\section{Introduction}
All the models discussed in previous chapters are for link flows, in which all vehicles have the same contribution to the flow dynamics.  When one considers a network flow, he/she has to deal with vehicles of different origins, destinations, or other attributes. These attributes of vehicles play a role in determining their choice of routes, and therefore affect the flow dynamics. Hence, to model a network flow, one has to take these attributes into account and disaggregate traffic flow into different components. Such a network flow model considering disaggregated traffic flow is called a multi-commodity model.

In literature, there have been several models for multi-commodity flow, including the model by Vaughan, Hurdle and Hauer (\nocite{vaughan84}1984), Jayakrishnan's model (\nocite{jayakrishnan91}1991) and Daganzo's cell transmission model (\nocite{Daganzo94}1994; \nocite{Daganzo95}1995).

Vaughan, Hurdle and Hauer (\nocite{vaughan84}1984) suggested a continuous model consisting of a `local equation' and a `history equation'. In their model, the basic element of traffic flow is a vehicle, and for each vehicle a label for its trajectory is introduced. In this model, the local equation ensures traffic conservation, and the history equation computes the trajectory of each vehicle. It has been shown that this model is consistent with the traditional LWR model at the aggregate level.  

Jayakrishnan (\nocite{jayakrishnan91}1991) introduced another multi-commodity model. This is a discrete model, in which each link is partitioned into a number of zones. In each zone, vehicles close to each other and with the same origin, destination or other common commodity-specific characteristics are considered as a ``macroparticle". To determine the position of a macroparticle at next time step,  Jayakrishnan considered its travel speed and the length of the zone where it stays. This model doesn't always preserve traffic conservation; i.e., this model may not be consistent with the LWR model, since the macroparticles are considered separately. 

Daganzo (\nocite{Daganzo94}1994; \nocite{Daganzo95}1995) introduced a multi-commodity model based on his {\em cell transmission model}. In this discrete multi-commodity model, traffic flow in a zone is also disaggregated into macroparticles. In every zone, the macroparticles are ordered according to the waiting time. In this order, the macroparticles are moved, and those macroparticles can proceed to a downstream zone if their waiting times are greater than a threshold minimum waiting time. In Daganzo's model, the minimum waiting time is set to the waiting time of a macroparticle if the total number of vehicles which are in the same zone and in front of this macroparticle is equal to the number of vehicles moving into the downstream zones. In this model, the number of vehicles moving into the downstream zones are computed as the boundary flux times the length of a time step based on solutions to the Riemann problem for the LWR model, which has been discussed in Chapter 5.

Of these three models, Jayakrishnan's and Daganzo's are simpler in computation since a macroparticle instead of a vehicle is considered as the basic element of traffic flow. 

Both the model by Vaughan et al. and Daganzo's model consider the dynamics of a network flow at two levels: aggregate level and disaggregated level. At aggregate level, vehicles in a network flow is considered to be identical particles. At this level, both models are consistent with the LWR model, and therefore they satisfy traffic conservation. In the model by Vaughan et al., the continuous LWR model is used, and while Daganzo's model uses the discrete form of the LWR model.

In these three models the vehicles are ordered. In the model by Vaughan et al. vehicles are ordered according to the trajectory labels, Jayakrishnan ordered macroparticles according to location and Daganzo ordered the macroparticles according to time. They all assume that the vehicles observe the First-In-First-Out (FIFO) discipline when they travel on the roadway. In particular, the model by Vaughan et al. assumes the vehicle trajectories don't cross each other at any time $t$ and any location $x$, Jayakrishnan's model assumes that vehicles always keep the order in location, and Daganzo's model assumes that vehicles always keep the order in time. Of all these models, the one by Vaughan et al. uses the full information of order in time and location, while the other two use part of the order information of either time or location. 

In this chapter we introduce a new multi-commodity model. Our model is based on the concept of ``macroparticle", and has also a two-level structure. In our model, we use the discrete form of the LWR model for traffic flow at the aggregate level. In our new model, we will interpret the FIFO discipline in a new way so that we can simplify the dynamics at disaggregated level.

This chapter is organized as follows. In section 2, we discuss the new multi-commodity model and define the network structure, data structure and program structure. In section 3, we present some numerical simulations. In section 4, we discuss the future plans on this topic.
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