Jing Lu and Carolina Osorio

# On the approximation of queue-length distributions in transportation
networks

Jing Lu
Operations Research Center
Massachusetts Institute of Technology  USA
Carolina Osorio
Department of Decision Sciences
HEC Montréal  Canada
Jing Lu, Ph.D., Corresponding Author
<EMAIL_ADDRESS>
##  1 Abstract

This paper focuses on the analytical probabilistic modeling of vehicular
traffic. It formulates a stochastic node model. It then formulates a network
model by coupling the node model with the link model of [1], which is a
stochastic formulation of the traffic-theoretic link transmission model. The
proposed network model is scalable and computationally efficient, making it
suitable for urban network optimization. For a network with $r$ links, each of
space capacity $\ell$, the model has a complexity of $\mathcal{O}(r\ell)$. The
network model yields the marginal distribution of link states. The model is
validated versus a simulation-based network implementation of the stochastic
link transmission model. The validation experiments consider a set of small
network with intricate traffic dynamics. For all scenarios, the proposed model
accurately captures the traffic dynamics. The network model is used to address
a signal control problem. Compared to the probabilistic link model of [1] with
an exogenous node model and a benchmark deterministic network loading model,
the proposed network model derives signal plans with better performance. The
case study highlights the added value of using between-link (i.e., across-
node) interaction information for traffic management and accounting for
stochasticity in the network.

Keywords: Probabilistic node modeling, Computational efficiency, Stochastic
network loading, Network optimization

##  2 Introduction

This paper formulates an analytical stochastic (i.e., probabilistic) network
model. It formulates a node model, and couples the node model with an existing
stochastic link model. Our focus on stochastic analytical modeling is
motivated by an interest in developing macroscopic traffic models that can
both provide a more detailed distributional description of network performance
and can be used for large-scale network analysis and optimization. The
increase in the quantity, quality and resolution of traffic data available
allows to validate models that provide a probabilistic description of traffic.
Such models can be used to enhance the reliability and the robustness of our
transportation networks. Major agencies in the US and in Europe are focusing
on measuring and improving such reliability and robustness performance
metrics. Our research in probabilistic network modeling is to formulate
traffic models that are suitable for the efficient analysis and optimization
of large-scale networks.

A stochastic network model consists of two main components: a link model,
which describes traffic dynamics within a homogeneous road segment, and a node
model, which describes traffic dynamics between links. There is an increasing
and recent interest in the formulation of probabilistic traffic-theoretic link
models. Recent reviews include [2] and [3]. Probabilistic link models
formulated with traffic-theoretic foundations include [4, 5, 6, 7, 8, 9, 3,
10, 11, 1, 12]. The main approaches have been the introduction of randomness
in the cell transmission model, the extensions of the variational theory of
[13] and the use of probabilistic queueing theory [14, 15, 16, 8, 11, 1].
Another stream of research in the estimation of queue length distribution is
based on observed data collected from probe vehicles [17, 18, 19, 20].

The model of [11] is a queueing-theoretic model that is a stochastic
formulation of the deterministic link-transmission model of [21], which itself
is an operational formulation of Newell’s simplified theory of kinematic waves
[22]. For a link with space capacity $\ell$, the model complexity is in the
order of $\mathcal{O}(\ell^{3})$. [1] have extended the formulation of [11],
leading to a model with complexity in the order of $\mathcal{O}(\ell)$. This
is a formulation that scales linearly with the link’s space capacity, making
it efficient and appropriate for large-scale network analysis. The model in
Chapter 3 of [12] further reduces the model complexity to a constant that no
longer depends on the link’s space capacity. This makes the proposed model
suitable for large-scale network optimization or situations where a large
number of model evaluations is required. In this paper, we focus on the
network model development with the analytical probabilistic link model of [1]
where the transient distribution of the queue-length is produced.

The formulation of traffic-theoretic node models is limited to deterministic
formulations (e.g., [23, 24, 25, 26, 27, 28, 29]). These node models satisfy
supply and demand restrictions at the link boundaries, mass balance
requirements on flows across the node and some further considerations in the
distribution of demands to different outgoing links to reflect destination
desires of drivers. Most deterministic node models impose flow maximization to
facilitate the physical requirement that traffic-holding does not exist. It is
shown in a recent work of [30] that flow maximized solutions are only
sufficient conditions for holding-free solutions, especially in complex
congested urban intersections. Other themes of node modeling include the
derivation of formulas for calculating node flows that satisfies the
aforementioned restrictions [31]. Nevertheless, one challenge of network
modeling that persists today is the analytically intractability due to the
coupling with link dynamics and node models that are typically stated as
optimization problems.

The development of stochastic node model is quite limited and mostly focused
on simple network topology. The model of [8] includes a stochastic node model
for a simple two queue network. [32] proposes an analytical model to
approximate the transient aggregate joint queue-length distribution of tandem
links, the aggregation-disaggregation technique and network decomposition
methods are used to reduce the model complexity. [33] propose an analytical
approximation of the aggregate stationary joint queue-length distributions for
arbitrary size and it can be extended to general topology by coupling with the
method introduced in [34].[35] develops an analytical stochastic link
transmission model (SLTM) for networks that produces the transient joint
queue-length distribution and demonstrated its usage on linear, diverge and
merge node models within the SLTM framework. Nevertheless, there is a lack of
stochastic node models that apply for general network topology and can be
integrated with SLTM for potentially large-scale network analysis.

In this article, we formulate a general stochastic node model that can be
combined with various types of stochastic link models for efficient network
analysis. We couple the proposed stochastic node model with the link model of
[1]. Section 3 formulates the proposed stochastic network model. The model is
validated in Section 4. It is used to address a signal control problem in
Section 5. This case study shows the added value of accounting for between-
link (i.e., across node) interactions for traffic management. Conclusions and
future work are presented in Section 6.

##  3 Network model formulation

We present a general node model formulation. The proposed node model can be
coupled with a variety of probabilistic link models to yield a network model.
We then couple the node model with the link model of [1], which produces the
transient distribution of the link’s queue length distribution. Section 3.1
briefly describes the link model, which is, hereafter, referred to as the
mixture model. The node model formulation is given in Section 3.2.

