Traffic density on corridors subject to incidents: models for long-term congestion management

Abstract

The purpose of this research is to provide a faster and more efficient method to determine traffic density behavior for long-term congestion management using minimal statistical information. Applications include road work, road improvements, and route choice. To this end, this paper adapts and generalizes two analytical models (for non-peak and peak hours) for the probability mass function of traffic density for a major highway. It then validates the model against real data. The studied corridor has a total of 36 sensors, 18 in each direction, and the traffic experiences randomly occurring service deterioration due to accidents and inclement weather such as snow and thunderstorms. We base the models on queuing theory, and we compare the fundamental diagram with the data. This paper supports the validity of the models for each traffic condition under certain assumptions on the distributional properties of the associated random parameters. It discusses why these assumptions are needed and how they are determined. Furthermore, once the models are validated, different scenarios are presented to demonstrate traffic congestion behavior under various deterioration levels, as well as the estimation of traffic breakdown. These models, which account for non-recurrent congestion, can improve decision making without the need for extensive datasets or time-consuming simulations.

Appendices

Appendix 1: Proof of Proposition 1

We will derive the probability mass function of the random number of cars on a segment from Eqs. 1, 4, 5, 9, 10, and 11.

$$\begin{aligned}&P\{X=k\}\\&\quad =P\{X_\phi +Y=k\} = \sum _{q=0}^{k}P\{X_\phi =k-q\}\cdot P\{Y=q\}\\&\quad =P\{X_\phi =k\}\cdot P\{Y=0\}\\&\qquad +\sum _{q=1}^{k}P\{X_\phi =k-q\}\cdot (p \cdot P\{Y_1=q\} +(1-p)\cdot P\{Y_2=q\})\\&\quad =\frac{e^{-\lambda /\mu }\cdot (\lambda /\mu )^k}{k!}\cdot \left[ p\cdot \left( 1+\frac{\varGamma (b)}{\varGamma (a)}(e^{-c}-1)\right) +(1-p)e^{-c}\right] \\&\qquad +\sum _{q=1}^{k}\frac{e^{-\lambda /\mu }\cdot (\lambda /\mu )^{k-q}}{(k-q)!}\left[ p\cdot \left( \frac{\varGamma (b)}{\varGamma (a)}\frac{(e^{-c}c^q)}{q!}\right) +(1-p)\frac{(e^{-c}c^q)}{q!}\right] \\&\quad =\frac{e^{-\lambda /\mu }\cdot (\lambda /\mu )^k}{k!}\cdot \left[ e^{-c}\left( 1+p\left( \frac{\varGamma (b)}{\varGamma (a)} - 1\right) \right) +p\left( 1-\frac{\varGamma (b)}{\varGamma (a)}\right) \right] \\&\qquad +\left( 1+p\left( \frac{\varGamma (b)}{\varGamma (a)}-1\right) \right) \sum _{q=1}^{k}\frac{e^{-\lambda /\mu }\cdot (\lambda /\mu )^{k-q}}{(k-q)!}\cdot \frac{e^{-c}c^q}{q!}\\&\quad =\frac{e^{-\lambda /\mu }\cdot (\lambda /\mu )^k}{k!}\cdot \left[ e^{-c}\left( 1+p\left( \frac{\varGamma (b)}{\varGamma (a)}-1\right) \right) +p\left( 1-\frac{\varGamma (b)}{\varGamma (a)}\right) \right] \\&\qquad +\left( 1+p\left( \frac{\varGamma (b)}{\varGamma (a)}-1\right) \right) \sum _{q=1}^{k}\frac{e^{-\lambda /\mu }\cdot (\lambda /\mu )^{k-q}}{(k-q)!}\cdot \frac{e^{-c}c^q}{q!}. \end{aligned}$$

Substituting $\left( 1-p\left( 1-\frac{\varGamma (b)}{\varGamma (a)}\right) \right) = m$, we have

$$\begin{aligned} P\{X=k\}=&\frac{e^{-\lambda /\mu }\cdot (\lambda /\mu )^k}{k!}\cdot \left[ e^{-c}m +(1-m)\right] +m\\&\cdot \frac{e^{-(\lambda /\mu +c)}[(\lambda /\mu +c)^k-(\lambda /\mu )^k]}{k!}\\ =&(1-m)\cdot \frac{e^{-\lambda /\mu }\cdot (\lambda /\mu )^k}{k!}\\&+m\left( \frac{e^{-(\lambda /\mu +c)}[(\lambda /\mu +c)^k-(\lambda /\mu )^k]}{k!}+\frac{e^{-\lambda /\mu +c}\cdot (\lambda /\mu )^k}{k!}\right) \\ =&(1-m)\cdot \frac{e^{-\lambda /\mu }\cdot (\lambda /\mu )^k}{k!}+m\left( \frac{e^{-(\lambda /\mu +c)}(\lambda /\mu +c)^k}{k!}\right) . \end{aligned}$$

Therefore

$$\begin{aligned} P\{X=k\}=p\left( 1-\frac{\varGamma (b)}{\varGamma (a)}\right) X_g+\left( 1-p\left( 1-\frac{\varGamma (b)}{\varGamma (a)}\right) \right) X_b. \end{aligned}$$

Now we will show that $p\left( 1-\frac{\varGamma (b)}{\varGamma (a)}\right)$ is approximately equal to $\frac{r}{r+f}$ for small a and b.

