IV.3. Model
IV.3.4. Traffic Speed-Density Model
To consider traffic speed on each road segment as a variant with traffic density, we include traffic speed as a function of traffic density to MAS model. We use the classic Speed-Density Model: Greenberg Model. Greenberg (1959) proposed a model for real time traffic flow. By assuming the traffic flow as a continuous fluid, he used fluid dynamic principles to conduct the relationships among traffic speed, traffic density, and the traffic flow. The model is as follows:
(Greenberg) kt = kje(−vt/vo) (4.1)
where vo is the traffic speed at qijE. qijE is the road capacity. vt is the traffic speed at time t. kj is the jam density when traffic speed is zero. kt is the traffic density at time t. Then, traffic speed can be a function of traffic density as follows:
(Greenberg) vt = −vo ln(kt/kj) (4.2)
From the above equation, we can easily get the derivations as follows. If the traffic density kt decreases, the traffic speed vt can increase. If the traffic density kt is less than the jam density kj, the traffic speed is greater than zero; otherwise, the traffic speed is zero. However, in our simulation, we give the jam speed a real small value but not zero to make sure the vehicles in system can move. In the case of kt< kj, if kt≤ e−1kj, vt is greater or equal to vo; otherwise, vt is less than vo. If kt = qEij, vt = vo based on the definition of vo, and if kt/kj = e−1, vt = vo. Thus, kj = q
E ij
e−1.
IV.4. Computational Study
In this section, we conduct five experiments to study the effects of five factors on the performance of evacuation process. We evaluate the performance of evacuation process in four perspectives: total evacuation time, individual traveling time, traffic conditions, and transportation cost.
In the first experiment, we test the effect of the probability, at which evacuees follow the optimal evacuation route, on the performance of evacuation process. We benchmark the performance of the evacuation in which evacuees may not follow the optimal evacuation route exactly (i.e. evacuees may have 70% probabilities to follow the optimal evacuation routes, or 30% probabilities, or even 0% probabilities) against the case in which evacuees follow the optimal evacuation route exactly. Through this experiment, we analyze the effect of probability of evacuees following the optimal evacuation routes, and we prove the robustness and the effectiveness of the strategic evacuation plan proposed by SEND model.
Second, we conduct an experiment to test the influence of evacuees leaving in groups with different leaving time on the performance of evacuation process. Evac-uees leave in groups in time sequence and each group has its own leaving time. We assume that these leaving times are not overlapped. Since the total population is constant, more leaving groups means less population in each group. Also, a wide range of leaving time causes a rare population density evacuating at one time unit.
However, a big number of groups or/and a wide range of leaving time may cause the groups, which are scheduled at the rear part of the sequence, leave at a late time, and result the evacuees, which leave in the late groups, in a risky situation. In this experiment, we analyze and evaluate the influence of this interesting part on the performance of evacuation process.
Third, we design an experiment to study the effect of information sharing in the decentralized MAS problem. We study two types of information shared in MAS model. The first type of the shared information is the messages, which are sent from evacuees to their connections (e.g. their friends, relatives, colleagues etc.), about real time traffic conditions (i.e. which road segments have slow traffic). The second type of shared information is the broadcast, which is sent from a radio station to all evacuees, about real time traffic conditions (i.e. which road segments have traffic jam) and the status of shelters (i.e. shelters are full occupied or not). Slow traffic is defined as the traffic flow with a speed less than vo, which value is defined as 40 mph; traffic jam is defined as the traffic flow with a density equal or greater than kj. From the derivations in subsection IV.3.4, kj = q
E ij
e−1, where qijE is defined as 2000*1.5=3000 cars/per lane/per hour to keep consistent with the type I parameters in SEND model. We define the traffic speed in traffic jam as vj (i.e. the value for v when k = kj), which value is 5 mph but not 0 mph, to insure that the evacuees do not stop before they arrive at shelters, which have available spaces. The traffic speed and traffic density parameters, which are used in all five experiments in this chapter, are reported in Table 16. Through this experiment, we analyze and evaluate the effect of information sharing on the performance of evacuation process.
Table 16 Parameters for Traffic Speed and Traffic Density
vo vj qijE kj
40 mph 5 mph 3000 cars/per lane/per hour qEij/e−1
Fourth, we test the performance of the evacuation network which has extra edges’ capacities added to some specific road segments. The locations where extra
edges’ capacities are added are a part of the optimal solution of SEND model. We benchmark the performance of the evacuation in which the network has extra edge-capacities against the case in which the network has no extra edge-edge-capacities. By comparing these two cases, we analyze the effect of road capacities on the perfor-mance of evacuation process, and we prove the effectiveness of the construction of extra edge-capacities which is proposed by SEND model.
Last, we conduct an experiment to test the performance of the evacuation routes proposed by SEND model. We benchmark the performance of evacuation in which evacuees follow their own favorable routes (i.e. the shortest paths from their origins to the shelters which are recommended by SEND model) against the case in which evacuees follow the designated routes proposed by SEND model. In this experiment, we prove the effectiveness of the evacuation routes, which are proposed by SEND model, by analyzing its effect on the performance of evacuation process.
We develop all experiments based on the same traffic network, which is used in the SEND problem in Chapter III. The network size for all experiments in this section is Class 3, which is introduced in Table 7 in § III.4.1. We choose one instance of SEND problem in Class 3 with type I parameters as a benchmark instance(BISP) for MAS problem. The optimal routes and the optimal shelters (OROS), which are a part of the optimal solution of BISP, are used as the designated routes and the designated shelters to guide evacuees. To consider computer memory issue, we down-scale the population in evacuating areas by 500 to run all instances in simulation.
That means we consider 500 vehicles as one agent in MAS model, comparing 1 vehicle considered as a unit in SEND model (i.e. assuming each vehicle has 4 passengers).
However, to compare and contrast the solutions of SEND model and MAS model, we simulate the case, in which evacuees follow OROS exactly, on the same scale level (i.e down-scale population by 500). We use JAVA to code our MAS model
in Repast Simphony environment. All machines used have 2.4 GHZ Intel Core 4 CPU processors with 8 GB RAM. All spatial analysis is conducted using ArcGIS 10 on the same machines. The remainder of this article is organized as follows. From subsection IV.4.1 to subsection IV.4.5, experiment I to experiment V are presented respectively, and their solutions are also analyzed respectively.
IV.4.1. Experiment for Effects of Varying Degrees of Compliance to the