###  3.1 Link model

We briefly outline here the main ideas of the link model of [1]. For a more
detailed description, we refer the reader to [1]. The mixture model considers
an isolated single-lane link. It assumes a triangular fundamental diagram. The
link is parameterized as follows: free flow velocity $v$, backward wave speed
$w$ (negative), flow capacity $\hat{q}$, jam density $\hat{\varrho}$, and link
length $L$. The link model approximates the following process. Upon arrival to
the link, vehicles are delayed $L/v$ time units (known as the forward lag).
After incurring this delay, vehicles are then ready for departure at the
downstream end of the link. Upon departure from the link, there is a delay of
$L/|w|$ time units (known as the backward lag) before the newly available
space is made available at the upstream end of the link. These delays or lags
capture the time it takes forward, and backward, kinematic waves to traverse
the link. The mixture model is a continuous-space discrete-time model. The
lags are rounded to the nearest integer: $L/(v\delta)$ is rounded to
$k^{\text{fwd}}$, and $L/(|w|\delta)$ is rounded to $k^{\text{bwd}}$, where
$\delta$ is the incremental time.

The vehicular flow process described above is modeled with two queues.
Vehicles that are ready for departure enter the downstream queue, denoted
$DQ$. In other words, $DQ$ captures the downstream boundary conditions of the
link. Spaces that are occupied (i.e., they are either occupied by vehicles on
the link, or by vehicles that have departed the link less than $L/(|w|\delta)$
time units ago) are represented by the upstream queue, denoted $UQ$. In other
words, $UQ$ captures the upstream boundary conditions of the link.

The model of [1] considers two univariate independent models: a univariate
$UQ$ model, which captures the link’s upstream boundary conditions, and a
univariate $DQ$ model, which captures the link’s downstream boundary
conditions. Each of these two models approximates two distributions: (i) the
distribution of the link’s upstream queue $UQ$, and (ii) the distribution of
the link’s downstream queue $DQ$.

Then, the model of [1] is defined as a mixture of these two models. For this
discrete time model, let $UQ(k)$ denote the number of vehicles (or spaces
occupied) in $UQ$ at the end of discrete time interval $k$. We use similar
notation for the downstream queue $DQ(k)$. The mixture model yields the
marginal distributions of the downstream queue and of the upstream queue.
These are given by:

$P(UQ(k))=\hat{w}P^{UQ}(UQ(k))+(1-\hat{w})P^{DQ}(UQ(k))$ (1)
$P(DQ(k))=\hat{w}P^{UQ}(DQ(k))+(1-\hat{w})P^{DQ}(DQ(k)),$ (2)

where $P^{UQ}(UQ(k))$ (resp. $P^{DQ}(UQ(k))$) denotes the $UQ$ distribution
obtained from the univariate $UQ$ (resp. $DQ$) model, $P^{UQ}(DQ(k))$ (resp.
$P^{DQ}(DQ(k))$) denotes the $DQ$ distribution obtained from the univariate
$UQ$ (resp. $DQ$) model, and $\hat{w}$ denotes an exogenous weight parameter
calculated according to Equation (37) of [1].

The mixture model has linear complexity in the links space capacity $\ell$
because the two univariate models are independent. In other words, the
distributions $P^{UQ}(UQ)$ and $P^{DQ}(UQ)$ are evaluated independently.
Similarly, the distributions $P^{UQ}(DQ)$ and $P^{DQ}(DQ)$ are evaluated
independently.

Each univariate model is then formulated as a system of nonlinear equations.
For a given time interval $k$, the expected inflow into, and the expected
outflow from, the link are assumed constant, and the solution of the system of
nonlinear equations yields the transient (i.e., time-dependent) distribution
of $UQ$ (or of $DQ$) within the time interval $k$ of duration $\delta$. At the
end of each time interval, the expected inflows and outflows are then updated
and used for the next time interval.

The stochastic processes that govern the link dynamics are as follows. The
link is assumed to have an inhomogeneous Poisson arrival process with
exogenous arrival rate $\lambda(k)$, and independent and exponentially
distributed service times at the downstream end of the link with exogenous
downstream flow capacity $\mu(k)$. It is shown in Chapter 3 section 3.4.2 of
[12] that with properly calibrated time-varying arrival rate and service rate
(i.e., $\lambda(k)$ and $\mu(k)$) and time increment $\delta$, a link’s
boundary conditions for realistic traffic situation such as signalized links
and platoon arrival patterns can be estimated accurately compared to a details
microscopic traffic simulator.

###  3.2 Node model

We now formulate the node model. The analysis hereafter is for a given
discrete time interval $k$. Hence, we use interchangeably $UQ$ and $UQ(k)$, as
well as $DQ$ and $DQ(k)$. Consider a general node with $m$ upstream links and
$n$ downstream links, let $M$ be the set of upstream links and $N$ be the set
of downstream links. We assume exogenous turning probabilities $\\{p_{ij},i\in
M,j\in N\\}$. Each link $i$ has an exogenous downstream flow capacity
$\mu_{i}$ and space capacity $\ell_{i}$. The rate at which vehicles can start
trips at link $i$ is exogenous and is denoted $\gamma_{i}$.

For a given upstream link $i$ ($i\in M$), the number of vehicles ready for
departure is $DQ_{i}$. For a given downstream link $j$ ($j\in N$), the number
of spaces available at the upstream end of link $j$ (to accommodate demand
from upstream links) is $(\ell_{j}-UQ_{j})$.

We assume that a vehicle from $i$ can transition to $j$ if the following two
conditions hold. First, the link $j$ is not full so that there is space
available to receive demand from upstream, i.e., $UQ_{j}<\ell_{j}$. Second,
the vehicle is not blocked by vehicles in front that intend to leave link $i$
to other downstream links other than link $j$. A sufficient, but not
necessary, condition to ensure that vehicles from link $i$ can transition to
link $j$ is that there is space available to receive vehicles in all
downstream links, i.e., $\forall n\in N,UQ_{n}<\ell_{n}$. Additionally, if
downstream links are not full, then the instantaneous departure rate from link
$i$ corresponds to its flow capacity, $\mu_{i}$. This leads to the following
expression for the expected flow rate from link $i$ to link $j$ during time
interval $k$ of length $\delta$:

$q_{i\rightarrow j}^{out}(k)=p_{ij}\mu_{i}P(DQ_{i}(k)>0,UQ_{n}(k)<\ell_{n}\
\forall n\in N).$ (3)

This proposed approximation may underestimate the true expected flow rate,
since $\\{DQ_{i}(k)>0$, $UQ_{n}(k)<\ell_{n}\ \forall n\in N\\}$ is a
sufficient, but not a necessary, condition for flow transmission. For
instance, there are cases where a downstream link is full, yet it does not
hinder outflow from all upstream links. This can happen if there are no
vehicles upstream that intend to enter the full downstream link. However,
since the it is assumed that there is always a constant non-zero proportion of
outflows $p_{ij}$ from upstream link $i$ intent to enter the downstream link
$j$ that is full, link $i$ will eventually be blocked, e.g., eventually the
most downstreamed vehicle will become one that intends to join link $j$ but
not able to and hence blocked the outflow of the single-lane link $i$. The
validation results of the next section indicate, this approximation has a
minor effect on the accuracy of the model.