Claim: $p\left( 1-\frac{\varGamma (b)}{\varGamma (a)}\right) \approx \frac{r}{r+f}$ for small a and b.

For $x>0,$

$$\begin{aligned} \varGamma (x) = \frac{\varGamma (x+1)}{x}. \end{aligned}$$

The above relation for $x=a=\frac{f}{\mu }$ and $x=b=\frac{f}{\mu } + \frac{r}{\mu '},$ implies

$$\begin{aligned} \frac{\varGamma (b)}{\varGamma (a)} = \frac{\frac{f}{\mu }}{\frac{f}{\mu } + \frac{r}{\mu '}} \cdot \frac{\varGamma (b+1)}{\varGamma (a+1)}. \end{aligned}$$

Using the approximation (8) for $z=1$, since $a \ll 1,$$b\ll 1$, and $b-a \ll 1$, one can deduce that $\frac{\varGamma (b+1)}{\varGamma (a+1)}$ is approximately equal to 1. Thus

$$\begin{aligned} \frac{\varGamma (b)}{\varGamma (a)} \approx \frac{\frac{f}{\mu }}{\frac{f}{\mu } + \frac{r}{\mu '}}. \end{aligned}$$

Because $p= \frac{r + f\frac{\mu '}{\mu }}{r+f}$, the result follows

$$\begin{aligned} p\left( 1-\frac{\varGamma (b)}{\varGamma (a)}\right) \approx&\left( \frac{r + f\frac{\mu '}{\mu }}{r+f} \right) \left( 1- \frac{\frac{f}{\mu }}{\frac{f}{\mu } + \frac{r}{\mu '}}\right) \\ =&\left( \frac{r\mu + f\mu '}{\mu (r+f)} \right) \left( \frac{\frac{r}{\mu '}}{\frac{f}{\mu } + \frac{r}{\mu '}}\right) = \frac{r}{r+f}. \end{aligned}$$

As a result, the probability mass function of density tends to the mixture $P\{X=k\}=\frac{r}{r+f} X_g+\frac{f}{r+f}X_b,$ where $X_g$ follow a Poisson with parameter $(\lambda /\mu )$ and $X_b$ follow a Poisson with parameter $(\lambda '/\mu ')$.

The error of the approximation can be derived similarly to the previous proof

$$\begin{aligned} Error&= p\left( 1-\frac{\varGamma (b)}{\varGamma (a)}\right) -\frac{r}{r+f}\\&=\left( \frac{r\mu + f\mu '}{\mu (r+f)} \right) \left( 1-\frac{\frac{f}{\mu }}{\frac{f}{\mu } + \frac{r}{\mu '}} \cdot \frac{\varGamma (b+1)}{\varGamma (a+1)}\right) -\frac{r}{r+f}\\&=\left( \frac{r\mu + f\mu '}{\mu (r+f)} \right) - \left( \frac{f \mu '}{\mu (r+f)} \cdot \frac{\varGamma (b+1)}{\varGamma (a+1)}\right) -\frac{\mu r}{\mu (r+f)}\\&=\frac{f}{r+f}\cdot \frac{\mu '}{\mu } \cdot \left( 1-\frac{\varGamma (b+1)}{\varGamma (a+1)}\right) . \end{aligned}$$

One can see that when f and r are smaller in comparison to $\mu$ and $\mu '$, respectively, this approximation of the weight of the mixture holds. Figure 17 shows the error for different ratios between f and $\mu$ and r and $\mu '$. We can see that when they are around $1\%$ of the $\mu$ and $\mu '$, the error is on the order of $10^{-3}$, and it decreases to lower than $10^{-5}$ when the ratios drop to $0.01\%$.

Fig. 17

Error between the algebraically found weight and $\frac{r}{r+f}$ for different ratios of f and $\mu$, and r and $\mu '$

Appendix 2: Tables containing AIC for each sensor

See Appendix Tables 9 and 10.

Table 9 Full list of the adapted AIC comparison between model and commonly used distributions during non-peak hours

Table 10 Full list of the adapted AIC comparison between model and commonly used distributions during peak hours