# A Traffic Equilibrium Model for Multi-Modal Networks with Uncertain Demands
## 1 Introduction  
In recent years, significant advances have been realized in the formulation and analysis of traffic equilibrium models for multi-modal networks. [PERSON] et al. (1994) [1] developed a mathematical programming formulation for traffic equilibrium models in bimodal networks. [PERSON] and [PERSON] (2003) [2] investigated travellers' behaviors under a multi-modal network with motorized and non-motorized transport modes and formulated an equilibrium model for a variational inequality problem. [PERSON], [PERSON], and [PERSON] (2004) [3], considering car, bus, and subway as transport modes in the network, proposed a three-level Negst logit model to describe combined-mode, transfer location, and route choices and formulated the equilibrium model as a nonlinear complementarity problem. [PERSON] and [PERSON] (2012) [4] proposed a binary logit model for the mode choice and a multinomial probit model to capture route choice, and formulated the equilibrium model as a fixed-point problem. [PERSON] et al. (2016) [5] adopted a nested and a cross-nested logit model to capture the mode similarity and route overlap, respectively, and formulated the equilibrium model as a mathematical programming problem. [PERSON] et al. (2018) [6] proposed a combined multinomial logit model for mode choice and a paired combinatorial logit model for route choice and developed the MNL-PCL model as a mathematical programming problem. [PERSON] et al. (2020) [7] adopted a dogit model to capture travellers' mode choice and a path-size logit model to express route overlapping effects and formulated a combined model as an equivalent entropy-based mathematical programming problem. However, in these studies, the travel demands were assumed to be perfectly known. In reality, travel demands fluctuate randomly, which is caused by factors such as time of day, day of the week, or seasonal effects. These factors are particular cases of travel demand fluctuations that cause traffic flow to differ significantly from its typical pattern. Theyalso directly contribute to the variability of network travel cost (e.g., travel time, crowd discomfort, and other attributes), which will influence travellers' choices. Uncertain demands have been considered in pure-mode network equilibrium frameworks. The effects of travel time variability on travellers' route choice have been discussed in several surveys, where uncertain demands were modelled using random variables. In travel behavior modelling, a trip can be successfully fulfilled within a given travel time range, which is referred to as travel time reliability. This gives rise to the concept of the travel time budget, which is defined as the average travel time plus an extra buffer time as an acceptable margin, which has been the theme of reliability-based user equilibrium models ([PERSON], [PERSON], and [PERSON] (2006)) [8]. Subsequently, later efforts were devoted to the improvement of this model and some applications were discussed ([PERSON] et al. (2006) [9]; [PERSON], [PERSON], and [PERSON] (2006) [10]; [PERSON] and [PERSON] (2008) [11]). The effect of the unreliability aspect of travel time variability affects the travellers' route choice decision process when a trip time longer than expected is considered as 'unreliable' or 'unacceptable' ([PERSON], and [PERSON] (2003) [12]). [PERSON] (2003) [13] proposed \"late arrival penalised\" as the route choice criterion and measured the penalty cost of delays. [PERSON] and [PERSON] (2010) [14] proposed a new model called the \(\alpha\)-reliable mean-excess travel time user equilibrium model, which considered both the reliability and unreliability aspects of travel time variability. Following a similar approach, [PERSON], [PERSON], and [PERSON] (2012) [15] proposed the budget-excess travel time user equilibrium model under stochastic capacity and demands, which assumed that travel demands obey a Gamma distribution while the link capacity obeys a uniform distribution.