Combining the above expression with a link model requires the computation of
the joint probability $P(DQ_{i}(k)>0,UQ_{n}(k)<\ell_{n}\ \forall n\in N)$. In
general, a straightforward, but also inefficient, way to compute this is to
formulate a network model that jointly tracks the state of all links in the
network. For a network with $r$ links, each with space capacity $\ell$, the
model complexity would be in the order of $\mathcal{O}(\ell^{r})$. Instead, we
propose an approach with complexity $\mathcal{O}(r\ell)$. The main idea is to
track the distribution of each link in the network independently, and to
capture the dependencies between links by adjusting the arrival rates and the
effective service rates of the links according to the expected flows within
the network.

Section 3.2.1 formulates an approximation for the above joint probability,
$P(DQ_{i}>0,UQ_{n}<\ell_{n}\ \forall n\in N)$, which we refer to, hereafter,
as the flow transmission probability. Section 3.2.2 then presents how the link
model parameters (arrival rates and effective service rates) are approximated
at every time step. Section 3.2.3 summarizes the method, and presents it as an
algorithm.

####  3.2.1 Flow transmission probability

In this section, we present an analytical approximation of the flow
transmission probability that only requires knowledge of the marginal
univariate distributions of $DQ$ and of $UQ$.

We use the inclusive-exclusive principle (Section 2.1 of [36]) to rewrite
$P(DQ_{i}>0,UQ_{n}<\ell_{n}\ \forall n\in N)$ as follows:

$\displaystyle P(DQ_{i}$ $\displaystyle>0,UQ_{n}<\ell_{n}\ \forall n\in N)=$
(4) $\displaystyle 1$ $\displaystyle-P(DQ_{i}=0)-\sum_{n\in
N}P(UQ_{n}=\ell_{n})$ $\displaystyle+\sum_{n\in
N}P(DQ_{i}=0,UQ_{n}=\ell_{n})+\sum_{n\neq m;n,m\in
N}P(UQ_{n}=\ell_{n},UQ_{m}=\ell_{m})$
$\displaystyle+...+(-1)^{|N|+1}P(DQ_{i}=0,UQ_{n}=\ell_{n}\ \forall n\in N).$

In Equation (4), the distribution of a single link state (e.g., $P(DQ_{i}=0)$,
$P(UQ_{n}=\ell_{n})$) can be obtained directly from the link model. Let us now
describe how we approximate the joint probabilities that appear in (4). Each
joint probability is approximated by rewriting it as a conditional probability
conditioned on the sum of the corresponding variables. For example:

$\displaystyle P(DQ_{i}$ $\displaystyle=0,UQ_{n}=\ell_{n}\ \forall n\in N)$
(5) $\displaystyle=P\left(DQ_{i}=0,UQ_{n}=\ell_{n}\ \forall n\in N\mid
DQ_{i}+\sum_{\forall n\in N}UQ_{n}=\sum_{\forall n\in N}\ell_{n}\right)$
$\displaystyle\cdot P\left(DQ_{i}+\sum_{\forall n\in N}UQ_{n}=\sum_{\forall
n\in N}\ell_{n}\right).$

In order to derive an approximate expression for the right-hand side of
Equation (5), we make two approximations. First, we approximate
$DQ_{i}+\sum_{\forall n\in N}UQ_{n}$ as a Poisson arrival process with rate
$q(k)$. Hence,

$P\left(DQ_{i}(k)+\sum_{\forall n\in N}UQ_{n}(k)=\sum_{\forall n\in
N}\ell_{n}\right)=[\delta\cdot q(k)]^{\sum_{\forall n\in
N}\ell_{n}}\frac{e^{-\delta\cdot q(k)}}{(\sum_{\forall n\in N}\ell_{n})!},$
(6)

where $\delta$ is the time step length. The rate is given by the sum:

$q(k)=q^{DQ_{i}}(k)+\sum_{\forall n\in N}q^{UQ_{n}}(k),$ (7)

where $q^{DQ_{i}}(k)\delta$ (resp. $q^{UQ_{n}}(k)\delta$) represents the
expected state of $DQ_{i}$ (resp. $UQ_{n}$) for discrete time interval $k$.
These can be obtained directly from the link model (see Equations (7) and (12)
of [1]). Let us briefly explain how they are calculated. The expression for
$q^{UQ_{n}}(k)$ is:

$q^{UQ_{n}}(k)=\sum_{r=0}^{k-1}q_{n}^{\text{in}}(r)-\sum_{r=0}^{k-k^{\text{bwd}}-1}q_{n}^{\text{out}}(r),$
(8)

where $q_{n}^{\text{in}}(k)$ (resp. $q_{n}^{\text{out}}(k)$) denotes the
instantaneous inflow (resp. outflow) rate of link $n$ at the end of time
interval $k$. This expression is obtained from the observation that the
expected state of $UQ_{n}$ represents the difference between the accumulated
inflows to link $n$ during the last $k-1$ time steps and the accumulated
outflows from link $n$ during the last $k-k^{\text{bwd}}-1$ time steps.
Similarly, the expression for $q^{DQ_{i}}(k)$ is:

$q^{DQ_{i}}(k)=\sum_{r=0}^{k-k^{\text{fwd}}-1}q_{i}^{\text{in}}(r)-\sum_{r=0}^{k-1}q_{i}^{\text{out}}(r).$
(9)

Equation (9) considers the expected flow in $DQ_{i}$ as the difference
between: (i) the sum of all of the expected inflows into link $i$ from time
$0$ to time $k-k^{\text{fwd}}-1$ (i.e., omitting the flows that have still not
arrived to the downstream end of the link) and (ii) the sum of all expected
outflows out of link $i$ (i.e., outflow from time $0$ to time $k-1$).

