Keywords
3. AGENT-BASED MODEL
Each simulated agent represents an individual and is given a simple behavioural rule which is followed in order to reduce
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Average Cost
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B
Equilibrium A
Social Optimum
Figure 3: Average cost for 2000 homogeneous agents with a 5% reviewing rate at each iteration.
the cost of the journeys undertaken. An agent who reviews his departure time calculates the cost of a randomly cho-sen departure time, and changes if this cost is sufficiently cheaper than his current cost. Simulations were performed with agents who had various sensitivities to cost.
Agents have a certain probability, or reviewing rate, of chang-ing their departure time at each iteration. This rate (the percentage of agents who review their departure time at each iteration) is an important parameter of the model that must be carefully calibrated for operational models. The qualitative effects of changing this parameter are discussed below.
At each iteration the agents who review their departure time are chosen randomly, and the new departure time tested is chosen randomly from a flat distribution around the cur-rent departure time. This distribution is the same size as the domain of the simulation. All the agents who review their departure time calculate the cost of the new departure time assuming that no other agent changes his/her depar-ture time.
In all the graphs that follow that show the value of a quan-tity that evolves from day to day the number of iterations is normalised. After one iteration the total number of times that individual agents have reviewed their departure times is equal to the number of agents. In a simulation with 2000 agents after one normalised iteration there have been 2000 reviews of departure time. This was done in order to facili-tate comparisons between different reviewing rates.
3.1 Homogeneous Agents
Homogeneous agents all wish to arrive at the same time,t∗, and follow the same rules. For every agent the travel cost per unit time was, α = 2, the cost of arriving early was β= 1 and the cost of arriving late wasγ= 4.
The first simulation was performed with 2000 agents who began with their departure times distributed so that overall the departure rate function was that of the social optimum.
5% of agents reviewed their departure times each iteration.
The within day time was broken into 2000 discrete units of time. The agents who reviewed their departure times chose
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Within Day Time
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Departure Rate
A - max B - min Equilibrium
Figure 4: The departure rate functions at points A and B of figure (3)
a new departure time, at random, from a flat distribution between 1000 time units before and 1000 units after their current departure time. The average cost for such a simula-tion can be seen in figure (3). The agents in this simulasimula-tion and in all other simulations presented here were assumed to be infinitely sensitive to reductions in cost. When the re-duction in cost required for agents to change their departure time increased to 20 percent the only effect was to slow the overall evolution of the system, instability was uneffected.
The average cost does not converge but oscillates below 800 which is the value of the equilibrium. Figure (4) shows two departure rates, one for which the average cost is close to the equilibrium value A, and one for which the average cost is significantly below the equilibrium value, B. The departure rate at A has a form closer to that of the equilibrium.
3.1.1 Explanation of the Oscillations
The fundamental reason for the oscillations is that an agent who changes to reduce his own cost regularly has a much greater effect on the collective cost, often causing it to in-crease. When an agent changes from a departure time where he suffers no congestion to one where he encounters a traffic jam, he increases the travel time for all who join the traffic jam after him. An agent who changes to avoid the traffic jam reduces the cost for all who joined the traffic jam after him.
The trajectory of the global cost depends on the average evolution of the departure times:
• The effect of an agent who leaves earlier during rush hour is to increase the congestion experienced by those who leave between his new and old departure times.
• The effect of an agent who leaves later during rush hour is to decrease the congestion experienced by those who leave between his old and new departure times.
3.2 Heterogeneous Agents
Homogeneous agents is a very strong assumption to make and as we have seen leads to instability in the system. In order to add more realism and hopefully find a more sta-ble global system we introduced heterogeneous agents to
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Value of Time
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Number of Agents
Figure 5: Graph of the number of agents for each value of time for 6000 agents. The values of time are distributed following a log-normal distribution with σ= 1,775and m= 2,423.
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Average Cost
Figure 6: The evolution of the average cost for 6000 agents with the distribution of the values of time in figure (5)
the simulation. The first heterogeneity introduced was in the schedule delay costs, that is the costs of arriving either early or late. Later the agents were given a distribution of preferred arrival times.
