A stochastic model of traffic flow: Gaussian

Transportation Research Part B 47 (2013) 15–41

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Transportation Research Part B journal homepage: www.elsevier.com/locate/trb

A stochastic model of traffic flow: Gaussian approximation and estimation Saif Eddin Jabari, Henry X. Liu ⇑ University of Minnesota, Department of Civil Engineering, 500 Pillsbury Drive S.E., Minneapolis, MN 55455, United States

a r t i c l e

i n f o

Article history: Received 29 December 2011 Received in revised form 5 September 2012 Accepted 6 September 2012

Keywords: Stochastic traffic flow Queueing processes Macroscopic traffic flow Gaussian approximation Observability Traffic state estimation

a b s t r a c t A Gaussian approximation of the stochastic traffic flow model of Jabari and Liu (2012) is proposed. The Gaussian approximation is characterized by deterministic mean and covariance dynamics; the mean dynamics are those of the Godunov scheme. By deriving the Gaussian model, as opposed to assuming Gaussian noise arbitrarily, covariance matrices of traffic variables follow from the physics of traffic flow and can be computed using only few parameters, regardless of system size or how finely the system is discretized. Stationary behavior of the covariance dynamics is analyzed and it is shown that the covariance matrices are bounded. Consequently, Kalman filters that use the proposed model are stochastically observable, which is a critical issue in real time estimation of traffic dynamics. Model validation was carried out in a real-world signalized arterial setting, where cycleby-cycle maximum queue sizes were estimated using the Gaussian model as a description of state dynamics. The estimated queue sizes were compared to observed maximum queue sizes and the results indicate very good agreement between estimated and observed queue sizes. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction One of the main challenges pertaining to the use and analysis of stochastic models of macroscopic traffic flow is tractability. In a deterministic context, it is well known that prominent models, namely the model of Lighthill and Whitham (1955) and Richards (1956) (LWR) and higher order extensions thereof, are ill-posed; adding uncertainty only exacerbates analytical complications. For this reason, many authors have resorted to the simplest of stochastic extensions: the addition of Gaussian noise. Examples of such models include (Gazis and Knapp, 1971; Szeto and Gazis, 1972; Gazis and Liu, 2003; Wang and Papageorgiou, 2005; Wang et al., 2007; Boel and Mihaylova, 2006; Khoshyaran and Lebacque, 2009; Sumalee et al., 2011), which, with the exception of Khoshyaran and Lebacque (2009), operate in discrete time and space and were developed for traffic state estimation purposes. The rationale behind adding Gaussian noise is that Gaussian models are fully characterized by their first two moments alone, a desirable feature from an analytical tractability point of view. However, Jabari and Liu (2012) argued that the arbitrary addition of (Gaussian) noise terms could lead to two problems: (i) the possibility of producing negative sample paths and (ii) mean dynamics that do not coincide with the original deterministic dynamics to which noise was added due to nonlinearity of the dynamic equations. In fact, nonlinear functions of Gaussian noise typically produce non-Gaussian, non-zero mean random variables. Other approaches to stochastic modeling of traffic flow in the literature include Botlzmann-like models of traffic flow (e.g., Prigogine and Herman, 1971; Paveri-Fontana, 1975), Markovian/ queueing network approaches (e.g., Davis and Kang, 1994; Kang, 1995; Di et al., 2010; Osorio et al., 2011; Jabari and Liu, 2012), and cellular automaton (CA) based models (e.g. Nagel and Schreckenberg, 1992; Gray and Griffeath, 2001; Sopasakis and Katsoulakis, 2006; Sopasakis, 2012). In general, these approaches do not suffer the two problems cited above (in some

⇑ Corresponding author. Tel.: +1 612 625 6347; fax: +1 612 626 7750. E-mail addresses: [email protected] (S.E. Jabari), [email protected] (H.X. Liu). 0191-2615/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.trb.2012.09.004

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S.E. Jabari, H.X. Liu / Transportation Research Part B 47 (2013) 15–41

cases minor modification may be needed), but are generally intractable. Specific to problems of real-time traffic state estimation, successive Monte-Carlo or particle filter based methods are required for these models, which could be computationally prohibitive. On the other hand, extended Kalman filters, which are fast by comparison to particle filters, rely on first order approximations of the nonlinear dynamics. This limits the applicability of extended Kalman filtering to differentiable dynamics and precludes some of the most prominent traffic flow dynamics, such as the cell transmission model (CTM) (Daganzo, 1994, 1995b) and its generalization Daganzo (1995a) due to disjunctive flux functions (i.e., involving extrema of traffic variables). Another problem that arises in this context is related to computing covariance matrices of traffic state variables, a crucial component of Kalman filters. To accurately capture queue build-up and dissipation dynamics, finer discretization of space and time are required. This results in large numbers of traffic flow variables and, consequently, large system covariance matrices. Furthermore, since flows across boundaries of cells (discrete space intervals) depend on traffic states on either side of the boundaries, dependencies between traffic state variables arise and assumptions of diagonal covariance matrices (typically made in the traffic state estimation literature) are not valid. Estimation of such large time varying covariance matrices may be prohibitive. More generally, for traffic estimation problems, finer discretization results in larger numbers of variables and increases the sparsity of the available measurements. This could lead to observability issues. That is, introducing more variables, we have fewer observations. This type of observability pertains to the mean dynamics of the system. A critical issue related to the stochastic features of the system, from an estimation point of view, is stochastic observability. Stochastic observability is related to the behavior of the estimated covariance matrices of the system, which provide ‘‘a statistical description of the errors associated with the estimated state mean vector’’ (Bageshwar et al., 2009). Since the objective of any filter is to compute a minimum variance estimate, estimated covariance matrices must be bounded. This boundedness property is what defines stochastic observability. In this paper, we develop a tractable Gaussian approximation of the queueing model proposed by Jabari and Liu (2012). The main idea is to determine, approximately, the distribution of the deviation between the (stochastic) queueing model and its (deterministic) fluid limit, which we show to be a Gaussian process. This Gaussian deviation, in essence, provides a second-order (stochastic) refinement to the fluid process. A crucial component in our development is a recipe for computing time-varying covariance matrices using only few parameters (namely, pertaining to the fundamental diagram). These covariance matrices do not assume independence of traffic variables and need not be diagonal as is typically assumed in the traffic estimation literature. The Gaussian approximation is the sum of a deterministic process and a zero-mean Gaussian process ^ integral with deterministic integrand), which means that the expected dynamic is simply the deterministic (namely, an It o process itself. The stationary behavior of the covariance matrix is a crucial ingredient in the proposed stochastic model, which ensures the non-negativity of the traffic variables and that traffic densities do not exceed the jam density. Another crucial ingredient is the boundedness of the covariance matrices: when mean traffic conditions do not change with time, variability of the traffic densities should not vary as well. These issues are demonstrated via numerical examples. The derived mean and covariance dynamics of the Gaussian model are first-order deterministic differential equations that depend on the expected values of traffic flow variables, not the stochastic traffic variables themselves. This allows for implementation of a standard Kalman filter for purposes of traffic state estimation and prediction, which results in computational tractability and permits real-time implementation of the proposed model. The continuous time setting in which our model is derived offers the flexibility of using different computational time scales for the state measurement equations of the Kalman filter. That is, we do not require the availability of measurements at regular time intervals to run the filter. In general, due to the sparsity of measurements, observability is difficult to establish. However, under certain traffic flow conditions, such as free-flow conditions, the presence of traffic sensors on either end of a road section will allow for reconstruction of initial mean traffic densities within the road section. To overcome observability issues, this study uses a warm-up period where the initial conditions are observable free-flow traffic conditions. Furthermore, the number of cells used has no impact on whether mean traffic conditions are observable or not. In terms of stochastic observability, a crucial contribution of this research is that the covariance matrices of the Gaussian model are bounded. Thus, Kalman filters built using the proposed model are stochastically observable. This paper is organized as follows: in Section 2, we motivate our proposed framework by looking at the simpler case of non-homogeneous Poisson processes representing cumulative flows across cell boundaries. This simple setting illustrates the difficulties that arise when using classical Markovian approaches to analyze stochastic flow models. Section 3 provides the intuition behind the proposed framework by working out the second-order approximation for the simpler setting of Section 2. Section 4 develops the second-order approximation for the general, state-dependent model. Here, we address issues pertaining to Lipschitz continuity and differentiability of the disjunctive flux functions, which are critical ingredients in developing the second-order approximation of the process. The intuition developed in Section 3 is carried over to the general case in Section 4. Section 5 investigates the stationary behavior of the covariance dynamics under various traffic regimes, boundedness of the covariance matrices, and numerical computation of the covariance dynamics. Section 6 gives a numerical example of the proposed model illustrating the numerical ingredients and the properties of the covariance matrix under various traffic conditions. In Section 7, we present the state and measurement equations used for Kalman filtering and discuss issues related to observability. Section 8 provides a real-world traffic state estimation example as a means to validate the proposed model and Section 9 concludes the paper. Appendices A and B, respectively, prove the Lipschitz continuity of the disjunctive flux function and illustrate the derivation of a matrix differential equation used to compute the covariance matrices.


17

2. Motivation In this section, we motivate the proposed approximation and explain the intuition behind it by applying it to a simpler modeling scenario. The intuition carries over to the more sophisticated (state-dependent) setting, which is the main interest in this paper. We consider throughout this section and the following sections a homogeneous roadway without sources or sinks, which is divided into cells. As in Jabari and Liu (2012), let Qðx; tÞ denote a stochastic counting process describing the cumulative number of vehicles that have crossed the downstream boundary of cell x 2 C at time t 2 [0, U], where C is a countable set and U < 1 is a horizon time. The conservation of traffic density in x is written as:

1 lx

qðx; tÞ ¼ qðx; 0Þ þ ðQðx 1; tÞ Qðx; tÞÞ;

ð1Þ

where q(x, t) is the (random) traffic density in cell x at time t and lx is the length (or size) of cell x. We wish to characterize q(x, t) probabilistically. For the sake of illustration, let’s assume that q(x, 0) = 0 almost surely for all x and that the cell lengths are equal and normalized to 1; that is lx = 1 for all x 2 C. Then the traffic density in x is characterized by the two counting processes Qðx 1; tÞ and Qðx; tÞ. This assumption leads to the following conservation equation:

qðx; tÞ ¼ Qðx 1; tÞ Qðx; tÞ:

ð2Þ

Again, for the sake of illustration, let’s assume that fQðx; tÞgx2C may be represented by independent (non-homogeneous) Poisson processes with time-varying deterministic rates fkðx; tÞgx2C ; that is, we assume here that the instantaneous flow rates are given time varying constants. We then obtain a non-homogeneous birth and death process for each of the cells, or an Mt/Mt/1 queueing system for each cell. Then the Markovian approach to characterizing the probabilistic nature of fqðx; tÞgx2C consists of solving the following system of (birth and death) equations for each x:

@ p ðk; tÞ ¼ kðx 1; tÞpx ðk 1; tÞ þ kðx; tÞpx ðk þ 1; tÞ ðkðx 1; tÞ þ kðx; tÞÞpx ðk; tÞ; @t x

ð3Þ

for k = 1, 2, . . . and

@ p ð0; tÞ ¼ kðx; tÞpx ð1; tÞ kðx 1; tÞpx ð0; tÞ; @t x

ð4Þ

for k = 0, where px ðk; tÞ Pðqðx; tÞ ¼ kÞ. The Markovian approach, which considers non-stationary dynamics (i.e., solving (3) and (4)), can be difficult, and a closed form solution is rarely available explicitly (see Mandelbaum and Massey (1995) and references therein). In fact, even when closed form expressions are available, their complexity could render their use and further analysis prohibitive. As an example, in the simplest of cases when kðx; tÞ do not vary with time, the solution involves modified Bessel functions; see for example, Masey (1981). For this reason, one resorts to asymptotic analysis of the probabilities; that is, the probabilities which arise when t ? 1. In this case, the left-hand sides of (3) and (4) are zero (i.e., @px(k, t)/@t = 0) and the differential equations become difference equations, which are easier to solve and typically deliver simpler solutions. However, while it may not be difficult to obtain/analyze the stationary probabilities corresponding to the long-term probabilistic behavior of the system, this is of little use to a traffic engineer concerned with the transient features of the system (e.g., changes in traffic characteristics from one traffic light cycle to the next). To quote Mandelbaum and Massey (1995): ‘‘approximating the behavior of the system in the here and now by its behavior at time infinity is typically futile’’. Then, instead of attempting to explicitly solve Eqs. (3) and (4), an alternative approach is to approximate (1) by a more tractable stochastic model. Specifically, if one could approximate (1) by a Gaussian process for which it is only necessary to compute the mean (deterministic) behavior of the system and the time-varying covariance, then the problem becomes much simpler. Indeed, this approach first appeared in Newell (1968a,b) for the Mt/M/1 queue and made rigorous in Masey (1981) and Massey (1985) for the Mt/Mt/1 queue. In particular, the latter introduced the idea of ‘‘uniform acceleration’’ as an approximation method that preserves the transient features of the queueing process. Using ‘‘uniform acceleration’’, Mandelbaum and Massey (1995) extended the work of Massey (1985) to the asymptotic analysis of the sample paths of the Mt/Mt/1 queue. This is the approach taken in this paper, which we illustrate below for the simple Mt/Mt/1 queue given above. It is crucial to note that the difficulties are substantially exacerbated when the flow rates k(x, t) depend on the (random) traffic densities, which is the case for macroscopic traffic flow considered in this paper.