Second, we approximate $DQ_{i}+\sum_{\forall n\in N}UQ_{n}$ as a sum of
independent Poisson processes. Hence, the conditional variate
$\\{DQ_{i},UQ_{n},\ \forall n\in N\\}$ given $\\{DQ_{i}+\sum_{\forall n\in
N}UQ_{n}=\sum_{\forall n\in N}\ell_{n}\\}$ follows a multinomial distribution
with parameters $\left(\sum_{\forall n\in
N}\ell_{n};q^{DQ_{i}}/q,q^{UQ_{n}}/q\ \forall n\in N\right)$ (cf., for
instance, Section 2.12.4 of [37]). This leads to:

$P\left(DQ_{i}=0,UQ_{n}=\ell_{n}\ \forall n\in N\mid DQ_{i}+\sum_{\forall n\in
N}UQ_{n}=\sum_{\forall n\in N}\ell_{n}\right)=\frac{(\sum_{\forall n\in
N}\ell_{n})!}{\prod_{n\in N}\ell_{n}!}\prod_{n\in
N}\left(\frac{q^{UQ_{n}}}{q}\right)^{\ell_{n}}.$ (10)

Note that although we ignore the temporal dependencies among the queues, the
multinomial distribution parameters ($q^{DQ_{i}}/q$ and $q^{UQ_{n}}/q$) are
highly temporally dependent. With this same technique, we approximate each of
the joint probabilities that appear in Equation (4).

####  3.2.2 Link model arrival and effective service rates

We now present how we use the expected flow between links (Equation (3)) to
approximate the arrival rates and the effective service rates of a link.

Consider a node with an upstream link $i$. The expected outflow rate from link
$i$ is given by:

$\displaystyle q_{i}^{out}(k)$ $\displaystyle=\mu_{i}\left(1-\sum_{j\in
N}p_{ij}\right)P(DQ_{i}(k)>0)+\sum_{j\in N}q_{i\rightarrow j}^{out}(k)$ (11)
$\displaystyle=\mu_{i}\left(1-\sum_{j\in
N}p_{ij}\right)P(DQ_{i}(k)>0)+\left(\sum_{j\in
N}p_{ij}\right)\mu_{i}P(DQ_{i}(k)>0,UQ_{n}(k)<\ell_{n}\ \forall n\in N).$ (12)

Equation (11) states that the expected outflow rate from link $i$ is the
summation of the expected flow rate that leaves the network through link $i$
and the expected outflow from link $i$ to all its downstream links $j$.
Equation (12) is obtained by inserting the expression for $q_{i\rightarrow
j}^{out}(k)$ of Equation (3).

Additionally, let $\hat{\mu}_{i}(k)$ denote the effective service rate of link
$i$ at time step $k$. The effective service rate, $\hat{\mu}_{i}$, differs
from the service rate, $\mu_{i}$, in that the former accounts for the impact
of downstream spillbacks on the departure rate of link $i$, while the latter
is an exogenous parameter that assumes the link is isolated and represents its
downstream flow capacity.

By definition, the expected outflow and the effective service rate are related
as follows:

$q_{i}^{out}(k)=\hat{\mu}_{i}(k)P(DQ_{i}(k)>0).$ (13)

We can combine Equations (12) and (13) to obtain an expression for the
effective service rate:

$\hat{\mu}_{i}(k)=\frac{q_{i}^{out}(k)}{P(DQ_{i}(k)>0)}=\mu_{i}\left[\left(1-\sum_{j\in
N}p_{ij}\right)+\left(\sum_{j\in N}p_{ij}\right)P(UQ_{n}(k)<\ell_{n}\ \forall
n\in N\mid DQ_{i}(k)>0)\right].$ (14)

Let us now describe how the expected flow between links is used to approximate
the arrival rate of a link. Consider a node with downstream link $j$. The
expected inflow rate to link $j$ is given by:

$\displaystyle q_{j}^{in}(k)$
$\displaystyle=\gamma_{j}P(UQ_{j}(k)<\ell_{j})+\sum_{i\in M}q_{i\rightarrow
j}^{out}(k)$ (15) $\displaystyle=\gamma_{j}P(UQ_{j}(k)<\ell_{j})+\sum_{i\in
M}p_{ij}\mu_{i}P(DQ_{i}(k)>0,UQ_{n}(k)<\ell_{n}\ \forall n\in N)$ (16)

where $\gamma_{j}$ is the exogenous arrival rate to link $j$ (i.e., it
represents the rate with which trips start at link $j$), Equation (15) states
that the expected inflow to link $j$ is the summation of the expected flow
entering the network through link $j$ and the expected flow from all upstream
links $i$ to link $j$.

By definition, the expected inflow rate and the arrival rate,
$\lambda_{j}(k)$, are related as follows:

$q_{j}^{in}(k)=\lambda_{j}(k)P(UQ_{j}(k)<\ell_{j}).$ (17)

We combine Equations (16) and (17) to obtain an expression for the arrival
rate of link $i$ at time step $k$:

$\lambda_{j}(k)=\frac{q_{j}^{in}(k)}{P(UQ_{j}(k)<\ell_{j})}=\gamma_{i}+\sum_{i\in
M}p_{ij}\mu_{i}P(DQ_{i}(k)>0,UQ_{n}(k)<\ell_{n}\ \forall n\neq j,n\in N\mid
UQ_{j}(k)<\ell_{j})$ (18)

####  3.2.3 Algorithm

Algorithm 1 summarizes the numerical evaluation of the proposed network model
for all links $i$ in the set of links $\mathcal{L}$. The state distribution of
link $i$ is obtained by evaluating the mixture model of [1] as defined in
Equations (1) and (2). Notice that steps 7(a) and 7(b) in the algorithm can be
run simultaneously to further enhance the runtime. Additionally, for both
steps 7(a) and 7(b) the calculations for each link can be done in parallel
(i.e., given their boundary conditions, links are updated independently).