3.2.1 Distribution of Schedule Delay Costs
We assumed that the agents would not all have the same aversion to arriving early or late. Studies [4, 6] have shown that there is a log-normal like distribution to the value of time among commuters. A study undertaken in Lyon [6]
calculated this distribution to be given by the parameters m= 2,423 (mean) andσ= 1,775 (variance). The value of time of the 6000 agents were assigned randomly from such a distribution, figure (5). The schedule delay costs β and γ, see equation (4), for each agent were multiplied by the agent’s value of time. It should be noted thatαthe cost of congestion, see equation (4), was the same for all agents.
The schedule delay costs were calibrated so that on average the cost when the agents were distributed at the social opti-mum was close to that for homogeneous agents i.e. 400. We can see from figure (6) that the average cost payed by the agents, who began with their departure times distributed
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Average Cost
Homog. Agents Heterog. Agents
Figure 7: Comparison of average costs for homoge-neous agents and agents with a Gaussian distribu-tion of preferred arrival times of variance 100
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Congestion Cost
Homog. Agents Heterog. Agents
Figure 8: Comparison of congestion costs for ho-mogeneous agents and agents with a distribution of preferred arrival times of variance 100
so that the overall rate of departure was that of the social optimum, reduces. This is because the agents with high schedule delay costs arrive near the desired time while the other agents avoid the high levels of congestion around the preferred arrival time. There are still significant oscillations that would render it very difficult to calibrate the model with observations.
3.2.2 Distribution of Preferred Arrival Times
It is clearly unrealistic that everybody wishes to arrive at exactly the same time. In order to correct this, each agent was given a preferred arrival time chosen randomly from a normal distribution around t∗,Many different variances of the normal distribution were tested, see below.
The amplitude of oscillations for a normal distribution of variance 100 were significantly less than those found for ho-mogeneous agents, see figure (7). The average costs are of roughly the same magnitude, though slightly reduced for heterogeneous agents. More importantly, the cost for het-erogeneous agents is much more stable. The stability of the average cost for heterogeneous agents comes from the sta-bilisation of the congestion cost, figure (8).
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N. of Changes
Homog. Agents Heterog. Agents
Figure 9: The total number of changes of departure time for heterogeneous and homogeneous agents.
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Average Change
Homog. Agents Heterog. Agents
Figure 10: The cumulative average magnitude of changes for both types of agents.
The stability of the average cost is a result of the greater reluctance of the heterogeneous agents to change their de-parture times and the fact that the changes they do make are smaller in magnitude, see figures (9 & 10 ). The agents tend to find a niche, a small range of departure times, that give consistently the lowest cost.
3.2.3 Heterogeneity and Reviewing Rate
In order to calibrate any model it is necessary to understand the qualitative effects of important parameters. In which regions of parameter space do we find macro level behaviour that resembles observations. Two parameters of this simple model that have important effects are the reviewing rate, the proportion of agents that try a new departure time at each iteration, and the level of heterogeneity, the variance of the distribution of preferred arrival times.
The stability was taken as the standard deviation between 100 normalised iterations and 800 normalised iterations, dur-ing this time the agents have, on average, tried a new de-parture time 700 times.
From figure (13) we can see that the reviewing rate has a straightforward effect on the stability. When the reviewing rate is increased the system becomes more unstable. The dependence on the variance of the distribution of preferred
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Average Cost variance = 128
variance = 192 variance = 288
Figure 11: The average cost, with a reviewing rate of 4%, for a range of variances of the distribution of preferred arrival times, the straight lines of the same colour as the average cost curves are the averages between 100 and 800 iterations. The first curve is for homogeneous agents.
arrival times is more complex.
Figures (11 & 12) show the average costs and standard devi-ations for a range of variances of the distribution of preferred arrival times for a reviewing rate of 4%. Figure (12) is sim-ply a slice taken from figure (13). We see that for a range of variances that the average cost is relatively stable, with a standard deviation of less than ten for average cost values of the order of 500 to 600.
Figure (13) shows some structure in the variation of the sta-bility with increased heterogeneity of the agents. Figures (11
& 12) show that in the region where the instability increases as the heterogeneity increases the average cost oscillates in a different manner, that is with much longer periods. The increase in the standard deviation for variances of the dis-tribution of preferred arrival times greater than 250 occurs in a region where the distribution of PAT becomes unreal-istically large. The region in which the instability begins to decrease and then increases for rising heterogeneity at large reviewing rates, is too unstable to be a viable part of parameter space.
The parameter space of stable behaviour is roughly for re-viewing rates less than 10% and variances of the distribution of preferred arrival times between 75 and 250.