3. The proposed methodology: interpretation and illustration 3.1. Scaled time headways and accelerated counting Let Xi(x) denote the (random) time headway of the ith vehicle departing cell x, then the departure time of vehicle k is, denoted by Ak ðxÞ:

18


Ak ðxÞ ¼

k X X i ðxÞ;

ð5Þ

i¼1

where A0 ðxÞ 0. Consequently, the cumulative flow is defined as

Qðx; tÞ maxfk : Ak ðxÞ 6 tg;

ð6Þ

or, alternatively, the counting process may be characterized by the events:

fQðx; tÞ ¼ kg () fAk ðxÞ 6 t \ Akþ1 ðxÞ > tg;

ð7Þ

where , means ‘‘if and only if’’. Both cases, (6) and (7), are interpreted as: ‘‘ Qðx; tÞ is the index of the most recent departure from cell x at time t’’. In order to perform an asymptotic analysis of Qðx; tÞ, i.e., determine the stationary flow rate: Qðx;tÞ as t t ? 1, one counts a large number of time headways. As letting k go to infinity provides the future behavior of the process, this is of little use. Instead, we count fractions of time headways and as we let the fraction size get smaller we obtain a larger number of fractions to which we may apply asymptotic analysis without loosing the transient information in the process. This is illustrated as follows: suppose we wish to count k whole vehicles by looking at the ‘‘index of the most recent fractional arrival’’. That is, suppose we divide the headways uniformly into n 2 N fractions, then the crossing time of the kth whole vehicle is the crossing time of the (nk)th fraction: nk X X i ðxÞ i¼1

n

¼

1 Ank ðxÞ; n

ð8Þ

where the right-hand side follows from (5). Suppose this occurs by some time t, but the crossing time of the next fraction occurs after time t; that is, the following event takes place:

1 1 Ank ðxÞ 6 t \ Ankþ1 ðxÞ > t ¼ fAnk ðxÞ 6 nt \ Ankþ1 ðxÞ > nt g n n

ð9Þ

From (7), this implies that

fQðx; ntÞ ¼ nkg ¼

1 Qðx; ntÞ ¼ k : n

ð10Þ

As n ? 1, we count a larger and larger number of smaller and smaller jumps across x, which in the limit resemble a fluid process. Qðx; ntÞ may then be interpreted as the process Qðx; tÞ with its time varying rates accelerated uniformly by a factor of n (n times as many vehicles per unit time). Dividing Qðx; ntÞ by n serves as a reminder that we are counting fractions (of size 1/n) of vehicles. When Qðx; tÞ is a non-homogeneous Poisson process (as given above), one writes:

Qðx; tÞ ¼ N x

Z

t

kðx; uÞdu ;

ð11Þ

0

where N x ðÞ is a unit rate Poisson process (i.e., averaging one event per unit time) for vehicles leaving cell x and a re-scaling of the time axis in accordance with the time varying rates.

Rt 0

kðx; uÞdu is

Example 1. A homogeneous Poisson process with rate k counts (on average) k vehicles per unit time. This is equivalent to a process that counts (on average) one vehicle k times per unit time. This can also be interpreted as stretching (or contracting) the time axis a constant rate of k and counting (on average) one vehicle per unit ‘‘scaled’’ time. With time-varying rates, the ‘‘stretching/contracting’’ of the time axis is not constant. Uniformly accelerating Qðx; tÞ a rate n and ‘‘aggregating’’, we get:

Z t 1 1 Qðx; ntÞ ¼ N x n kðx; uÞdu : n n 0

ð12Þ

Scaling the time axis allows us to obtain results about the asymptotic behavior of sophisticated processes from simpler processes. For instance, it is well known that the scaled unit rate Poisson process satisfies the following (functional) strong law of large numbers; see derivation in (Jabari and Liu, 2012, Appendix B):

1 N x ðntÞ ! t n!1 n

almost surelyða:s:Þ:

In fact, this holds for any t. Indeed, since

ð13Þ

Rt 0

kðx; uÞdu is deterministic, we have that:

Z t Z t 1 Nx n kðx; uÞdu ! kðx; uÞdu a:s: n!1 0 n 0

ð14Þ

The limiting (fluid) behavior in (14) is deterministic. This provides a first-order approximation of the stochastic process Qðx; tÞ, which serves as a good way of characterizing the qualitative behavior of the stochastic model. The fluid limit in


19

(14) should not be surprising for a Poisson process; that is, it converges to its expectation. However, it is useful to note this also holds for more general counting processes, where the fluid limit and the expectation do not coincide (see Section 4.3). 3.2. Second order approximation The fluid limit, in essence, captures the mean behavior of the process. Our proposed second order refinement can simply be thought of as a measure of the deviation of the stochastic model from its mean. Then, considering both the mean behavior and the probabilistic deviation from the mean, we obtain a second order (stochastic) approximation of the process. Consider the scaled unit rate Poisson process, ð1=nÞN x ðnÞ: when n is large, the random variable N x ðntÞ (when t is some constant) is approximately equal to the sum of bntc independent and identically distributed (i.i.d.) Poisson random variables with mean 1, where bntc is the largest integer less than or equal to nt; that is, let fnj gnj¼1 i:i:d. Poisson (1). Then:

N x ðntÞ

bntc X nj ;

ð15Þ

j¼1

where equality holds in the limit. Consider the re-scaled centered process (centered at its fluid limit):

pffiffiffi 1 n N x ðntÞ t : n

ð16Þ

By (15), this is approximated by

1 X pffiffiffi ðnj 1Þ n j¼1 bntc

ð17Þ

By Donsker’s theorem (Billingsley, 1999, Theorem 14.1), (17) converges weakly (i.e., in distribution) to standard Brownian motion, denoted by Wx(t), for vehicles leaving cell x. That is:

pffiffiffi 1 D n N x ðntÞ t W x ðtÞ; n

ð18Þ

D

where means ‘‘approximately equal, in distribution, to’’. Define the scaled time ~t nt and multiply both sides of (18) by pffiffiffi n, we get: D

N x ð~tÞ ~t

~t D pffiffiffi nW x ¼ W x ð~tÞ; n

ð19Þ

D

where equivalence in distribution ð¼Þ follows from the scale invariance property1 of Brownian motion. We have thus obtained that the deviation of the scaled Poisson process from its mean, N x ðtÞ t, behaves, in terms of distribution, like standard Brownian motion, Wx(t). Since this holds for any time t, this result is immediately extended, via time scaling, to the non-homogeneous Poisson process describing vehicle counts across cell boundaries:

Nx

Z

t

Z t Z t D kðx; uÞdu kðx; uÞdu W x kðx; uÞdu :

0

0

ð20Þ

0

The time changed Brownian motion on the right-hand side of (20) can be understood, for a fixed time ranR t, as a normal Rt t dom variable with mean zero and variance 0 kðx; uÞdu. Allowing k(x, t) to vary continuously with t, W x 0 kðx; uÞdu may also ^ integral with deterministic integrand (i.e., a Gaussian process)2: be written as an Ito

Z t pffiffiffiffiffiffiffiffiffiffiffiffiffiffi kðx; uÞ dW x ðuÞ

ð21Þ

0

Returning to the scenario given at the beginning of the section, we see that the traffic density q(x, t), given by Eq. (2) with Qðx 1; tÞ and Qðx; tÞ represented by independent non-homogeneous Poisson processes, may be approximated by:

q~ ðx; tÞ ¼

Z

t

kðx 1; uÞdu 0

Z

t

kðx; uÞdu þ

Z t pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z t pffiffiffiffiffiffiffiffiffiffiffiffiffiffi kðx 1; uÞ dW x1 ðuÞ kðx; uÞ dW x ðuÞ;

0

0

ð22Þ

0

pffiffiffi 1 It is this property which motivates the choice of n as a re-scaling factor. In essence, multiplying the difference in (16) by this re-scaling factor plays the role of an amplification of the deviation, which otherwise converges to zero. 2 ^ integral with deterministic integrand is defined as follows: Let 0 = t0, t1, . . . , tm = t be any partition of the interval [0, t], Dt = max{jti ti1j, i = 1, . . . , m} An Ito be the mesh of the partition, g() be some deterministic (integrable) process, and W() denote standard Brownian motion, then the process

Z

t

gðuÞdWðuÞ limDt!0 0

m X gðti1 ÞðWðti Þ Wðt i1 ÞÞ i¼1

^ integral with deterministic integrand. is referred to as an Ito

20


~ ðx; tÞ denotes approximated traffic density. This process has mean where q

~ ðx; tÞ ¼ Eq

Z

t

ðkðx 1; uÞ kðx; uÞÞdu

ð23Þ

0

and variance

~ ðx; tÞ Eq ~ ðx; tÞÞ2 ¼ Eðq

Z t pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi kðx 1; uÞ þ kðx; uÞ du:

ð24Þ

0

We note that the variance given by (24) grows without bound so as long as vehicles continue to flow through the cells. This is an undesirable feature from both physical and application standpoints. That the variance grows without bound means that the probability of negative traffic states and traffic densities that exceed jam density increase with time, even under free-flow traffic conditions. From an estimation stand-point, if the variances grow with out bound and one has sparse measurements, there is little than can be done to find an optimal (minimum variance) estimate. 4. State dependent flow rates 4.1. Preliminaries Consider now the scenario where the instantaneous flow rates across cell boundaries depend on cell traffic densities via an appropriate flux function. Here we only consider concave fundamental relations without discontinuities, which include many fundamental relations typically used in the traffic flow literature; see Del Castillo and Benitez (1995) for a thorough discussion of fundamental relations. Flow rates across cell boundaries depend on traffic densities on both sides of the boundary. Let y(x, t) = [q(x, t)q(x + 1, t)]T denote the vector of ‘‘relevant’’ traffic conditions at the downstream boundary of cell x, which include the (random) traffic densities in the two cells adjacent to the boundary. Let Qe(q) denote an equilibrium (fundamental) flow-density relation, then the flux function may be written as: k(y(x, t), b), where b is a vector of parameters. We shall assume throughout that b is pre-determined and shall concisely write the flux function as k(y(x, t)), where dependence on parameters is understood to be implicit. As in Jabari and Liu (2012), we consider the flux function proposed by Daganzo (1995a):

kðyðx; tÞÞ ¼ minfSe ðqðx; tÞÞ; Re ðqðx þ 1; tÞÞg;

ð25Þ

where Se and Re denote sending and receiving functions, respectively. They are depicted in Fig. 1 (when the traffic densities are deterministic). We note that when y(x, t) is random, then so is k(y(x, t)), which means that our counting processes shall possess stochastic rates. In general, in order to extend our results in Section 3 to this state-dependent case, the following properties are needed: 1. For all y(x, t) 2 [0, qjam] [0, qjam], the flux functions are bounded and non-negative (qjam denotes the jam density). 2. For all y(x, t) 2 [0, qjam] [0, qjam], the flux functions are Lipschitz continuous. 3. The flux function is differentiable in both elements of y(x, t). The first property is immediate for flux functions obtained from the fundamental diagram; see Fig. 1 for illustration. The second property, on the other hand, is not obvious, since the flux function involves taking the minimum of two functions. However, since both the sending and the receiving functions are concave, it is simple to show that the minimum of the two is Lipschitz continuous in the vector y(x, t); see Appendix A for proof. The flux function (25) is non-differentiable in the classical sense as noted in Work et al. (2008) and Blandin et al. (2012). To demonstrate this, let a and b denote arbitrary traffic densities in two adjacent cells; noting that

minfSe ðaÞ; Re ðbÞg ¼

(a)

1 ððSe ðaÞ þ Re ðbÞÞ jSe ðaÞ Re ðbÞjÞ; 2

ð26Þ

(b)

(c)

Fig. 1. (a) A typical concave fundamental relationship; (b) the sending function; (c) the receiving function.