Algorithm 1 Network model algorithm

  1. 1.

set values for the exogenous parameters
$\hat{\varrho}_{i},v_{i},w_{i},\ell_{i}$ and $\delta$

  2. 2.

compute
$k_{i}^{fwd}=\lceil\frac{\ell_{i}}{\hat{\varrho}_{i}v_{i}\delta}\rceil$ and
$k_{i}^{fwd}=\lceil\frac{\ell_{i}}{\hat{\varrho}_{i}|w_{i}|\delta}\rceil$

  3. 3.

compute $\hat{w}_{i}$ according to Eq. (37) of [1]

  4. 4.

set initial conditions: $q_{i}^{in}(0)$, $q_{i}^{out}(0)$,
$P^{UQ}(UQ_{i}(0))$, $P^{DQ}(DQ_{i}(0))$, $q^{UQ_{i}}(0)$, $q^{LLO_{i}}(0)$,
$q^{LLI_{i}}(0)$, $q^{DQ_{i}}(0)$, $\lambda_{i}(0)$ and $\hat{\mu}_{i}(0)$.

  5. 5.

set $q_{i}^{in}(r)=0$ and $q_{i}^{out}(r)=0$ for $r<0$

  6. 6.

repeat the following for time intervals $k=1,2,...$

    1. (a)

for all links $i$, evaluate the univariate UQ model (i.e., run Algorithm 1 of
[1]) with initial conditions $P^{UQ}(UQ_{i}(k-1))$, $\hat{\mu}_{i}(k-1)$ and
$\lambda_{i}(k-1)$, this yields $P^{UQ}(UQ_{i}(k))$, $P^{UQ}(DQ_{i}(k))$

    2. (b)

for all links $i$, evaluate the univariate DQ model (i.e., run Algorithm 2 of
[1]) with initial conditions $P^{DQ}(DQ_{i}(k-1))$, $\hat{\mu}_{i}(k-1)$ and
$\lambda_{i}(k-1)$, this yields $P^{DQ}(DQ_{i}(k))$, $P^{DQ}(UQ_{i}(k))$

    3. (c)

calculate $P(DQ_{i}(k)>0,UQ_{n}(k)<\ell_{n}\ \forall n\in N)$ according to Eq.
(4)

    4. (d)

calculate $q_{i\rightarrow j}^{out}(k)$ according to Eq. (3)

    5. (e)

calculate $q_{i}^{in}(k)$, $q_{i}^{out}(k)$ according to Eq. (15) and (11),
respectively.

    6. (f)

calculate $\hat{\mu}_{i}(k)$ and $\lambda_{i}(k)$ according to Eq. (14) and
(18), respectively.

##  4 Model validation

In this section, we validate the proposed network model formulation. We
consider the following network topologies: a three-link merge network (Section
4.1), a three-link diverge network (Section 4.2) and a network with 8 links
with intricate traffic dynamics (Section 4.3). The latter is the same network
at that considered in Section 3 of [38]. We carry out experiments with time-
varying demand and varying levels of congestion.

The outputs of the analytical model are compared to estimates obtained from a
discrete-event simulator of the stochastic link transmission model. This
simulator has been used for validation in [11]. The simulator samples
individual vehicles each with exact forward and backward lags, and it
satisfies FIFO. For each experiment, the simulated estimates are obtained from
$10^{6}$ simulation replications. Estimates of 95% confidence intervals were
calculated, yet were barely visible. Hence, they are not displayed in the
figures.

For all networks, each link consists of a single lane. The lane parameters are
common to all experiments, and are displayed in Table 1. The link lengths are
fixed at $50$m (resp. space capacity $\ell=10$, forward lags $k^{fwd}\delta=5$
sec, and backward lags $k^{bwd}\delta=10$ sec). Each experiment starts with an
empty network and runs for $600$ seconds.

Table 1: Link Parameters Parameter | Value
---|---
$v$ |  $0.01$ km/sec
$w$ |  $-0.005$ km/sec
$\hat{\varrho}$ |  $200$ veh/km
$\hat{q}$ |  $2400$ veh/h = $0.67$ veh/sec
$\delta$ |  $0.1$ sec
L ($\ell$, $k^{fwd}$, $k^{bwd}$) |  $50$ m ($10$, $50$, $100$)

###  4.1 Three-link merge network

The topology of the three-link merge networks is shown in Figure 1. Vehicles
enter the network by joining link 1 or 2, and they leave the network through
link 3. The service rates $\mu_{i}$ for link $i$ ($i=1,2,3$) are exogenous,
and the arrival rates $\lambda_{i}$ to links 1 and 2 ($i\in\\{1,2\\}$) are
exogenous, whereas $\lambda_{3}$ is endogenous. We conduct 2 experiments with
time-varying demand for link 1 and constant demand for link 2. The
experimental design is detailed in Table 2. For each experiment, the arrival
rate is 0.15 veh/sec during time [0,200] seconds, 0.25 veh/sec during time
[200,400] seconds and 0.05 veh/sec during time [400,600] seconds.

For experiment 1, the congestion level of link 1 is expected to increase and
then decrease, while link 2 is expected to be uncongested. During the first
400 seconds a spillback from link 3 to links 1 and 2 is expected. For
experiment 2, link 1 is expected to be congested for the first 400 seconds and
uncongested thereafter. Link 2 is expected to be congested, and link 3 is
expected to be uncongested all the time. Figure 4.1 (resp. Fig. 4.1) displays
$E[UQ(T)]$ and $E[DQ(T)]$ as a function of time $T$ for experiment 1 (resp.
experiment 2).

$1$$2$$3$ Figure 1: Merge network of three links Table 2: Experimental design Experiment | network type | $\lambda_{1}$ | $\mu_{1}$ | $\lambda_{2}$ | $\mu_{2}$ | $\mu_{3}$
---|---|---|---|---|---|---
1 | merge | $0.15\rightarrow 0.25\rightarrow 0.05$ | $0.4$ | $0.05$ | $0.2$ | $0.2$
2 | merge | $0.15\rightarrow 0.25\rightarrow 0.05$ | $0.2$ | $0.15$ | $0.2$ | $0.6$
Experiment | network type | $\lambda_{1}$ | $\mu_{1}$ | $\mu_{2}$ | $\mu_{3}$ | $p_{12}$
3 | diverge | $0.2\rightarrow 0.4\rightarrow 0.1$ | $0.3$ | $0.05$ | $0.4$ | $0.2$
4 | diverge | $0.2\rightarrow 0.4\rightarrow 0.1$ | $0.6$ | $0.4$ | $0.4$ | $0.5$

The plots in columns 1, 2, and 3 display, respectively, the results for links
1, 2 and 3. For link 1, there is, as expected, an increase in expectation
during time [200, 400] seconds, which is caused by both the increase in demand
and the occurrence of spillback from link 3. Then, there is a sharp decrease
after time $T=400$ seconds caused by the decrease in demand and the spillback
dissipation.

