Newell, 1998], states D and U are connected by a straight line of slope v; see Fig. 1a. Therefore, if D is known one can obtain U by intersecting this line with the congested branch of the fundamental diagram, ie

, ,(2)

Where we have defined the function "overbar" that gives the complement of a dimensionless variable; eg, [c]=1-c.

The time-space diagram for one direction corresponding to our problem is presented in Fig. 1b. The length of the queue generated by a SV, L_{U}, and the length of the free-flow state upstream until the next SV, L_{D}, are determined at the beginning of the road section. These quantities are a consequence of the collision between the shock waves from the transitions A®U, with slope s, and A®D, with slope u; ie,

, (3)

We are interested in the non-trivial cases 0 ≤ s ≤ v, or equivalently

(4)

As in Fig 1a. When (4) is not satisfied trivial solutions are adopted: if q_{A} < q_{D} there is no congestion, L_{U}=0 and all fast vehicles travel at free-flow speed; when q_{A} > q_{U} a queue propagates upstream of the entrance and the entire segment operates in state U.

### The Model

It follows from the preceding section that the only unknown is q_{D}. Our goal is defining q_{D} endogenously in the model, rather than assuming exogenous values, as customary. Towards this end, we utilize the recursive nature of the problem; ie, that there exists a physical process, f(·), that describes q_{D} as a function of relevant variables, and that this process applies to both directions following the same rules. To identify a well-defined general relation among variables, one can use the Buckingham π-theorem [Buckingham, 1914], which suggests that our problem can be stated as the following system of equations

Where η and η' are adimensional constants. This paper proposes a functional form for f(·) based on the assumption that overtaking is possible only when traffic in the opposite direction is in free-flow. In these circumstances f(·) can be interpreted as the proportion of time that a SV sees state D in the opposing direction, and η as the mean overtaking dimensionless flow.

Figure 2: (a) A SV crossing two SVs on the opposite direction; (b) downstream state as predicted by eqns. (7)-(8).

(5)

Where c_{0} = c ^{2} (η+(η)¢)-c^{2} ηη¢, c_{1} = c ^{4} ((η) ^{2} + (η¢)^{2} - 1) - ηη¢2c ^{2}(c ^{2} (η+ η¢)+2cc + 1 ) -c ^{4} ηη¢ , c_{2} = 2(ηη¢+ 2 η(η)¢c + (η)¢(η) c ^{2})c η¢ and c_{3} = 2 (ηη¢+ 2 η¢(η) c + (η)¢(η) c ^{2})c η. The solution for q_{D¢} is identical to (7) but interchanging the primes. To ensure that (7) remains real in the region defined by (4), parameters η and η¢ are chosen such that the square root term vanishes at q_{D}=q_{A}; this gives

, (6)

Notice that for typical conditions c ≈ 1 and thus η ≈ 1-q_{A¢}. Thus, (8) can be interpreted, roughly, as the mean proportion of time a void is found in the opposite direction, which is consistent with our previous interpretation. Notice that (8) avoids introducing additional exogenous parameters that would require calibration and makes our model a function of c, q_{A} and q_{A¢} only.

Figure 2b shows the predictions of (7)-(8) as a function of q_{A} for several values of q_{A¢}. Recall that this model applies only when (4) is satisfied; otherwise trivial solutions are adopted, as explained earlier. It is straightforward to show that this condition defines a feasible region for (7)-(8) of triangular shape (white region in Fig. 2b) with boundaries q_{D}=0, q_{D}=q_{A} and q_{D}=(q_{A}-c)/c. The line q_{D}=q_{A} can be interpreted as an *uncongested branch* of the q_{D}-q_{A} diagram while a given opposing flow q_{A¢} defines a particular *congested branch*. As expected, q_{D} is a decreasing function of opposing demand q_{A¢}. Interestingly, inside a congested branch, q_{D} is also a decreasing function of same-direction demand q_{A}. This is precisely the phenomenon we wanted to capture: as q_{A} increases q_{D¢} decreases and the length of the queue in the opposing direction will increase. In turn, this will decrease passing opportunities in the same direction and thus q_{D} will decrease. Finally, notice that overtaking may become impossible only when q_{A¢} ≥ c; eg, if {q_{A}=c, q_{A¢}=c} then q_{D}=0.

## Performance measures

### Percent time spent following

The HCM 2000 defines the percent time spent following, PTSF, as the average percentage of travel time vehicles travel in platoons behind slower vehicles because of the inability to pass. Unfortunately, the surrogate estimate adopted by researchers is the proportion of headways less than some critical value, t_{c}, as observed at a fixed location. This estimation procedure is ill-defined because a headway of, say three seconds, maybe observed both in free-flow and congestion. As a consequence, the PTSF can be highly overestimated in practice.

To see the bias introduced by this surrogate estimate, let PTSF_{φ} be the PTSF measured along a vehicle trajectory, and PTSF_{x} the PTSF measured at location x. Along a vehicle trajectory the time spent following a single SV is t_{U}=L_{U}/(v_{U}-v) and the time in free-flow before reaching the queue of the next SV is t_{D}=L_{D}/(u-v). At a fixed location one would see that between two consecutive SVs vehicles pass queueing during the first t_{U}=L_{U}/v time units and in free-flow during the following t_{D}=L_{D}/v time units. In both cases the PTSF is given by t_{U}/(t_{U}+t_{D}), which after manipulation gives

, (7)

(8)

The differences between these two equations can be seen in Fig. 3a-b for different values of q_{D}. It is apparent how PTSF_{φ}^{3} PTSF_{x} under all conditions. Notice that this result is true in general (ie, independent of the proposed model). Part b of the figure includes empirical data from Dixon et al., [2002], where a critical value t_{c}=3 sec was assumed. It can be seen how the empirical values are well above PTSF_{x}, which illustrates the bias introduced by current field PTFS estimates.