21

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi we see that the partial derivatives are computed, using jSe ðaÞ Re ðbÞj ¼ ðSe ðaÞ Re ðbÞÞ2 , as:

@k dSe ðaÞ 1 Se ðaÞ Re ðbÞ ¼ @a da 2 2jSe ðaÞ Re ðbÞj @k dRe ðbÞ 1 Se ðaÞ Re ðbÞ ¼ þ @b db 2 2jSe ðaÞ Re ðbÞj

ð27Þ

Now, note that when Se(a) = Re(b), the partial derivatives in (27) do not exist. The set of points where this occurs is nonnegligible: for any a 2 [0, qjam] it is easy to see that there exists a point b 2 [0, qjam] such that Se(a) = Re(b) and vice versa. Furthermore, the derivatives are generally not equal for these cases.3 To overcome this problem, we apply the following intuition: when Se(a) = Re(b) we have that jSe(a) Re(b)j = 0 and the formula for the minimum in (26), for such points of discontinuity, reduces to:

minfSe ðaÞ; Re ðbÞg ¼

1 ðSe ðaÞ þ Re ðbÞÞ; 2

ð28Þ

and the derivatives, when Se(a) = Re(b), are then computed as:

@k 1 dSe ðaÞ ¼ ; @a 2 da @k 1 dRe ðbÞ ¼ : @b 2 db

ð29Þ

The derivatives above are in fact weak derivatives (see for example Howison (2005) for definition and further discussion). It is well known that the absolute value function jzj possesses the following weak derivative4:

8 if z > 0; >1 djzj < ¼ 0 if z ¼ 0; > dz : 1 if z < 0:

ð30Þ

We, hence, see that the weak derivatives of k in (26) can be written as:

8 0 >
Re ðbÞ; @k 1 dSe ðaÞ if Se ðaÞ ¼ Re ðbÞ; ¼ 2 da @a > : dSe ðaÞ if Se ðaÞ < Re ðbÞ; da

ð31Þ

and

8 dRe ðbÞ > < db

if Se ðaÞ > Re ðbÞ; @k ¼ 1 dRe ðbÞ if Se ðaÞ ¼ Re ðbÞ; @b > : 2 db 0 if Se ðaÞ < Re ðbÞ;

ð32Þ

which is consistent with (27) when Se(a) – Re(b) and (29) when Se(a) = Re(b). 4.2. State-dependent processes In this section, we consider general counting processes fQðx; tÞgx2C and seek to determine a second-order approximation of these processes and the corresponding traffic densities. We apply the same thinking presented in Section 3, while addressing the stochasticity in the flow rates. Let us first specify the counting processes: for each x 2 C, Qðx; tÞ is a point process with rates defined by the flux function k(y(x, t)). In the language of probability theory, we say that the point process Qðx; tÞ is measurable with respect to the sigmaRt field F t while its intensity, 0 kðyðx; uÞÞdu, is measurable with respect to F t , where F t # F t . The sigma field is, in essence, an event space for a random variable and when considering a stochastic process, a sequence of sigma-fields (a filtration) represents the history of the process and therefore grows with time. This technical detail, referred to as predictability in the stochastic calculus literature, is needed to ensure that the stochastic integrals we construct in the sequel possess the desired mathematical properties. The best way to illustrate this measurability condition in our traffic flow model is to consider how one would simulate the process: suppose one devises a discrete time approximation of our processes and computes traffic variables over small discrete time intervals of length Dt, which for the sake of illustration may be assumed fixed. One begins with initial traffic densities, {q(x, 0)}, which are possibly random. These traffic densities are used to compute/simulate flux functions over the time 3 That is, a simple treatment where one first determines the minimum of the two differentiable functions and then computes the derivative of that function is not appropriate. 4 An alternative intuition is: the absolute value function has a V shape. To approximate the derivative at z = 0, one may approximate jzj by a V with a very small parabola (of negligible size) at z = 0; the derivative at z = 0 is then 0.

22


interval [0, Dt), which are then used to simulate cumulative flows over the time interval (0, Dt], which in turn are used to compute the traffic densities at time Dt, {q(x, Dt)}. The procedure is then repeated for the interval [Dt,2Dt), and so on. We note that this is precisely a stochastic version of how both the Godunov scheme (Godunov, 1959; Lebacque, 1996) and the CTM operate; that is flows are computed using the most recent past values of traffic density and the new traffic densities are computed using these flows. These conditions along with those discussed in Section 4.1 allows us to represent Qðx; tÞ as a unit rate Poisson process Rt with the stochastic time change 0 kðyðx; uÞÞdu; that is,

Qðx; tÞ ¼ N x

Z

t

kðyðx; uÞÞdu :

ð33Þ

0

That any point process can be transformed via a time change to a Poisson process is an important result in point processes theory; a detailed exposition of the result can be found in Daley and Vere-Jones (2003, 2008). The Poisson process with the random time change is a doubly stochastic Poisson process and can be interpreted in the same manner as the time changed non-homogeneous Poisson process, except that the time axis is stretched/contracted in accordance with the crossing times fAi ðxÞgiP0 . We now write the scaled conservation equation as:

qn ðx; tÞ ¼ qn ðx; 0Þ þ

1 1 1 Qðx 1; ntÞ Qðx; ntÞ : lx n n

ð34Þ

As in Section 3, the scaled counting processes are given by:

Z t 1 1 Qðx; ntÞ ¼ N x n kðyn ðx; uÞÞdu : n n 0

ð35Þ

Note that flow rates are functions of the scaled traffic densities,5 yn(x, t) = [qn(x, t)qn(x + 1,t)]T. The interpretation of the scaled traffic density, qn(x, t), is: the traffic density in cell x which corresponds to scaled counting processes at the cell boundaries. That is, the traffic density itself is not scaled, but constructed from scaled counting processes. This is in contrast to the way scaling is typically applied in the basic sciences, where it is sometimes referred to as system size expansion (for an application to traffic flow, see for example (Karmeshu and Pathria, 1981)). 4.2.1. Initial conditions The initial traffic density q(x, 0) is assumed here to be a random variable, independent of the future of the process (but not ðx; 0Þ vice versa) and independent in x. This preserves causality in our model. Denote the mean and variance of q(x, 0) by q and r2(x, 0), respectively. The scaled initial traffic densities shall be assumed to converge to their mean values as n ? 1. Since we are only interested in the limiting behavior, this can be easily accomplished by construction. For example, let q1(x), q2(x), . . . , qn(x) be a sequence of i.i.d. random variables, identically distributed as q(x, 0), then

qn ðx; 0Þ

n 1X q ðxÞ n j¼1 j

ð36Þ

achieves the desired limit, as n ? 1 by the classical strong law of large numbers. Furthermore, under re-scaling and center ðx; 0Þ and variance r2(x, 0) by ing, we have that qn(x, 0) converges in distribution to a Normal random variable with mean q the classical central limit theorem. 4.2.2. The fluid limit Lipschitz continuity and boundedness of the flux functions k(y(x, u)) allow us, by appeal to the functional strong law of large numbers (Mandelbaum and Pats, 1995), to immediately obtain the following for each x 2 C:

ðx; 0Þ þ ðx; tÞ ¼ q qn ðx; tÞn!1 !q

1 lx

Z 0

t

ðx 1; uÞÞdu kðy

Z

t

ðx; uÞÞdu kðy a:s:

ð37Þ

0

ðx; tÞ and, consequently, y ðx; tÞ are deterministic. The limiting process is a continuous time version of the Godunov where q scheme, which means that the ‘‘average’’ behavior is consistent with first-order macroscopic traffic flow theory. Not only is (37) consistent with the main result in Jabari and Liu (2012), but also strengthens the type of convergence. 4.2.3. The second order approximation As in Section 3, we are interested in the limiting behavior of the re-scaled process:

rn ðx; tÞ

5

pffiffiffi n ðx; tÞÞ nðq ðx; tÞ q

In the immediate past, in accordance with the measurability (or predictability) condition.

ð38Þ


23

Expanding (38), we have:

rn ðx; tÞ ¼

pffiffiffi n ðx; 0ÞÞ nðq ðx; 0Þ q pffiffiffi Z t Z t n 1 ðx 1; uÞÞdu N x1 n kðyn ðx 1; uÞÞdu kðy þ lx n 0 0 pffiffiffi Z t Z t n 1 ðx; uÞÞdu Nx n kðyn ðx; uÞÞdu kðy lx n 0 0

ð39aÞ ð39bÞ ð39cÞ

As discussed above, the first difference in (39a) converges in distribution to a zero mean Normal random variable with variance r2(x, 0). The limiting behavior of the second and third differences, (39b) and (39c), are obtained in the same manner. Take the following process:

Z t Z t pffiffiffi 1 ðx; uÞÞdu n Nx n kðyn ðx; uÞÞdu kðy n 0 0 Rt

Adding and subtracting

0

ð40Þ

kðyn ðx; uÞÞdu, (40) can be expressed as the sum of the two differences:

Z t Z t pffiffiffi 1 kðyn ðx; uÞÞdu kðyn ðx; uÞÞdu n Nx n n 0 0

ð41Þ

and

Z t Z t pffiffiffi ðx; uÞÞdu n kðyn ðx; uÞÞdu kðy 0

ð42Þ

0

For (41), we first note that since the flux functions k() are Lipshitz continuous, we immediately obtain the following convergence from the fluid limit:

Z

t

kðyn ðx; uÞÞdu !

n!1

0

Z

t

ðx; uÞÞdu a:s: kðy

ð43Þ

0

by the continuous mapping theorem (Whitt, 2002). Then, again by the continuous mapping theorem, combined with Donsker’s theorem, (18), we obtain the following for (41):6

Z t Z t Z t pffiffiffi 1 ðx; uÞÞdu kðyn ðx; uÞÞdu kðyn ðx; uÞÞdu ! W x kðy weakly; n Nx n n!1 n 0 0 0

ð44Þ

where the Brownian processes Wx() are independent in x. This is precisely the limit obtained for the non-homogeneous Poisson process in Section 3. The dependence on traffic states between cells is captured by the limit of (42), which we derive now. pffiffiffi ðx; tÞ þ ð1= nÞv n ðx; tÞ, in accordance with (38). Now, (42) can be written Let vn(x, t) [rn(x, t) rn(x + 1,t)]T; then yn ðx; tÞ ¼ y as:

Z

t

0

pffiffiffi ðx; uÞ þ ð1= nÞv n ðx; uÞ kðy ðx; uÞÞ k y pffiffiffi du; kv n ðx; uÞk ð1= nÞkv n ðx; uÞk

ð45Þ

where kvn(x, t)k is the ‘‘length’’ of the vector vn(x, t). The term inside the brackets in (45) converges to the directional derivðx; tÞÞ along v ~ ðx; tÞ, which is the limit of vn(x, t). Consequently, (45) converges to7: ative of kðy

Z

t

ðx; uÞÞÞdu ðv~ ðx; uÞT rkðy

ð46Þ

0

Expanding the product in (46), we get:

Z t ðx; uÞÞ ðx; uÞÞ @kðy @kðy ~r ðx; uÞ þ ~rðx þ 1; uÞ du; ðx; uÞ ðx þ 1; uÞ @q @q 0

ð47Þ

where ~rðx; tÞ and ~r ðx þ 1; tÞ are the (weak) limits of rn(x, t) and rn(x + 1,t), respectively. We have, thus, found that the deviation of our stochastic traffic flow model from its fluid limit, for cell x, is obtained as the solution of the following stochastic differential equation (written in integral form):