For link 2 (middle plot), there is an increase in expectation during time
[200, 400] seconds. This increase is purely caused by the spillback from link
3. For link 3, there is an increase in expectation during time [200, 400]
seconds and a decrease after time $T=400$ seconds caused by the changes in
demand of link 1. During time [200, 400] seconds, $E[UQ(T)]$ almost reaches
the space capacity and causes a significant spillback effect from link 3 to
links 1 and 2. For all three plots of Figure 4.1, the proposed analytical
model yields approximations which accurately capture the trends of the
simulated estimates. However, the proposed model overestimates the spillback
of link 3, and its effect on links 1 and 2.

Figure 4.1 displays the results of experiment 2. It has the same structure as
Figure 4.1. The increase and decrease in demand of link 1 causes the increase
and decrease in expectation for both links 1 and 3. For link 2 with constant
demand, there is a subtle increase and decrease in expectation. Even this
subtle change is captured by the proposed model. Overall, the proposed model
captures the temporal variations of these performance metrics accurately.
Experiment 1 Experiment 2 $E[UQ(T)]$ and $E[DQ(T)]$

###  4.2 Three-link diverge network

The topology of the three-link diverge network is shown in Figure 2. Vehicles
enter the network by joining link 1 at a rate $\lambda_{1}$. From link 1, they
continue to either link 2 with turning probability $p_{12}$ or to link 3 with
probability $p_{13}$ (i.e., $p_{13}=1-p_{12}$). The two experiments that we
conduct consider time-varying demand for link 1 and different turning
probabilities. The experimental design is presented in Table 2. For each
experiment, the arrival rate is 0.2 veh/sec during time [0,200] seconds, 0.4
veh/sec during time [200,400] seconds and 0.1 veh/sec during time [400,600]
seconds.

For both experiments 3 and 4, the congestion level of all links is expected to
increase and then decrease over time. For experiment 3, the congestion levels
of downstream links 2 and 3 will differ. For experiment 4, both downstream
links have identical configuration, and the vehicles have equal probability of
choosing either link, hence we expect the congestion levels of links 2 and 3
to be identical.

$1$$2$$3$ Figure 2: Diverge network of three links

Figure 4.2 shows the results for experiment 3. The graphs from left to right
display $E[UQ(T)]$ and $E[DQ(T)]$ for links 1, 2, and 3, respectively. As a
first impression, the deviations between analytical and simulated expectations
are small. The shapes of expectation curves for links 2 and 3 are different.
The sharp increase in expectation during the second 200 seconds and the sharp
decrease in expectation after $400$ seconds for all three links are well
approximated by the proposed model. Figure 4.2 shows the results for
experiment 4. The proposed model yields identical expectation curves for links
2 and 3, as expected. The analytical approximate expectations match the
simulated estimates. The proposed model accurately captures the temporal
variations, in both shape and magnitude, of these performance metrics.
Experiment 3 Experiment 4 $E[UQ(T)]$ and $E[DQ(T)]$

###  4.3 Eight link network

$1$$6$$4$$2$$8$$3$$5$$7$ Figure 3: Toy network of eight links Figure 4:
$E[UQ(T)]$ and $E[DQ(T)]$ for the experiment on toy network

Finally, we consider an eight link network with intricate traffic dynamics.
The network topology is given in Figure 3. This network configuration is the
same as in [38]. Vehicles enter the network by joining links 1 or 2. They can
leave the network through links 3 and 4. Upon entrance to link 1 (resp. 2),
vehicles can either go straight to link 6 (resp. 8) and leave the network
through link 4 (resp. 3) or initiate a U-turn to link 5 (resp. 7) and leave
the network through link 3 (resp. 4). The concrete parameters used for this
network are as follows. Vehicles can enter the network through links 1 or 2.
For link 1, vehicles enter at a rate of 0.1 veh/sec during [0,200] seconds,
0.2 veh/sec during [200, 400] seconds and 0.15 veh/sec during [400,600]
seconds. For link 2, vehicles enter at a rate of 0.1 veh/sec during [0,200]
seconds, 0.15 veh/sec during [200, 400] seconds and 0.2 veh/sec during
[400,600] seconds. They continue straight with probability $p_{16}=p_{28}=2/3$
and perform a U-turn with probability $p_{15}=p_{27}=1/3$. The service rates
for link 1 and 2 are $\mu_{1}=\mu_{2}=0.25$ veh/sec and the service rates for
the links within the network are $\mu_{5}=\mu_{6}=\mu_{7}=\mu_{8}=0.4$
veh/sec, which is on average sufficient to serve the demand. The service rates
for exit links 3 and 4 are lower at $\mu_{3}=\mu_{4}=0.2$ veh/sec.

The difference in time-varying demands at the two source links 1 and 2 leads
to intricate traffic patterns. The congestion patterns that we expect to
observe based on this experimental design are as follows. Link 1 experiences
step-changes from uncongested to congested, and then again to uncongested
conditions. Link 2, on the other hand, experiences step-changes steadily from
uncongested to congested conditions. For the first 200 seconds, the exit link
3 (resp. 4) is uncongested, hence the congestion level of its upstream links 5
and 8 (resp. 6 and 7) are determined by the outflow from their respective
upstream links 1 (resp. 2). After 200 seconds, travel demand increases. Thus,
the occurrence of spillback on the exit link 3 (resp. 4) increases, this
impacts the congestion levels of its upstream links 5 and 8 (resp. 6 and 7).

Each plot of Figure 4 shows the evolution of $E[UQ(T)]$ and $E[DQ(T)]$ for one
of the eight links in the network. The line styles and colors are the same as
for the previous figures. The temporal variations of the analytical
approximations mimic well the simulated variations. However, it tends to
underestimate the congestion levels of links 6 and 8.

In summary, the proposed model captures the temporal evolution in expectations
for this network with intricate traffic patterns. Additionally, it does so
with a significant gain in computational efficiency. For instance, since each
link of this network has a space capacity of 10, the state space of the joint
network distribution has dimension $11^{8}$. The proposed model overcomes the
need to model the entire joint distribution. Instead, the memory required for
the proposed model is $2\times 11\times 8=176$ numbers. This represents a
significant gain in computational efficiency, and enables its use for large-
scale network analysis and optimization.