Figure 3: Performance measures: (a) PTSF_{φ} as a function of q_{A} for several values of q_{D}; (b) PTSF_{x} as a function of q_{A} for several values of q_{D} and for t_{c}= 3 sec; and (c) 2-lane average of the space-mean speed.

### Average Speed

The space-mean speed, [^v], can be computed as (L_{U}k_{U}v_{U}+L_{D}k_{D}u)/( L_{U}k_{U}+L_{D}k_{D}) , which gives

(9)

As expected, [^v]® u when q_{D}® q_{A}, and [^v] ® v when q_{D}® 0. Fig. 3c shows the predictions of (11) for the field data of tables 22 and 29 in Harwood et al., [1999], which presents the mean speed for both directions of travel combined, ie, ([^v]+[^v]¢)/2. As shown in the figure, using the reported speed limit u=85 km/hr and assuming w=15 km/hr, Q=1 500 vph and v=75 km/hr, gives an adequate fit to this empirical data.

### Overtaking rate

To obtain the number of overtakings per km-hr, F, we note that the passing rate along a SV trajectory is R=q_{D}(1-v/u) and that SVs are L=v/rq_{A} km apart. Therefore, F = R/L, which depends on r. To avoid r-dependencies we will use:

(10)

Figure 4a shows the predictions of model (7)-(8)-(12) assuming u=110 km/hr, w=17 km/hr, Q=1 700 vph, v=75 km/hr, compared to the field data of Fig. 2a in Enberg and Pursula, [1997]. This reference reports 15-min overtaking rates for one direction, the corresponding 15-min flow on that direction and the aggregate range of truck proportion during the measurement period (we use the mean value r=0.06). It also reports the extreme opposing flow values, q_{¢}=0.2 and q_{A¢}=0.5, which reduces the feasible region to the gray area in Fig. 4a. Reassuringly, the figure shows that almost all data points lay within the feasible region predicted by the model.

Figure 4b shows the predictions of the model assuming w=17 km/hr and Q=1 700 vph, for the field data in Hegeman, [2004]. This reference reports 3-hr aggregate overtaking rates and flows for both directions, together with u=u¢=110 km/hr, v=83 km/hr, v¢=87 km/hr, r=0.072 and r¢=0.052. Fig. 4b shows that the model predictions are in agreement with empirical data. It also reveals that although the flow in one direction is high, free-flow conditions prevailed during the measurement period in both directions. This is true because the data points lay close to the free-flow branches, and well below the congested branches imposed by opposing flow (solid curves in the figure).

Figure 4: Predicted and observed overtaking rate.

### Capacity

Capacity can also be computed using the proposed model. Given q_{A}, the maximum opposing flow, q_{A¢}^{*}, must satisfy the condition q_{A}=q_{U}(q_{A},q_{A¢}^{*}). A value bigger than q_{A¢}^{*} would induce queues in the direction of q_{A} that would propagate upstream of the entrance, and therefore the total throughput would be smaller than q_{A}+q_{A¢}^{*}. One can show that

(11)

Where a_{1}=c^{2}(c^{4}-4c^{3}+4c^{2}+4),a_{2}=2cc^{2}(c^{3}-5c^{2}+8c-2) and a_{3}=cc^{3}(c^{2}-5c+8). This equation defines the capacity frontier shown in Fig. 5b. Notice that total flow q_{A}+q_{A¢} is maximized at {q_{A}=c, q_{A¢}=c}; ie, when demand is symmetric and equal to the flow in the queue of a SV and there is no overtaking. We conclude that capacity equals 2c. This indicates that capacity of two-lane roads is not constant (as commonly assumed) but heavily dependent on the speed of SVs.

### References

[Dixon et al. 2002] Dixon, M., Sarepali, S., Young, K., 2002. Field evaluation of highway capacity manual 2000 analysis procedures for two-lane highways. Transportation Research Record 1802, 125-132.

[Enberg and Pursula 1997] Enberg, A., Pursula, M., 1997. Traffic flow characteristics on a three-lane rural highway in finland results from empirical studies and simulations. Transportation Research Record 1572, 33-42.

[Harwood et al. 1999] Harwood, D., May, A., Anderson, I., Leiman, L., Archilla, A., 1999. Capacity and quality of service of two-lane highways. Tech. rep., NCHRP Project 3-55(3).

[HCM 2000] HCM, 2000. Highway Capacity Manual 2000. Transportation Research Board, National Academy of Sciences, Washington D.C.

[Hegeman 2004] Hegeman, G., 2004. Overtaking frequency. IEEE International Conference on Systems, Man and Cybernetics, 4017-4022.

[Laval 2006] Laval, J. A., 2006. A macroscopic theory of two-lane rural roads. *Transportation Research Part B*, 40 (10): 937-944, 2006. Abstract / PDF

[Newell 1998] Newell, G. F., 1998. A moving bottleneck. Transportation Research Part B 32 (8), 531-537.