6 More accurately, this follows from a generalized version of the continuous mapping theorem (Whitt, 2002, Theorem 3.4.4) combined with the Skorohod representation theorem (Whitt, 2002, Theorem 3.2.2). 7 That the product of random variables converges to the product of their limits is established by appeal to Cramér’s theorem (Davidson, 1994), which is ðx; uÞÞ is deterministic. Convergence under the integral sign is allowed by the continuous mapping theorem. allowed since rkðy

24


Z ðx 1; uÞÞ ðx 1; uÞÞ 1 t @kðy @kðy ~r ðx 1; uÞ þ ~r ðx; uÞ du ðx 1; uÞ ðx; uÞ lx 0 @q @q Z t Z 1 @kðyðx; uÞÞ @kðyðx; uÞÞ 1 t pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx 1; uÞÞdW x1 ðuÞ ~r ðx; uÞ þ ~rðx þ 1; uÞ du þ kðy ðx; uÞ ðx þ 1; uÞ lx 0 @q @q lx 0 Z t pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðx; uÞÞdW x ðuÞ kðy lx 0

~rðx; tÞ ¼ ~rðx; 0Þ þ

ð48Þ

We note that the only stochastic terms in this equation are the ~rð; Þ’s and the standard Brownian motions Wx1() and Wx(). This stochastic differential equation (SDE) has a unique strong solution (see Oksendal (2007) for definition), which can be written explicitly and results in a linear Gaussian process. Moreover, we emphasize the following two merits of this derivation, which underscore the contribution of the proposed model: 1. The dependence of traffic densities in cell x on traffic densities in adjacent cells x 1 and x + 1 is preserved in our stochastic model; this constitutes a departure from assumptions of diagonal covariance matrices typically made in the traffic flow estimation literature (usually for tractability purposes). 2. By solving the SDE, explicit expressions are obtained for the covariance matrix of the system. The only parameters needed for this are those pertaining to the fundamental diagram. 4.2.4. Solution of the SDE Since for each cell we have dependence on adjacent cell traffic densities, we have a system of SDEs, which are to be solved simultaneously. We first write our system of equations in vector form and then present the solution. In vector form, we write (48) as follows:

~rðtÞ ¼ ~rð0Þ þ

Z

t

DðuÞ~rðuÞdu þ 0

Z

t

BCðuÞdWðuÞ;

ð49Þ

0

where ~rðtÞ and W(t) are, respectively, the jCj and jCj þ 1 dimensional vector valued processes ½~r ð1; tÞ ~rðjCj; tÞT and ½W 0 ðtÞ W jCj ðtÞT ; here, W0(t) represents flows into the first cell. jCj denotes the number of cells and is also used here to denote the index of the last cell. The jCj jCj matrix D(t) captures the dependence between cells (the second and third components on the right-hand side of (48)). For arbitrary cell x, the corresponding row in D(t) is:

1 lx

0

ðx 1; tÞÞ @kðy ðx 1; tÞ @q

ðx 1; tÞÞ @kðy ðx; tÞÞ @kðy ðx; tÞ ðx; tÞ @q @q

ðx; tÞÞ @kðy ðx þ 1; tÞ @q

0

;

ð50Þ

where the middle element lies in the xth column of the row. The constant jCj ðjCj þ 1Þ matrix B is given by

2

1=l1

6 0 6 6 6 B6 0 6 6 4 0

1=l1

0

0

0

0

1=l2

1=l2

0

0

0

0

1=l3

1=l3

0

0

0

0 .. . 1=ljCj

0

3 7 7 7 7 7 7 7 5

ð51Þ

1=ljCj

C(t) is the ðjCj þ 1Þ ðjCj þ 1Þ matrix of Ito^ integrands 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ð0; tÞÞ 0 0 kðy 0 6 7 .. 7; CðtÞ 6 4 5 . pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 0 kðyðjCj; tÞÞ

ð52Þ

ð0; tÞ can be taken to be q ð; Þ. This is both to ð1; tÞ. Note that in our vector notation, we have omitted dependence on y where y simplify our notation and to emphasize that these are deterministic (albeit time-varying) quantities. The vector stochastic integral Eq. (49) can also be written (symbolically) in differential form as:

d~rðtÞ ¼ DðtÞ~rðtÞdt þ BCðtÞdWðtÞ

ð53Þ

with the initial condition ~rð0Þ (a zero mean Gaussian random vector); this is a narrow sense linear SDE, since all coefficient matrices are (time varying) constant matrices that do not depend on ~rðtÞ. Eq. (53) is known to have a unique strong solution written explicitly as (see Arnold, 1974, Theorem 8.2.2):

Z t ~rðtÞ ¼ UðtÞ ~rð0Þ þ U1 ðuÞBCðuÞdWðuÞ ;

ð54Þ

0

^ integral with deterministic integrand; thus, ~rðtÞ is a Gaussian where U(t) is a fundamental matrix. The solution (54) is an Ito process.


25

~ ðtÞ its second-order approximation, and Let q(t) denote the jCj-dimensional vector of traffic densities, q ð0; tÞÞ kðy ðjCj; tÞÞT . We have, by reasoning in a similar way to (19), that: kðtÞ ½kðy D

qðtÞq~ ðtÞ ¼ q~ ð0Þ þ

Z

t

Z t BkðuÞdu þ UðtÞ ~rð0Þ þ U1 ðuÞBCðuÞdWðuÞ

0

ð55Þ

0

~ ðtÞ, which is characterized by it’s first two moments. It’s mean dynamic is given by Since ~rðtÞ is Gaussian, then so is q

~ ðtÞ ¼ q ð0Þ þ Eq

Z

t

BkðuÞdu;

ð56Þ

0

ð0Þ is the mean traffic density at time t = 0. Eq. (56) is simply the fluid limit written in vector form (a Godunov where q scheme in continuous time) and its covariance matrix is obtained by solving the following matrix differential equation:

dWðtÞ ¼ DðtÞWðtÞ þ WðtÞDðtÞT þ BCðtÞCðtÞT BT dt

ð57Þ

with initial covariance matrix W(0). Derivation of this equation is given in Appendix B. 4.3. A generalization Despite the fact that doubly stochastic Poisson processes generalize both time homogeneous and non-homogeneous Poisson processes with deterministic rates, the unit coefficient of variation may be too restrictive in traffic flow applications. For example, one could confront situations where traffic flow measurements are either under-dispersed or over-dispersed. We, thus, consider the generalization to time changed renewal processes. The technical machinery used to derive the second-order approximation generalizes to this case. Let c N x ðtÞ now denote a renewal process with i.i.d. time headways with mean 1 and coefficient of variation c. For this case, we also have that the scaled process converges almost surely to t (see Jabari and Liu, 2012, Appendix B):

1c N x ðntÞ ! t a:s: n!1 n

ð58Þ

Since this holds R almost surely, we obtain the same fluid limit as in (37) for the time changed counting processes t c N x 0 kðyðx; uÞÞdu . Furthermore, we have by the functional central limit theorem (FCLT) for counting processes (Chen and Yao, 2001, Theorem 5.11), that

pffiffiffi 1 c D n N x ðntÞ t cW x ðtÞ n

ð59Þ

In our derivation of the second-order approximation, this only impacts the limit given in (44). The new limit is:

cW x

Z

t

ðx; uÞÞdu kðy

ð60Þ

0

The new second-order approximation is the same as that derived in the previous section with one modification to the matrix C(t), which now becomes:

2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c kðy ð0; tÞÞ 0 0 6 .. CðtÞ 6 4 . 0

0

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðjCj; tÞÞ 0 0 c kðy

3 7 7 5

ð61Þ

The solution of the SDE remains the same Gaussian process. That is, with C(t) given by (61), the time varying means and covariances are obtained from (56) and (57). 5. Stationary behavior of the covariance matrix The stationary behavior of the covariance matrix gives an indication of what the covariance matrix tends to with time. This sheds light on whether or not the covariance grows without bound or is bounded. When the covariance matrix is bounded, the stationary behavior aids in designing numerical schemes for computing the covariance matrix; that is, it helps with choosing appropriate Dt. Moreover, boundedness of the covariance matrices is critical to estimation applications using Kalman filters. Boundedness ensures stochastic observability of the system. We discuss this further in Section 7. As discussed in Section 3, for the simple non-stationary Poisson model, variances grow without bound. This is not the case for the approximated state-dependent process developed in Section 4.2. To see this, we first point out that, for stationary mean traffic conditions, we have that the matrices D(t) = D and C(t) = C do not vary with time. Consequently, the fundamental matrix, U(t) can be written explicitly as U(t) = eDt and the solution of the stochastic differential equation for the deviation becomes:

26


~rðtÞ ¼ eDt ~rð0Þ þ

Z

t

eDðtuÞ BC dWðuÞ;

ð62Þ

0

for which the covariance matrix is: T

WðtÞ ¼ eDt Wð0ÞeD t þ

Z

t

T

eDðtuÞ BCCT BT eD

ðtuÞ

du:

ð63Þ

0

The behavior of the covariance matrix depends on the traffic conditions. We investigate the three cases: (i) free-flow traffic conditions (sub-critical mean traffic densities), (ii) capacity conditions (critical mean densities), and (iii) congested traffic conditions (super-critical mean traffic densities). 5.1. Free-flow mean traffic conditions ðx; tÞÞ < Re ðq ðx þ 1; tÞÞ, where q is the (subUnder free-flow traffic conditions, we have, for each cell boundary, that Se ðq critical) mean traffic density in the cells. Consequently, from (31) and (32), we have that

ðx; tÞÞ dSe ðq ðx; tÞÞ @kðy ¼ ðx; tÞ ðx; tÞ @q dq

ð64Þ

ðx; tÞÞ @kðy ¼0 ðx þ 1; tÞ @q

ð65Þ

and

Define

xx

ðx; tÞÞ 1 dSe ðq ðx; tÞ lx dq

We then see that D has the following structure:

2

x1

6 6 x2 6 6 D¼6 0 6 6 4 0

0

0

0

0

0

x2

0

0

0

0

x3

x3

0 .. .

0

0

0

0

0 xjCj

xjCj

3 7 7 7 7 7 7 7 5

ð66Þ

This can be written as the sum of two matrices D = D1 + D2, where D1 is a diagonal matrix and D2 is a nilpotent matrix.

t Assuming cell lengths are all equal,8 we see that D1D2 = D2D1 and, thus, eðD1 þD2 Þt ¼ eD1 eD2 . Since D2 is nilpotent, eD2 is a lower

triangular matrix with 1s along the diagonal, while eD1 is a diagonal matrix with diagonal elements exx . Consequently, eD1 eD2

t (tu)D has spectral radius strictly less than 1. Then as t ? 1, eD1 eD2 ! 0. Furthermore, for any u < t, e ? 0, and when u = t, e(tu)D = I. We have just established that:

WðtÞ ! BCCT BT dt; t!1

ð67Þ

which is bounded since B and C are bounded. 5.2. Capacity (critical) mean traffic condition ðx; tÞÞ ¼ Re ðq ðx þ 1; tÞÞ. Consequently, When mean traffic conditions are critical, we have that Se ðq

ðx; tÞÞ 1 dSe ðq ðx; tÞÞ @kðy ¼ ðx; tÞ ðx; tÞ @q 2 dq

ð68Þ

ðx; tÞÞ ðx þ 1; tÞÞ @kðy 1 dRe ðq ¼ ðx þ 1; tÞ 2 dq ðx þ 1; tÞ @q

ð69Þ

and

Assuming again equal cell lengths, let 8 This is easily generalized by considering upper and lower bounds established by substituting all cell lengths with the minimum and maximum cell lengths and seeing that both bounds converges to the same limit.

