**1. Introduction**

In recent years, there have been a number of interesting studies related to modeling the spread of traffic congestion propagation and traffic dissipation in urban network systems (e.g., see [1–5] in the context of macroscopic traffic model involving traffic flux and traffic density; see [6, 7] in the context of percolation theory; see [8] for results based on machine-learning methods; and see [9, 10] for studies based on queuing theory). In this paper, without attempting to give a literature review, we consider a dynamical model, based on the well-known susceptible-infected-recovered (SIR) model from mathematical epidemiology, with small random perturbation, that describes the spread of traffic congestion propagation and dissipation in an urban network system, i.e.,

$$\begin{split}d\boldsymbol{c}^{\boldsymbol{\varepsilon}}(\boldsymbol{t}) &= (-\boldsymbol{\mu} + \beta k(\mathbf{1} - \boldsymbol{r}^{\boldsymbol{\varepsilon}}(\boldsymbol{t}) - \boldsymbol{c}^{\boldsymbol{\varepsilon}}(\boldsymbol{t}))) \boldsymbol{c}^{\boldsymbol{\varepsilon}}(\boldsymbol{t})dt \\ &+ \sqrt{\boldsymbol{\varepsilon}} \sqrt{(\boldsymbol{\mu} + \beta k(\mathbf{1} + \boldsymbol{r}^{\boldsymbol{\varepsilon}}(\boldsymbol{t}) + \boldsymbol{c}^{\boldsymbol{\varepsilon}}(\boldsymbol{t})))} \boldsymbol{c}^{\boldsymbol{\varepsilon}}(\boldsymbol{t}) \boldsymbol{d} \boldsymbol{W}\_{1}(\boldsymbol{t}) \end{split} \tag{1}$$

$$dr^{\varepsilon}(t) = \mu r^{\varepsilon}(t) + \sqrt{\varepsilon} \sqrt{\mu r^{\varepsilon}(t)} \, d\mathcal{W}\_2(t) \tag{2}$$

$$\begin{split}d\boldsymbol{f}^{\varepsilon}(t) &= (-\beta k(\mathbf{1} - \boldsymbol{r}^{\varepsilon}(t) - \boldsymbol{c}^{\varepsilon}(t)))c^{\varepsilon}(t)dt \\ &+ \sqrt{\varepsilon}\sqrt{(\beta k(\mathbf{1} + \boldsymbol{r}^{\varepsilon}(t) + \boldsymbol{c}^{\varepsilon}(t)))}c^{\varepsilon}(t)d\mathcal{W}\_{3}(t) \end{split} \tag{3}$$

where


Notice that Eq. (1) describes the rate at which the fraction of congested links, i.e., *c<sup>ε</sup>* ð Þ*t* , changes over time given the propagation rate *β* and recovery rate *μ* considering that a fraction of congested links will eventually recover as the demand for the travel volume diminishes. Moreover, Eq. (2) describes the rate at which congested links normally recover given the recovery rate *μ*. Finally, Eq. (3) represents how the fraction of free flow links *f ε* ð Þ*t* in the network changes over time given *c<sup>ε</sup>* ð Þ*<sup>t</sup>* and *<sup>r</sup><sup>ε</sup>* ð Þ*t* . Note that, for a normalized SIR based traffic network dynamic model, the following mathematical condition *c<sup>ε</sup>* ð Þþ*<sup>t</sup> <sup>r</sup><sup>ε</sup>* ðÞþ*t f ε* ðÞ¼ *t* 1 holds true for all *t*>0, where *f ε* ð Þ*t* represents links that have remained in a free flow state starting from *t* ¼ 0 (e.g., see Saberi et al. [11] for detailed discussions related to deterministic models).

In this chapter, we provide the asymptotic probability estimate based on the Freidlin-Wentzell theory of large deviations for certain rare events that are difficult to observe in the simulation of urban traffic network dynamics. The framework considered in this study basically relies on the connection between the probability theory of large deviations and that of the values functions for a family of stochastic control problems, where such a connection also provides a desirable computational algorithm for constructing an efficient importance sampling estimator for rare event simulations of certain events associated with the spread of traffic congestions in the dynamics of the traffic network. Here, it is worth mentioning that a number of interesting studies based on various approximations techniques from the theory of large deviations have provided a framework for constructing efficient importance sampling estimators for rare event simulation problems involving the behavior of diffusion processes (e.g., [12–16] for additional discussions). The approach followed in these studies is to construct exponentially-tilted biasing distributions, which was originally introduced for proving Cramér's theorem and its extension, and later on it was found to be an efficient importance sampling distribution for

*Rare Event Simulation in a Dynamical Model Describing the Spread of Traffic Congestions… DOI: http://dx.doi.org/10.5772/intechopen.95789*

certain problems with various approximations involving rare-events (e.g., see [17–19] or [13] for detailed discussions). The rationale behind our framework follows in some sense the settings of these papers. However, to our knowledge, the problem of rare event simulations involving the spread of traffic congestions in an urban network system has not been addressed in the context of large deviations and stochastic control arguments in the small noise limit; and it is important because it provides a new insight for understanding the spread of traffic congestions in an urban network system.

This chapter is organized as follows. In Section 2, we provide an asymptotic estimate on the exit probability using the Freidlin-Wentzell theory of large deviations [20] (see also [21], Chapter 4) and the stochastic control arguments from Fleming [22] (see also [23]), where such an asymptotic estimate relies on the interpretation of the exit probability function as a value function for a family of stochastic control problems that can be associated with the underlying SIR based traffic network dynamic model with small random perturbations. In Section 3, we discuss importance sampling and the necessary background upon which our main results rely. In Section 4, we provide our main results for an efficient importance sampling estimator for rare event simulations of certain events associated with the spread of traffic congestions in the dynamics of the traffic network. Finally, Section 5 provides some concluding remarks.