##  5 Case study

In this section, the proposed network model is used to address an optimization
problem. The purpose of this section is to investigate the added value of the
proposed network model, and more specifically of the node model, for traffic
control. We consider a fixed-time signal control problem for the network shown
in Figure 6. The representation of the road network as a single-lane network
is presented in Figure 6. Section 5.1 formulates the problem and describes the
case study. Section 5.2 presents the numerical results.

We address the problem with the proposed network model. We also solve it with
the mixture model (described in Section 3.1) and with an exogenous node model
(described in Section 4.1 of [1]). Hereafter, we call the latter approach the
marginal model. In the marginal model, the total demand to each link is
exogenous. Hence, each link can be modeled independently. That is, the
marginal model accounts for the within-link temporal variations of congestion,
but does not capture any between-link (i.e., across-node) interactions. In
other words, both models (proposed and marginal) consider the same link model.
The proposed model has an endogenous node model (formulated in Section 3.2)
which captures the between-link interactions, while the marginal model omits
between-link interactions. We use both models to address the optimization
problem and compare their performances. This comparison will indicate the
added value of accounting for between-link interactions in traffic control.

Additionally, the deterministic intelligent link transmission model (ILTM)
[39], which is a complete network loading model that embeds the well-
established node model of Tampère et al. [40], is applied to address the same
problem. The comparison between the performance of the ILTM and the proposed
model will indicate the added value of accounting for stochasticity in traffic
control.

Figure 5: Network topology

Figure 6: Network model

###  5.1 Network configuration and problem formulation

We consider a synthetic network, which is displayed in Figure 6. The network
consists of $20$ single-lane roads and $4$ intersections, each with $2$
endogenous signal phases. The circles in Figure 6 are where the vehicles can
enter and leave the network. We consider a fixed-time signal control problem.
We refer the reader to Appendix A of [41] for a review of traffic signal
control terminology and formulations. Fixed-time signal plans are commonly
used for networks with sparse or unreliable real-time data, they are also used
by major cities, such as New York City [42], with high and uniformly
distributed congestion levels.

We consider a fixed-time signal control problem for a $30$ min period. The
signal plans of the $4$ intersections are jointly determined. The decision
variables are the green splits of the phases of the different intersections.
All other traditional control variables (e.g., cycle times, offsets, stage
structure) are assumed fixed. There are $2$ endogenous signal phases per
intersection. Hence, the dimension of the decision variable vector is $8$.

To formulate the signal control problem, we introduce the following notation:

$b_{d}$ | ratio of available cycle time to total cycle time for intersection $d$;
---|---
$x$ | vector of green splits;
$x(j)$ | green split of signal phase $j$;
$x_{LB}$ | vector of lower bounds for green splits;
$\mathcal{D}$ | set of intersection indices;
$\mathcal{P}_{D}(d)$ | set of endogenous signal phase indices of intersection $d$;
$\mathcal{L}$ | set of all links;
$\tilde{T}$ | the length of the time interval of interest [in min];
$|\mathcal{L}|$ | the number of links, i.e., cardinality of $\mathcal{L}$.

The problem is formulated as follows:

$\displaystyle\min_{x}f(x)=\frac{1}{\tilde{T}|\mathcal{L}|}\sum_{i\in\mathcal{L}}\sum_{\hat{t}=1}^{\tilde{T}}E[DQ_{i}(\hat{t};x)]$
(19)

subject to

$\displaystyle\sum_{j\in\mathcal{P}_{D}(d)}x(j)$ $\displaystyle=$
$\displaystyle b_{d},\quad\forall d\in\mathcal{D}$ (20) $\displaystyle x$
$\displaystyle\geq$ $\displaystyle x_{LB}.$ (21)

The decision vector, $x$, denotes the green splits of the signal controlled
links. Constraint (20) ensures that for each intersection, the sum of green
splits equals the available cycle time. Constraint (21) ensures lower bounds
for the green splits. The objective function represents the average, over time
and over links, of the expectation of $DQ$. $E[DQ_{i}(\hat{t};x)]$ denotes the
expectation of $DQ$ of link $i$ at integer time $\hat{t}$ (in minutes) under
signal plan $x$. It is calculated by evaluating either the proposed network
model or the marginal link model.

For the deterministic ILTM, the objective function is the average, over time
and over the network lanes, number of vehicles on the link, i.e.,

$f(x)=\frac{1}{\tilde{T}|\mathcal{L}|}\sum_{i\in\mathcal{L}}\sum_{\hat{t}=1}^{\tilde{T}}\left(c^{up}_{i}(\hat{t};x)-c^{down}_{i}(\hat{t};x)\right).$
(22)

where $c^{up}_{i}(\hat{t};x)$ (resp. $c^{down}_{i}(\hat{t};x)$) is the
cumulative number of vehicles that cross the upstream (resp. downstream) end
of link $i$ up until time $\hat{t}$ under signal plan $x$, which is the output
metric produced by the ILTM.

Note that the decision variables (i.e., the green splits) are related to the
link parameters as follows. Given a signal plan, the service rate, $\mu_{i}$,
of signalized link $i$ is determined by the following equation.

$\mu_{i}-\sum_{j\in\mathcal{P}_{I}(i)}x(j)s=e_{i}s$ (23)

where $s$ represents the saturation flow, $e_{i}$ represents the ratio of
fixed green time to cycle time and $\mathcal{P}_{I}(i)$ represents the set of
endogenous signal phases of link $i$. The ILTM also considers an average
outflow capacity as suggested by the developers. This is because the explicit
implementation of traffic cycles comes at a cost of more memory use (which can
be prohibitive in Matlab) and higher computation times.

Table 3 displays the values of the main exogenous model parameters. Note that
there are two sets of demand scenarios in Section 4 of [32], high demand and
medium demand, we are going to carry out the optimization for both demand
scenarios. The optimization problems are solved using the Active-set algorithm
of the fmincon routine of [43].