27


ðx 1; tÞÞ 1 dSe ðq ; ðx 1; tÞ 2lx dq ðx; tÞÞ dRe ðq ðx; tÞÞ 1 dSe ðq xm ; ðx; tÞ ðx; tÞ 2lx dq dq

xl

and

xr

ðx þ 1; tÞÞ 1 dRe ðq ðx þ 1; tÞ 2lx dq

Then, D has the following structure:

2

xm

xr 0 0 xm xr 0 xl xm xr

6 6 xl 6 6 D¼6 0 6 6 4 0

.. 0

0

0

0 0 0

.

xl

3 0 7 0 7 7 0 7 7 7 7 5

ð70Þ

xm

The matrix D in (70) is said to be of Toeplitz type, which is any tridiagonal matrix with equal elements along its diagonals. Let P be a matrix with its columns consisting of eigenvectors of the matrix D. It can be shown that (i) D has distinct eigenvalues, denoted fa1 ; . . . ; ajCj g; (ii) P diagonalizes D, i.e., D ¼ Pdiagða1 ; . . . ; ajCj ÞP1 ; and (iii) the eigenvalues are calculated as qffiffiffiffiffi jp (see, for instance Meyer (2000, Example 7.2.5) for detailed derivation). Consequently, aj ¼ xm þ 2xr xxrl cos jCjþ1

eD ¼ P diag ðea1 ; . . . ; eajCj ÞP1

ð71Þ

and

eDt ¼ Pdiag ea1 t ; . . . ; eajCj t P1

ð72Þ

jp jCjþ1

2 ð0; 1Þ for all From the definition of sending and receiving functions, we have that xm < 0, xr < 0, and cos j 2 f1; . . . ; jCjg. Hence, diag ðea1 t ; . . . ; eajCj t Þ ! 0, as t ? 1, then from (72), we have that eD t ? 0 as t ? 1. Likewise, for u < t, we have that, as t ? 1, e(tu)D ? 0 and when t = u, e(tu)D ? PIP1 = I. We have just established that the covariance matrix converges to the same limit as the free-flow case, (67), for the case of critical mean traffic densities. 5.3. Congested (super-critical) mean traffic conditions ðx; tÞÞ > Re ðq ðx þ 1; tÞÞ and, consequently, When mean traffic conditions are super-critical, we have that Se ðq

ðx; tÞÞ @kðy ¼0 ðx; tÞ @q

ð73Þ

ðx; tÞÞ ðx þ 1; tÞÞ @kðy dRe ðq ¼ ðx þ 1; tÞ ðx þ 1; tÞ @q dq

ð74Þ

and

Now, let

x

1 dRe ; lx dq

then D has the following structure:

3 x x 0 0 0 0 60 x x 0 0 0 7 7 6 7 6 0 0 x x 0 0 7 D¼6 7 6 2

6 4

..

0

0

0

0

.

7 5

ð75Þ

0 x

is super-critical, dRe =dq < 0, which implies that x < 0. Consequently, the limiting behavior of W(t) is also given by Since q (67), by the same argument given for the free-flow case.

28


5.4. Computing the covariance matrices Computing the mean dynamics is straight-forward: simply apply the Godunov scheme (or the CTM). To compute the time varying covariance matrix, one simply recursively computes a discrete time approximation of (57), for appropriate choice of Dt. We shall first take a look at the bounded behavior of the covariance matrix. 5.4.1. Boundedness The analyses in Sections 5.1–5.3 all suggest that the covariance matrix converges, for all traffic flow conditions, to BCCTT B dt. Furthermore, as the rate of convergence is proportional to the spectral radius, we see that the convergence rate is fast. This limiting behavior, thus, gives bounds on the covariance matrix. To see what these are, let’s take a look at the structure of the limiting covariance matrix. For arbitrary cell i at time instance t the elements of row i of the stationary covariance matrix are (j is the column index):

8 0 > > > c2 > > ðj; tÞÞdt kðy > > > lj li > < 2 c ði 1; tÞÞ þ kðy ði; tÞÞÞdt Wi;j ðtÞ ¼ ðkðy li > > > 2 > c > kðyði; tÞÞdt > > ll > > ij : 0

if j < i 1; if j ¼ i 1; if j ¼ i;

ð76Þ

if j ¼ i þ 1; if j > i þ 1;

where the middle element (j = i) corresponds to the variance of the traffic density in cell i at time t. Let lmin ¼ mini2C li denote the minimum cell length. Then the maximum variance is 2qmax/(lmin)2, where qmax is the capacity flow rate. Maximum variance occurs when the stationary mean traffic conditions are critical. The minimum variance is zero and occurs when either the stationary traffic densities are zero or at jam density. 5.4.2. Computation We now discuss numerical computation of the covariance matrix. Eq. (57) may be solved numerically, starting with W(0), by recursively computing:

Wðt þ DtÞ ¼ WðtÞ þ DðtÞWðtÞDt þ WðtÞDðtÞT Dt þ BCðtÞCðtÞT BT Dt

ð77Þ

In each discrete time interval, one computes the cell boundary fluxes (for the mean dynamics), which immediately give the matrix C(t) and allow for computation of the matrix D(t). The only issue that remains is the appropriate choice of the discrete time interval, Dt. We prescribe the smaller of the Dt prescribed by the Courant, Friedrichs, and Lewy (CFL) condition (LeVeque, 1992) and one that ensures with 95% confidence that the traffic density will be non-negative. We first note that the choice of Dt (provided it honors the CFL condition) has no bearing on the eventual traffic densities in the cells, only the number of time steps it takes to reach that traffic density. However, from (76), we see that the variance of the traffic density is impacted by the size of Dt, where

Z

tþDt

~ ðx; uÞ Eq ~ ðx; uÞj2 du ¼ Ejq

t

Z t

tþDt

2 2 c c ðx 1; uÞÞ þ kðy ðx; uÞÞÞdu ¼ ðx 1; tÞÞ þ kðy ðx; tÞÞÞDt ðkðy ðkðy lx lx

ð78Þ

and the last equality follows from the CFL condition. 2 9 For ease of notation, let’s denote this variance, which pffiffiffiffiffiis ffi the last term in (78), by r Dt and let q denote the stationary mean 1:96r Dt P 0 to ensure non-negative traffic densities with 95% confidence. It traffic density. One then chooses Dt so that q follows that

Dt 6

q 2 3:842r2

ð79Þ

Example 2. Suppose the flux functions are in accordance with a triangular fundamental relationship and c ¼ 1. For free-flow Þ 6 Re ðq Þ, for stationary mean traffic density q . From (78), we see that and capacity traffic conditions, we have that Se ðq

r2 ¼

2 ðlx Þ2

v f q

ð80Þ

and Dt is in accordance with

Dt 6

9

lx lx q 7:69 v f

Dependence on t is understood to be implicit.

ð81Þ


29

Þ > Re ðq Þ (in general, this would give a more The same calculation may be applied to congested traffic conditions with Se ðq relaxed bound). A rule of thumb that we propose for Dt is one-eight that prescribed by the upper bound of the CFL condition so that lx q is just one vehicle. This rule of (multiplied by c when c < 1). This was obtained by considering a traffic density q thumb does not guarantee non-negativity of traffic densities for all possible dynamics, but these violations are sufficiently short-lived that we shall not seek further improvement.

6. Numerical examples Consider a simple two cell setting, where both cells are of length 264 feet. Assume a triangular fundamental relationship (i.e., a CTM fluid model) with the following parameters: free-flow speed vf = 60 min/h, capacity qmax = 1800 veh/h, backward wave propagation speed w = 12 min/h, a jam density of qjam = 180 veh/min, and coefficient of variation of time headways of c ¼ 1. The simulation time horizon is U = 200 s. The boundary flow rates are in accordance with the following:

kðyð0; tÞÞ ¼ minfk0 ; 12ð180 qð1; tÞÞg;

ð82aÞ

kðyð1; tÞÞ ¼ minf60qð1; tÞ; 1800; 12ð180 qð2; tÞÞg;

ð82bÞ

kðyð2; tÞÞ ¼ minf60qð2; tÞ; 1800gðtÞg;

ð82cÞ

where k0 is a constant mean inflow rate into cell 1 and g(t) captures downstream capacity restrictions (e.g., a traffic signal). The mean traffic density is computed using the CTM and the variance is computed using (77) with Dt = 0.2 s (for both the mean and the covariance). The matrices B, C(t), and D(t) are computed according to 51, 52, and 50 as:

B¼

20 20 0 ; 0 20 20

ð83Þ

2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð0; tÞÞ kðy 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 CðtÞ ¼ 4 0 kðyð1; tÞÞ 0

0

0

3

7 5; 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2; tÞÞ kðy

ð84Þ

and

2

3 ð0;tÞÞ ð1;tÞÞ ð1;tÞÞ y @kðy @kðy 10 @kð 10 @ qð1;tÞ @ qð1;tÞ @ qð2;tÞ 6 7 5; DðtÞ ¼ 4 ð1;tÞÞ ð1;tÞÞ ð2;tÞÞ y @kðy @kðy 10 @kð 10 ð1;tÞ ð2;tÞ ð2;tÞ @q @q @q

ð85Þ

where all derivatives in (85) are to be interpreted in the weak sense given in Section 4.1. We shall illustrate this for the derivative in the first row, in the second column. We may re-write (82b) as:

ð1; tÞÞ ¼ minfSe ðq ð1; tÞÞ; Re ðq ð2; tÞÞg; kðy

ð86Þ

where

ð1; tÞÞ ¼ minf60q ð1; tÞ; 1800g; Se ð q ð2; tÞÞ ¼ minf1800; 20ð180 q ð2; tÞÞg; Re ðq

ð87aÞ ð87bÞ

This gives

8 ð2; tÞÞ; > 12 if 1800 > 12ð180 q ð2; tÞÞ < dRe ðq ð2; tÞÞ; ¼ 6 if 1800 ¼ 12ð180 q ð2; tÞ > dq : ð2; tÞÞ; 0 if 1800 < 12ð180 q

ð88Þ

and, consequently,

8 dR ðq ð2;tÞÞ e > > < dq ð2;tÞ

ð1; tÞÞ > Re ðq ð2; tÞÞ; if Se ðq ð1; tÞ; bÞ @kðy ¼ 1 dReðq ð2;tÞÞ if Se ðq ð1; tÞÞ ¼ Re ðq ð2; tÞÞ; 2 dqð2;tÞ ð2; tÞ > @q > : ð1; tÞÞ < Re ðq ð2; tÞÞ; 0 if Se ðq

ð89Þ

To illustrate the behavior of the covariance in the model, we look at five scenarios, in which the mean traffic densities converge to different stationary traffic densities: 1. Convergence to free-flow traffic conditions: k0 = 900 veh/h, g(t) = 1 for all t 2 [0, 200], and zero initial traffic densities and variances. 2. Convergence to critical traffic conditions: k0 = 1800 veh/h, g(t) = 1 for all t 2 [0, 200], and zero initial traffic densities and variances.

30


3. Convergence to congested traffic conditions: k0 = 1800 veh/h, g(t) = 0.5 for all t 2 [0, 200], and zero initial traffic densities and variances. 4. Convergence to zero traffic densities: k0 = 0 veh/h, g(t) = 1 for all t 2 [0, 200], and mean initial traffic densities of 105 veh/ min with standard deviations of 50 veh/min. 5. Convergence to jam traffic densities: k0 = 1800 veh/h, g(t) = 0 for all t 2 [0, 200], and zero initial traffic densities and variances. The traffic densities for the five scenarios along with 95% confidence intervals are shown in Fig. 2a–e, for cell 1; cell 2 behaves in a similar fashion. The figures illustrate both the stationary behavior of the variance of the traffic densities and the speed with which it adapts to changing mean densities. Notice that the variance converges to zero in Fig. 2d and e, corresponding to zero mean traffic density and mean jam density. This is interpreted as certainty of zero flows under these two traffic conditions, which indicates that the model will not predict negative traffic densities or traffic densities greater than jam density. In both these cases, the process degenerates and there is no uncertainty in the traffic dynamics. We now illustrate the transient behavior of the model by introducing a traffic light in the downstream boundary of cell 2, which turns red during the time interval [50, 70) and green during the remainder of the 200 s time period. Thus, g(t) = 0 if

(a)

(b)

(c)

(d)

(e) Fig. 2. Stationary traffic densities in cell 1 with 95% confidence intervals; (a) free-flow, (b) capacity flow, (c) congested, (d) zero traffic density, (e) jam traffic density.


(a)

31

(b)

Fig. 3. Traffic densities with 95% confidence intervals; (a) cell 1, (b) cell 2.

(a)

(b)

Fig. 4. Simulated traffic densities with 95% confidence intervals; (a) cell 1, (b) cell 2.

t 2 [50, 70) and g(t) = 1, otherwise. The behavior of the traffic densities are shown in Fig. 3 for both cells 1 and 2, along with 95% confidence intervals. Here, we shall assume capacity inflows: k0 = 1800 veh/h in order to create traffic congestion. It is also easy to simulate sample paths of the Gaussian process, which is done by simulating normal random variables for each of the time intervals using the computed means and variances. Two such sample paths are illustrated, for the traffic light scenario, in Fig. 4. As expected, we see in Fig. 4 that the two sample paths fall mostly within the computed 95% confidence intervals. A larger confidence interval (e.g., 99%) would entirely encapsulate the sample paths.

7. Traffic state estimation 7.1. State equations, measurement, and filtering Traffic state estimation is carried out using a discrete–continuous Kalman filter (Jazwinski, 1970). That is, we use a continuous time model of traffic state combined with discrete noisy measurements obtained from inductance loop detectors. This section specifies these ingredients. The ability to use a classical (as opposed to an extended) Kalman filter in our work stems from the linearity of our stochastic traffic flow model. This is due to dependence of the mean and the covariance on mean traffic states rather than the stochastic traffic states. The continuous time setting in which our models were derived offer the flexibility of using different computational time scales for the state equation and the measurement equation. That is, we do not require the availability of measurements at regular time intervals to run the filter. An important contribution of our model to the filtering problem is the ability to compute state covariance matrices using few parameters (parameters of the fundamental diagram and coefficient of variation of time headways) and capturing correlations between traffic variables in different cells. 7.1.1. State dynamics Our state dynamics include cell boundary flows and traffic densities. We shall use the second-order approximated e ðtÞ denote a vector of approximated cumulative cell boundary flows and dynamics to construct our state equation. Let Q

32


e ðtÞ, which is a diagonal matrix ^ gðtÞ ¼ ½q~ ðtÞT Qe ðtÞT T the traffic state vector. Further, let WðtÞ denote the covariance matrix of Q computed in a similar way to W(t). That is, for cell x, start with the centered difference

Z t Z t pffiffiffi 1 c ðx; uÞÞdu Nx n n kðyn ðx; uÞÞdu kðy n 0 0

on ðx; tÞ ¼

ð90Þ

and following the same derivation of Section 4.2, this converges to the solution of the following SDE:

~ðx; tÞ ¼ o

Z t 0

Z t pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx; uÞÞ ðx; uÞÞ @kðy @kðy c kðy ðx; uÞÞdW x ðuÞ; ~ðx; uÞ þ ~ðx þ 1; uÞ du þ o o ðx; uÞ ðx þ 1; uÞ @q @q 0

ð91Þ

~ðx; tÞ is the weak limit of on(x, t). The system of equations is written (symbolically) in vector form as: where o

b o ~ ðtÞ ¼ DðtÞ ~ ðtÞdt þ CðtÞdWðtÞ; do

ð92Þ

b ~ ðtÞ is a vector of dimension jCj þ 1. C(t) and W(t) retain their meaning given in Section 4.2. DðtÞ where o is the following ðjCj þ 1Þ ðjCj þ 1Þ matrix:

2

0 6 60 6 b DðtÞ 6 6 6 4 0

ð0;tÞÞ @kðy ð1;tÞ @q

0

0

0



0

0

.. 0

0

. ðjCj;tÞÞ @kðy ðjCj;tÞ @q

0

3 7 7 7 7 7 7 5

ð93Þ

Then, the mean dynamics are given by

e ðtÞ ¼ EQ

Z

t

ð94Þ

kðuÞdu; 0

and the time varying covariance matrix (the covariance function) is obtained by solving:

b d WðtÞ b WðtÞ b T þ CðtÞCðtÞT b b DðtÞ ¼ DðtÞ þ WðtÞ dt

ð95Þ

e ðtÞ E Q e ðtÞÞT Þ denote the covariance of q e ðtÞ. Following the same steps, we find that ~ ðtÞ Eq ~ ðtÞÞð Q ~ ðtÞ and Q Let HðtÞ ¼ Eððq H(t) is obtained by solving:

dHðtÞ b T þ BCðtÞCðtÞT : ¼ DðtÞHðtÞ þ HðtÞ DðtÞ dt

ð96Þ

Finally, the state vector, a Gaussian vector, is characterized as follows:

"

gðtÞ ¼

q~ ðtÞ e ðtÞ Q

"

# ;

EgðtÞ ¼

q ð0Þ þ Rt 0

Rt 0

BkðuÞdu

kðuÞdu

"

# ;

RðtÞ ¼

WðtÞ HðtÞT

HðtÞ b WðtÞ

# ð97Þ

7.1.2. Measurement equation Consider a countable sequence of noisy measurements of traffic flows (and/or traffic densities) at discrete time instants t1, t2, . . . , tu 2 [0, U]. Without loss of generality, we shall only consider traffic count (flow rate) measurements. Let z(tj) be a vector of such measurements made at time tj taken at point sensor locations that constitute a subset of cell boundaries; that is, for each tj, z(tj) is a vector of dimension jMj 6 jCj. To characterize the measurement sequence {z(ti)}, let H be an jMj jCj incidence matrix with a 1 in row i column j if the ith element in the measurement vector represents a measurement of the jth state variable, and a 0 in row i column j otherwise. Here, we shall assume that measurement error can be represented by Gaussian jMj-dimensional noise vectors {f(ti)} with zero mean and covariance matrices {N(ti)}. The true nature of measurement errors vary by sensor type, time of day, weather conditions, controller sensitivity settings, and problematic vehicle dynamics (such as lane changing). A thorough investigation of measurement error, while enlightening, is beyond the scope of this study and our assumption of Gaussian noise is in accord with the convention in most of the literature on traffic state estimation. Thus, our measurement equation is written as:

zðti Þ ¼ Hgðt i Þ þ fðti Þ;

i 2 f1; 2; . . . ; ug

ð98Þ


33

Eqs. (97) and (98) are all that is needed to run the Kalman filtering algorithm. For the sake of completeness we include the algorithm below. Algorithm 5.1. Discrete–continuous Kalman filter 1: Initialization: ~ ð0Þ Eq q ð0Þ 2: Egð0Þ ¼ ¼ e ð0Þ 0 EQ Wð0Þ Hð0Þ 3: Rð0Þ ¼ b Hð0ÞT Wð0Þ 4: Iteration: 5: for i = 1 ? u 6: Predict: 7: 8: 9: 10: 11: 12: 13: 14: 15:

Predict mean Egðt i jti1 Þ :¼

"

q ðti1 Þ þ R ti

R ti

t i1

BkðuÞdu

#

t i1 kðuÞdu Wðt i jti1 Þ Hðti jt i1 Þ Predict covariance Rðti jti1 Þ :¼ T b Hðti jt i1 Þ Wðti jt i1 Þ Update: Measurement residual mðt i Þ :¼ zðt i Þ H Egðti jt i1 Þ Residual covariance P(ti) :¼ HR(tijti1)HT + N(ti) Kalman gain K(ti) :¼ R(tijti1) HTP(ti)1 Updated (posterior) mean Egðti jti Þ :¼ Egðti jt i1 Þ þ Kðti Þpðti Þ Updated (posterior) covariance Rðti jt i Þ :¼ ðI Kðti ÞHÞRðt i jt i1 Þ end for

7.2. Observability A system is said to be observable if the available measurement sequence, Z(t), is adequate to construct the initial conditions of the (mean) state vector, Egð0Þ (Jazwinski, 1970; Stengel, 1994). Since the proposed research involves dividing a roadway into (potentially many) cells, observability becomes an important question, as this discretization results in larger numbers of state variables and increases the sparsity of the measurements.

Fig. 5. Characteristic lines of an unobservable scenario; (a) larger queue; (b) smaller queue.

34


Consider the scenario where a road section is instrumented at either end, so that traffic densities at the upstream and downstream ends of the section and flow rates into the section and out of the section are observed. In the literature, several articles have noted that scenarios where, at time t = 0, the traffic density at the upstream end of the road section is sub-critical, while the traffic density at the downstream end is super-critical, result in unobservable traffic densities within the road section (Stanková and De Schutter, 2010; Sun et al., 2004; Muñoz et al., 2006, 2003). These unobservable scenarios occur, for instance, when a rarefaction fan meets a shockwave within the road section, creating a wave front with varying slope. Such scenarios occur frequently along congested signalized arterials, where queues can start to build-up in the downstream (creating a shockwave that propagates upstream) before a queue in the upstream from a previous cycle has fully dissipated. The characteristic lines of such scenarios are illustrated in Fig. 5. While in both Fig. 5a and b traffic conditions throughout the road section at time tv are the same and despite seeing the same traffic conditions at positions xl and xr, the initial traffic conditions (at time t0) are different. In Fig. 5a, the shockwave meets the rarefaction fan farther upstream at time t = 0 than in Fig. 5b. The position ^ xð0Þ represents the back of a queue, which cannot be observed using sensors at positions xl and xr. In general, the ability to observe traffic conditions within the road section using only observations at its ends is only possible when there is a single wavefront/rarefaction fan within the road section at time t = 0. As an example, suppose, the observed traffic densities at the upstream and downstream positions, denoted, respectively, by ql and qr, are such that ql < qr < qcrit, where qcrit denotes the critical traffic density as depicted in Fig. 1. Then there is a shockwave that was initiated at a position ^ xð0Þ at time t = 0 and has a speed of vs = (Qe(ql) Qe(qr))/(ql qr). If the system is observable then at some time tv, the shockwave will be detected in the downstream (position xr). In essence, this is the time that the traffic density at xr changes from qr to ql. With tv and vs known, ^ xð0Þ can be calculated. This is illustrated in Fig. 6a. The arrows in Fig. 6a and b indicate how the position of the shockwave at time 0, ^ xð0Þ, indicated by the solid dots in the figure, are determined from the position of the shockwave at a later time. The directions of the arrows are not to be interpreted as the direction of the shockwave. In both scenarios, the initial traffic densities throughout the road section are those observed at the ends; then, estimating the initial traffic densities is a question of determining how far downstream of the entrance of the road section are the traffic densities equal to those at the entrance. The same is true for the case where qcrit < ql < qr, except now the wave is detected at the upstream end of the road section (i.e., position xr), which is depicted in Fig. 6b. In both situations, using a larger number of cells results in a more accurate estimate of the position ^ xð0Þ and, hence, more accurate initial traffic densities. It is important to note that the scenarios depicted in Fig. 6 are ideal and not likely to arise in the real world, even under free-flow conditions. In general, queue build-up and dissipation dynamics as described by continuum traffic flow theory are not observable in the classical sense (i.e., reconstruction of initial conditions). To overcome this issue in this research, the

Fig. 6. Example shockwave observability; (a) ql < qr < qcrit; (b) qcrit < ql < qr.


35

start time of the estimation period is chosen such that free-flow traffic conditions prevail. Then, at some finite time in the future, traffic conditions observed at the upstream end of the road section will have propagated through the road section, while vehicles that were present within the section at time t = 0 will have left. In essence, this constitutes a warm-up period for the estimation process. If estimation of peak period is desired, the start time of the warm-up period should be chosen well before the onset of traffic congestion. Observability, thus far, has been discussed in terms of the mean dynamics. This is the classical way defining of observability. For stochastic systems, a more relevant issue is stochastic observability, which is defined in terms of the behavior of the covariance matrix. A system is said to be stochastically observable if there exists a time, to < 1 such that the largest element of the estimated covariance matrix R(tijti) is bounded for all ti P t0 (see Bageshwar et al. (2009) and references b therein). The boundedness of W() was established in Section 5. Following the same steps, it can be shown that WðÞ and H() are also bounded, since these two matrices are very similar in structure to W(). Since the residual covariance matrix, P(ti), and the Kalman gain matrix, K(ti), are both calculated directly from the predicted covariance matrix, R(tijti1), which is bounded, the estimated covariance matrix, R(tijti), is also bounded (see step 14 of Algorithm 5.1). This establishes stochastic observability of the system.

8. Model testing and validation

Rhode Island Ave.

In this section, validation of the proposed model is carried out using real-world traffic data. In this test, maximum queue sizes are estimated along the westbound direction of Trunk Highway 55, a high-speed signalized arterial in Minnesota. The maximum queues sizes were estimated on a cycle by cycle basis for the (actuated) signalized intersection of Rhode Island Avenue (the west-most intersection in Fig. 7) and during the morning peak period of 7:00–9:00 AM on December 10th, 2008. Inductance loop detector data and signal timing information were obtained using the SMART-Signal system described in Liu and Ma (2009). A 30 min warm-up period starting at 6:30 AM was used, where initial traffic densities were assumed to be zero throughout the study road section. Data that was used for estimation consists primarily of source and sink counts and advanced loop detector counts. Queues typically build up beyond the locations of the advanced detector and thus cannot be observed using the loop detector counts. For this reason, a manual queue size data collection effort was carried out by a Minneapolis-based transportation consulting

North

Advanced loop detectors 400 ft. from stop bar

Source

TH 55 1776.7 ft (541 .5 m)

Source /sink

2643.7 ft (805 .8 m)

Fig. 7. Data collection site and detector locations.