Table 3: Parameters for the case study Parameter | Value
---|---
$\tilde{T}$ |  $15$ min
$|\mathcal{L}|$ |  $20$ links
$\delta$ |  $0.1$ sec
$x_{LB}$ |  $4$ sec
$v$ |  $50$ km/h
$w$ |  $-15$ km/h
$\hat{\varrho}$ |  $200$ veh/km
$s$ |  $1800$ veh/h
$\mu$ | varies by signal plans
$\ell$, $\gamma$ | exogenous values from section 4 of [32]
$p_{ij}$ | turning factions is shown in Figure 6
$k^{fwd}$ | $k^{fwd}=\lceil\frac{\ell}{\hat{\varrho}v}\rceil$
$k^{bwd}$ | $k^{fwd}=\lceil\frac{\ell}{\hat{\varrho}|w|}\rceil$

###  5.2 Numerical analysis

To initialize the algorithm, we randomly and uniformly draw a point from the
feasible space (Equations (20)-(21)). We use the code of [44] to perform this
uniform sampling. We initialize the proposed model, the marginal model and the
ILTM with the same random initial point. We then solve Problem (19)-(21) with
each of the models. Each model yields an optimal signal plan. To evaluate the
performance of a given signal plan, we use a microscopic traffic simulation
model implemented with the Aimsun simulator [45]. For a given signal plan, we
embed it within the simulator and run $50$ simulation replications. Each
simulation replication simulate a $30$ min period. Each replication yields one
observation of the main performance measures: average link queue-length and
average trip travel time. We then compare the cumulative distribution
functions (cdf’s) obtained from the $50$ replications of these performance
measures.

Figures 7 and 7 consider the average link queue-length for high and medium
demand scenarios, respectively. For each demand scenario, it displays four cdf
curves: the initial signal plan (dashed line), the signal plans derived by the
proposed network model (solid line), by the marginal model (dot-dashed line)
and by the ILTM (crossed line). The average link queue-length is an average
obtained over time and over all links in the network. The $x$-axis displays
the average link queue-length. For given $x$ value, the $y$-axis displays the
proportion of simulation replications (out of $50$) that have average queue-
lengths smaller than $x$. Therefore, the more a cdf curve is to the left, the
better the performance of the corresponding signal plan.

For high demand scenario, all models yield signal plans with improved
performance compared to the initial plan. Additionally, the cdf curve of the
proposed model is to the left of both cdf curves from the ILTM and the
marginal model. This indicates the signal plan derived by the proposed model
outperforms that derived by the ILTM and by the marginal model. The signal
plan of the proposed model yields a $25\%$ improvement in the mean average
link queue-length compared to the signal plan of the marginal model ($2.31$
versus $3.06$) and a $14\%$ improvement compared to the ILTM ($2.31$ versus
$2.69$).

For medium demand scenarios, the ILTM does not detect objective function
improvement direction at the initial point and hence the optimization stopped
with the initial point randomly generated. Therefore, in Figure 7, the curve
of the ILTM completely overlaps with that of the initial signal plan. This is
because the ILTM calculates the deterministic average inflow and outflow
capacity of the link over time and hence the model is not sensitive to the
change of outflow capacity as long as it is large enough to allow all inflow
to pass through. On the other hand, the stochastic models are sensitive to
changes in outflow capacity (i.e., $\mu$), especially when traffic intensity
is near 1. The cdf curves of both the proposed and the marginal model are to
the left of that of the ILTM. This indicates the signal plans derived by the
proposed and marginal models outperform that derived by the ILTM.
Additionally, the proposed model outperforms the marginal model and yields a
$55\%$ improvement in the mean average link queue-length ($0.97$ versus
$2.13$).

(a) High demand scenario

(b) Medium demand scenario

Figure 7: Cumulative distribution functions of the average, over all lanes in
the network, link queue-length in vehicles.

Figures 8 and 8 compare the performance of the four signal plans considering
the average trip travel times (in minutes) for high and medium demand
scenarios, respectively. The $x$-axis shows average trip travel time. the
$y$-axis displays the proportion of simulation replications (out of $50$) that
have average trip travel times smaller than $x$. Figure 8 has a similar layout
as Figure 7. The same conclusions as for Figure 7 hold. All models yield
signal plans no worse than the initial signal plan in both demand scenarios.
The proposed model yields signal plans that outperform those derived by the
ILTM and the plan derived by the marginal model in both demand scenarios. The
signal plan of the proposed model yields an $35\%$ improvement in the mean
average trip travel time compared to the signal plan of the marginal model
($0.99$ minutes versus $1.53$ minutes) in medium demand scenario and $19\%$
improvement in high demand scenario ($1.48$ minutes versus $1.82$ minutes).
The ILTM performances better than the marginal model in high demand scenario
but not in medium demand scenario.

This case study shows the proposed network model outperforms the marginal
model and the ILTM. It demonstrates the added value of the proposed node model
to address the between-link interactions. Moreover, by considering
stochasticity, the proposed network model, compared to the ILTM, is able to
identify signal plans with better and more robust performance in various
demand level.

(a) High demand scenario

(b) Medium demand scenario

Figure 8: Cumulative distribution functions of the average trip travel time in
minutes.

##  6 Conclusion

This paper formulates an analytical stochastic node model. It formulates a
full network model by coupling the node model with the link model of [1]. The
proposed network model is scalable and tractable. In particular, the node
model is not based on the joint modeling of all upstream and downstream links
adjacent to a node. Instead the joint probabilities are approximated from the
univariate distributions of the upstream and downstream links. The node model
is formulated for nodes with an arbitrary configuration, it satisfies flow
conservation properties. The proposed network model is validated versus
stochastic simulation results, using a simulator of the stochastic link
transmission model. We consider a set of small networks with intricate time-
varying congestion patterns. The boundary conditions of every link in the
network are well approximated by the proposed network model. The proposed
model is used to address a signal control problem for a synthetic network with
different demand level. We compare the performance of the proposed approach to
that of a model with the same link model but without a node model (i.e.,
between-link interactions are not captured) as well as a deterministic ILTM
with well-established deterministic node model. The proposed network model
yields signal plans that outperform the marginal model as well as the
deterministic ILTM. It shows both the added value of accounting for the
across-link dynamics for traffic management and the added value of taking
stochasticity in consideration

Ongoing work investigates formulations to approximate the full joint network
distribution. For instance, we plan to combine the proposed approach with
ideas from the network decomposition method of [38]. We are also investigating
the use of aggregation-disaggregation techniques to approximate the full joint
network distribution in a scalable and tractable way. Aggregation-
disaggregation ideas address the curse of dimensionality by providing an
aggregate description of network states [32, 33].

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