Fig. 8. Fitted flow-density relationship.

36

S.E. Jabari, H.X. Liu / Transportation Research Part B 47 (2013) 15–41 Table 1 Fitted flow-density relation parameters.

vf

qmax

qjam

40 min/h (64.37 km/h)

1800 veh/h/ln

174 veh/min (108.12 veh/km)

Cumulative Probability (%)

99 95 90 80 70 60 50 40 30 20 10 5 1 0.1

1.0

10.0

100.0

Time Headway (sec) Fig. 9. Probability plot of free-flow time headways with 95% confidence intervals.

firm, Alliant Engineering Inc., during the morning peak period on December 10th, 2008 (Liu et al., 2009). This data set was used for comparison with the model estimates. 8.1. Model parameters Model parameters were fitted prior to estimation. The model parameters include fundamental diagram parameters and the coefficient of variation of time headways. For the fundamental diagram, a triangular mean relationship was assumed. The parameters are the free-flow speed vf, the capacity qmax, and the jam density qjam. These parameters were fitted using a minimum least squares estimate. The results are illustrated in Fig. 8, and summarized in Table 1. The dataset used to fit the fundamental diagram parameters consists of individual vehicle arrival times to the detector station and detector occupation times. Flow rates and occupancies in the scatter-plot in Fig. 8 were averaged over 10-vehicle batches as follows: ðobsÞ

ki

¼

10 ðobsÞ

Ti

ð99Þ

;

ðobsÞ

ðobsÞ

where ki is the observed flow rate for the ith 10-vehicle batch and T i is the difference between the departure time of the first vehicle in the batch and the last vehicle in the batch. Observed occupancies were computed as: ðobsÞ

Oi

ðobsÞ

¼

10 1 X ðobsÞ Oj ; ðobsÞ

Ti

ð100Þ

j¼1 ðobsÞ

where Oj is the occupancy time of the jth vehicle in the batch and Oi is the average occupancy for the ith 10-vehicle batch (i.e., the percentage of time the detector is occupied). The reason for aggregating was to average out the effect of unreasonable time headways that are due to measurement error (e.g., over-counting or under-counting when vehicles change lanes at the detector locations). The choice of ten as the batch size was arbitrary. TH-55 is high-speed arterial with a speed limit of 50 mph; the small fitted value of 40 min/h is a result of lower speeds due to snow accumulation on the road on the day the data was collected. Note that the heavy scatter seen in Fig. 8 is characteristic of traffic flow data that is averaged over short time intervals. Consequently, choosing a different shape can only provide slight improvements to the overall fit and only negligible improvements to the queue size estimates. To fit the coefficient of variation parameter, c, sets of consecutive vehicle time headways were collected and binned according to their associated vehicle occupancy times (to check dependency of time headways on traffic density). Probability plots were developed using the statistical software package Minitab in order to determine the appropriate time headway distributions. Fig. 9 is a probability plot for time headways associated with small occupancies (free-flow traffic conditions)

37

S.E. Jabari, H.X. Liu / Transportation Research Part B 47 (2013) 15–41 Table 2 Fitted free-flow headway parameters. Fitted parameters

Goodness of fit

Location

Scale

Shift

Anderson–Darling

0.8971

0.9069

4.001

0.626

99.9

Cumulative Probability (%)

99 95 90 80 70 60 50 40 30 20 10 5 1 0.1 0.1

1.0

10.0

Time Headway (sec) Fig. 10. Probability plot of congested time headways with 95% confidence intervals.

Table 3 Fitted congested headway parameters. Fitted parameters

Goodness of fit

Location

Scale

Shift

Anderson–Darling

0.4772

0.5624

0.9723

0.348

fitted to a 3-parameter log-normal distribution with c ¼ 0:593. The fitted distribution parameters are given in Table 2 along with the Anderson–Darling goodness of fit statistic.10 Fig. 10 is a probability plot for time headways associated with large occupancies (congested traffic conditions) also fitted to a 3-parameter log-normal with c ¼ 0:544 and Table 3 summarizes the fitted parameters. In general, the 3-parameter log-normal distribution was found to give good fits for a variety of time headway datasets; the data points mostly fall within the 95% confidence region as illustrated in the probability plots. The Anderson– Darling statistics in both cases indicate a good fit as well.11 Despite requiring different distribution parameters, a coefficient of variation of 0.5–0.6 seemed reasonable for a variety of traffic conditions.

8.2. Estimated cycle-by-cycle maximum queue sizes With the parameters of the fundamental diagram and the coefficient of variation estimated, the traffic state dynamics (97) are fully characterized. For the measurement equation, it was assumed that measurement errors are uncorrelated and assumed a standard deviation of 5% of the measured value to describe the measurement covariance. This value is based on studies carried out in Minnesota and other parts of the US, which indicate that loop detectors, when operational, provide accurate counts and speeds (ranging 1–9% depending on facility type) (Martin et al. (2003)). The two links were divided into 90 cells of length 50 ft (15.24 m), except boundary cells to ensure a correct total length. The Kalman filtering algorithm presented in Section 7.1 was then used to estimate cell densities and state covariance matrices. Queue sizes are not state parameters, so the estimated queues were computed from estimated cell densities. Cells with average speeds less than or equal to 5 min/h (8.05 km/h) were considered to be part of the queue and the estimated queue size was simply treated as the average density in consecutive cells that meet the criterion multiplied by the total length of 10 The Anderson–Darling statistic provides a measure of the difference between the hypothesized distribution and the empirical distribution of the data while assigning relatively higher weights to the tails. 11 That is, we cannot reject the hypotheses that the data points are distributed according to 3-parameter log-normals.

38


(a)

(b) Fig. 11. Comparison between estimated and measured queue sizes; (a) with 5 min/h criterion (b) with 5 ± 2 min/h (8.05 ± 3.22 km/h) criterion.

the cells. The 5 min/h criterion was established by the data collection team (Alliant Engineering Inc.) as a low speed, at and below which vehicles were considered to be part of the queue and were otherwise considered to be coasting freely. The queue size is, hence, an average over random variables with known means and covariance matrix, so that a corresponding mean queue size and variance can be readily calculated. A comparison between the estimated queue sizes along with a 95% confidence interval and the measured queue sizes is shown in Fig. 11a. 8.3. Discussion The results shown in Fig. 11a reveal a very good match between estimated and observed queue sizes. All except two observed queue sizes fall within the confidence region. For the queue in cycle no. 10, the model seems to have trapped vehicles in the upstream intersection, which is revealed by the larger estimated queue size in cycle no. 11. The discrepancy in cycle no. 17 however cannot be explained in the same way; here, observed measurement error is a plausible cause. Observers were instructed to follow the 5 min/h (8.05 km/h) criterion, but did not possess an accurate speed measurement apparatus. If the criterion is changed slightly, by ±2 min/h (±3.22 km/h), in the estimated queue calculation, the discrepancy disappears as shown in Fig. 11b. That is, the criterion was allowed to vary between 3 and 7 min/h between cycles. Allowing for possibly larger error (e.g. ±3 min/h) could also take care of the discrepancy in cycle No. 10 in the sense that the observations would fall within the confidence interval, but the qualitative difference between estimated and observed maximum queue sizes in cycles 10 and 11 (in terms of mean) would remain. An interesting exercise is estimation of both traffic states and parameters simultaneously via an adaptive filtering algorithm. This would shed light on whether the proposed estimation framework would be capable of detecting the drop in estimated free-flow speed due to snowy conditions. Since the purpose of this validation exercise was to test the Gaussian model, not the filtering algorithm, this was not carried out as part of the present research, but should be considered in any future research concerned with the estimation problem itself. 9. Conclusion and future research This paper, a sequel to Jabari and Liu (2012), proposes a second-order Gaussian approximation of the Markovian queueing model that was presented in the prequel. The proposed Gaussian model serves as a tractable stochastic representation of first-order traffic dynamics described by finite difference solutions to the LWR model of traffic flow. Differentiability of


39

disjunctive flux functions and their Lipschitz continuity are analyzed, which provide the necessary mathematical properties for deriving the Gaussian approximation. Linearity of the model allows for using a standard Kalman filter for estimating traffic conditions. As part of the derivation, a recipe is given for computing potentially large system covariance matrices using few parameters and allowing for dependency between traffic variables (namely, traffic densities) in adjacent cells. The Gaussian model was tested in a real-world setting using high-resolution traffic data from a high-speed actuated signalized arterial in Minnesota, where maximum cycle-by-cycle queue sizes were estimated using a standard Kalman filter. The results show good agreement between observed and estimated queue sizes. Future research could be carried out in various directions. From an application standpoint, the proposed model could be used in a variety of traffic control contexts, including ramp metering, adaptive signal control, and real-time traveler information systems. The proposed model can, in effect, be used in any application that uses the CTM. The probabilistic nature of the model lends itself to a variety of extensions that incorporate measures of uncertainty, reliability, and robustness, all measures that are usually established on probabilistic grounds. Another direction for future research is the investigation of stochastic models of traffic flow in general network settings. As in deterministic models, this would be accomplished by first investigating merge and diverge dynamics. Applications of the latter include real-time estimation of turning counts at network junctions. A stochastic model which includes merge and diverge dynamics could also be used to develop stochastic dynamic traffic assignment models, which allows for incorporating both the detailed dynamics and travel time reliability measures in the investigation of route choice dynamics at the network level. As another example of future research, a more thorough investigation into observability could shed light on questions related to sensor placement along road networks and how mobile sensors could be used to overcome observability issues. As a parallel line of research, issues related to controllability and stochastic controllability using the proposed model, in a variety of traffic management applications can be pursued; these include ramp metering and adaptive signal timing. From a theoretical standpoint, stochastic conservation laws of traffic flow could be developed as continuous time and continuous space models (stochastic partial differential equations) to investigate wave propagation dynamics in a probabilistic context. Such models could have the potential to unify the various macroscopic traffic flow phenomena currently modeled using disparate deterministic theories, such as stop-and-go waves and traffic hysteresis. These different phenomena would arise as sample paths of the unifying stochastic framework and may be studied probabilistically. Appendix A. Lipschitz continuity of flux functions A.1. Lipschitz continuity To see that (25) is Lipschitz, first note that k:H1 ? H2 is a map between two metric spaces (H1, d1) and (H2, d2) where H1 R2þ and H2 Rþ , both bounded, d1([a1 a2]T, [b1 b2]T) = max{ja1 b1j, ja2 b2j}, and d2(k1, k2) = jk1 k2j. Then, for k to be Lipschitz continuous, a constant 0 6 K < 1 must exist such that the following holds:

d2 ðkð½a1 a2 T Þ; kð½b1 b2 T ÞÞ 6 Kd1 ð½a1 a2 T ; ½b1 b2 T Þ T

ðA:1Þ

T

for all [a1 a2] ,[b1 b2] 2 H1. We proceed as follows:

d2 ðkð½a1 a2 T Þ; kð½b1 b2 T Þ ¼ j minfSe ða1 Þ; Re ða2 Þg minfSe ðb1 Þ; Re ðb2 Þgj 1 jðSe ða1 Þ þ Re ða2 ÞÞ jSe ða1 Þ Re ða2 Þj ðSe ðb1 Þ þ Re ðb2 ÞÞ þ jSe ðb1 Þ Re ðb2 Þjj 2 1 1 1 6 jSe ða1 Þ Se ðb1 Þj þ jRe ða2 Þ Re ðb2 Þj þ jjSe ðb1 Þ Re ðb2 Þj jSe ða1 Þ Re ða2 Þjj 2 2 2 1 1 1 6 jSe ða1 Þ Se ðb1 Þj þ jRe ða2 Þ Re ðb2 Þj þ jSe ðb1 Þ Re ðb2 Þ Se ða1 Þ þ Re ða2 Þj 2 2 2 1 1 1 1 6 jSe ða1 Þ Se ðb1 Þj þ jRe ða2 Þ Re ðb2 Þj þ jSe ðb1 Þ Se ða1 Þj þ jRe ða2 Þ Re ðb2 Þj 2 2 2 2 ¼ jSe ða1 Þ Se ðb1 Þj þ jRe ða2 Þ Re ðb2 Þj ¼

ðA:2aÞ

where the second line in (A.2a) relies on (26) and all bounds in (A.2a) follow from the triangle inequality. Now, from concavity of the sending and receiving functions (linear approximation over-estimates a concave function), we have that

jSe ða1 Þ Se ðb1 Þj þ jRe ða2 Þ Re ðb2 Þj 6 max

dSe ða1 Þ dSe ðb1 Þ dRe ða2 Þ dRe ðb2 Þ ja1 b1 j þ max ja2 b2 j ; ; da1 db1 da2 db2

6 2v f maxfja1 b1 j; ja2 b2 jg;

ðA:3Þ

where vf denotes the free-flow speed (i.e., the largest slope in the fundamental diagram). This proves the Lipschitz continuity of the flux function given in (25). We note that this is easily extended to any flux functions (possibly non-concave) where the sending and receiving functions are Lipschitz continuous; here one simply replaces vf with the maximum of the two Lipschitz constants.

40


Appendix B. Derivation of covariance matrix differential equation We begin with SDE

d~rðtÞ ¼ DðtÞ~rðtÞdt þ BCðtÞdWðtÞ

ðB:1Þ

with the initial condition ~rð0Þ. The solution is written as:

Z t ~rðtÞ ¼ UðtÞ ~rð0Þ þ U1 ðuÞBCðuÞdWðuÞ ;

ðB:2Þ

0

where U() is a jCj jCj fundamental matrix. That is, U() solves the matrix differential equation:

dUðtÞ ¼ DðtÞUðtÞ; dt

Uð0Þ ¼ I

ðB:3Þ

where I is a jCj jCj identity matrix From (B.2), we have the traffic density process

q~ ðtÞ ¼ q~ ð0Þ þ

Z

t

Z t BkðuÞdu þ UðtÞ ~rð0Þ þ U1 ðuÞBCðuÞdWðuÞ

0

ðB:4Þ

0

with mean dynamic

~ ðtÞ ¼ q ð0Þ þ Eq

Z

t

BkðuÞdu

ðB:5Þ

0

The covariance matrix, denoted by W(t), is

e ðtÞ E q e ðtÞÞð q e ðtÞ E q e ðtÞÞT Þ ¼ Eð~rðtÞ~rðtÞT Þ WðtÞ ¼ Eðð q Z t ¼ UðtÞ Wð0Þ þ U1 ðuÞBCðuÞCðuÞT BT U1 ðuÞT du UðtÞT

ðB:6Þ

0

Taking the first order derivative of W(t) with respect to time and using (B.3), we see that W(t) is obtained by solving the linear matrix differential equation

dWðtÞ ¼ DðtÞWðtÞ þ WðtÞDðtÞT þ BCðtÞCðtÞT BT dt

ðB:7Þ

with initial covariance matrix W(0) is given by the variances of the initial cell traffic densities. References Arnold, L., 1974. Stochastic Differential Equations: Theory and Applications. Wiley, New York. Bageshwar, V., Gebre-Egziabher, D., Garrarc, W., Georgiou, T., 2009. Stochastic observability test for discrete-time Kalman filters. Journal of Guidance, Control, and Dynamics 32 (4), 1356–1370. Billingsley, P., 1999. Convergence of Probability Measures, second ed. Wiley, New York. Blandin, S., Couque, A., Bayen, A., Work, D., 2012. On sequential data assimilation for scalar macroscopic traffic flow models. Physica D: Nonlinear Phenomena 241 (17), 1421–1440. Boel, R., Mihaylova, L., 2006. A compositional stochastic model for real-time freeway traffic simulation. Transportation Research Part B 40 (4), 319–334. Chen, H., Yao, D., 2001. Fundamentals of Queuing Networks. Springer, New York. Daganzo, C., 1994. The cell transmission model: a dynamic representation of highway traffic consistent with the hydrodynamic theory. Transportation Research Part B 28 (4), 269–287. Daganzo, C., 1995a. A finite difference approximation of the kinematic wave model of traffic flow. Transportation Research Part B 29 (4), 261–276. Daganzo, C., 1995b. The cell transmission model – Part II: Network traffic. Transportation Research Part B 29 (2), 79–93. Daley, D., Vere-Jones, D., 2003. An introduction to the theory of point processes, . Elementary Theory and Methods, second ed., vol. I. Springer, New York. Daley, D., Vere-Jones, D., 2008. An introduction to the theory of point processes, . General Theory and Structure, second ed., vol. II. Springer, New York. Davidson, J., 1994. Stochastic Limit Theory: An Introduction for Econometricians. Oxford University Press, Oxford. Davis, G., Kang, J., 1994. Estimating destination-specific traffic densities on urban freeways for advanced traffic management. Transportation Research Record: Journal of the Transportation Research Board 1457, 143–148. Del Castillo, J., Benitez, F., 1995. On the functional form of the speed–density relationship. I: General theory, II: Empirical investigation.. Transportation Research Part B 29 (5), 373–406. Di, X., Liu, H., Davis, G., 2010. Hybrid extended Kalman filtering approach for traffic density estimation along signalized arterials. Transportation Research Record: Journal of the Transportation Research Board 2188, 165–173. Gazis, D., Knapp, C., 1971. On-line estimation of traffic densities from time-series of flow and speed data. Transportation Science 5 (3), 283–301. Gazis, D., Liu, C., 2003. Kalman filtering estimation of traffic counts for two network links in tandem. Transportation Research Part B 37 (8), 737–745. Godunov, S., 1959. A difference scheme for numerical computation of discontinuous solutions of hydrodynamic equations. Matematicheskii Sbornik 47 (3), 271–306. Gray, L., Griffeath, D., 2001. The ergodic theory of traffic jams. Journal of Statistical Physics 105 (3), 413–452. Howison, S., 2005. Practical Applied Mathematics: Modelling, Analysis, Approximation. Cambridge University Press, Cambridge. Jabari, S., Liu, H., 2012. A stochastic model of traffic flow: theoretical foundations. Transportation Research Part B 46 (1), 156–174. Jazwinski, A., 1970. Stochastic Processes and Filtering Theory. Academic Press, New York.


41

Kang, J., 1995. Estimation of Destination-Specific Traffic Densities and Identification of Parameters on Urban Freeways Using Markov Models of Traffic Flow. Ph.D. Thesis, University of Minnesota. Karmeshu, Pathria, R., 1981. A stochastic model for highway traffic. Transportation Research Part B 15 (4), 285–294. Khoshyaran, M., Lebacque, J., 2009. A stochastic macroscopic traffic model devoid of diffusion. In: Appert-Rolland, C., Chevoir, F., Gondret, P., Lassarre, S., Lebacque, J., Schreckenberg, M. (Eds.), Traffic and Granular Flow ’07. Springer, New York, pp. 139–150. Lebacque, J., 1996. The Godunov scheme and what it means for first order traffic flow models. In: Lesort, J.B. (Ed.), Proceedings of the 13th International Symposium on Transportation and Traffic Theory. Elsevier, Lyon, France, pp. 647–677. LeVeque, R., 1992. Numerical Methods for Conservation Laws, second ed. Birkhäuser, Berlin. Lighthill, M., Whitham, G., 1955. On kinematic waves. I: Flood movement in long rivers, II: A theory of traffic flow on long crowded roads. Proceedings of the Royal Society (London) A229, 281–345. Liu, H., Ma, W., 2009. A virtual vehicle probe model for time-dependent travel time estimation on signalized arterials. Transportation Research Part C 17 (1), 11–26. Liu, H., Wu, X., Ma, W., Hu, H., 2009. Real-time queue length estimation for congested signalized intersections. Transportation Research Part C 17 (4), 412– 427. Mandelbaum, A., Massey, W., 1995. Strong approximations for time-dependent queues. Mathematics of Operations Research 20 (1), 33–64. Mandelbaum, A., Pats, G., 1995. State-dependent queues: approximations and applications. In: Kelly, F., Williams, R. (Eds.), IMA Volumes in Mathematics and Its Applications. Springer, Berlin, pp. 239–282. Martin, P., Feng, Y., Wang, X., 2003. Detector Technology Evaluation. Mountain-Plains Consortium.. Masey, W., 1981. Non-Stationary Queues. Ph.D. Thesis, Stanford University. Massey, W., 1985. Asymptotic analysis of the time dependent M/M/1 queue. Mathematics of Operations Research 10 (2), 305–327. Meyer, C., 2000. Matrix Analysis and Applied Linear Algebra. SIAM, Philadelphia. Muñoz, L., Sun, X., Horowitz, R., Alvarez, L., 2003. Traffic density estimation with the cell transmission model. In: Proceedings of the 2003 American Control Conference, pp. 3750–3755. Muñoz, L., Sun, X., Horowitz, R., Alvarez, L., 2006. Piecewise-linearized cell transmission model and parameter calibration methodology. Transportation Research Record: Journal of the Transportation Research Board 1965, 183–191. Nagel, K., Schreckenberg, M., 1992. A cellular automaton model for freeway traffic. Journal de Physique I 2 (12), 2221–2229. Newell, G., 1968a. Queues with time-dependent arrival rates. I: The transition through saturation.. Journal of Applied Probability 5 (2), 436–451. Newell, G., 1968b. Queues with time-dependent arrival rates. II: The maximum queue and the return to equilibrium, III: A mild rush hour. Journal of Applied Probability 5 (3), 579–606. Oksendal, B., 2007. Stochastic Differential Equations, sixth ed. Springer, Berlin. Osorio, C., Flötteröd, G., Bierlaire, M., 2011. Dynamic network loading: a stochastic differentiable model that derives link state distributions. Transportation Research Part B 45 (9), 1410–1423. Paveri-Fontana, S., 1975. On Boltzmann-like treatments for traffic flow: a critical review of the basic model and an alternative proposal for dilute traffic analysis. Transportation Research 9 (4), 225–235. Prigogine, I., Herman, R., 1971. Kinetic Theory of Traffic Flow. Elsevier, New York. Richards, P., 1956. Shock waves on the highway. Operations Research 4 (1), 42–51. Sopasakis, A., 2012. Lattice free stochastic dynamics. Communications in Computational Physics 12 (3), 691–702. Sopasakis, A., Katsoulakis, M., 2006. Stochastic modeling and simulation of traffic flow: asymmetric single exclusion process with Arrhenius look-ahead dynamics. SIAM Journal on Applied Mathematics 66 (3), 921–944. Stanková, K., De Schutter, B., 2010. On freeway traffic density estimation for a jump Markov linear model based on Daganzo’s cell transmission model. In: Proceedings of the 13th International IEEE Conference on Intelligent Transportation Systems, pp. 13–18. Stengel, R., 1994. Optimal Control and Estimation. Dover, New York. Sumalee, A., Zhong, R., Pan, T., Szeto, W., 2011. Stochastic cell transmission model (SCTM): a stochastic dynamic traffic model for traffic state surveillance and assignment. Transportation Research Part B 45 (3), 507–533. Sun, X., Muñoz, L., Horowitz, R., 2004. Mixture Kalman filter based highway congestion mode and vehicle density estimator and its application. In: Proceedings of the 2004 American Control Conference. IEEE, pp. 2098–2103. Szeto, M., Gazis, D., 1972. Application of Kalman filtering to the surveillance and control of traffic systems. Transportation Science 6 (4), 419–439. Wang, Y., Papageorgiou, M., 2005. Real-time freeway traffic state estimation based on extended Kalman filter: a general approach. Transportation Research Part B 39 (2), 141–167. Wang, Y., Papageorgiou, M., Messmer, A., 2007. Real-time freeway traffic state estimation based on extended Kalman filter: a case study. Transportation Science 41 (2), 167–181. Whitt, W., 2002. Stochastic Process Limits. Springer, New York. Work, D., Tossavainen, O., Blandin, S., Bayen, A., Iwuchukwu, T., Tracton, K., 2008. An ensemble Kalman filtering approach to highway traffic estimation using GPS enabled mobile devices. In: Proceedings of the 47th IEEE Conference on Decision and Control, pp. 5062–5068.