=Paper=
{{Paper
|id=Vol-2215/paper1
|storemode=property
|title=Between Avoidance and Imitation: Plausible Wayfinding in Pedestrian Agent-Based Models
|pdfUrl=https://ceur-ws.org/Vol-2215/paper_1.pdf
|volume=Vol-2215
|authors=Luca Crociani,Giuseppe Vizzari,Stefania Bandini
|dblpUrl=https://dblp.org/rec/conf/woa/CrocianiVB18
}}
==Between Avoidance and Imitation: Plausible Wayfinding in Pedestrian Agent-Based Models==
https://ceur-ws.org/Vol-2215/paper_1.pdf
Between Avoidance and Imitation:
Plausible Wayfinding in Pedestrian
Agent-Based Models
Luca Crociani, Giuseppe Vizzari, Stefania Bandini
CSAI - Complex Systems & Artificial Intelligence Research Center,
University of Milano-Bicocca, Milano, Italy
name.surname@disco.unimib.it
Abstract—Extending the range of pedestrian decision making In previous works we defined a model for the simulation
activities represented in a simulation model represents a serious of wayfinding decisions, especially considering the possibility
challenge: different decisions are taken at distinct levels of of considering the perceivable congestion in agents’ path
abstraction, employing different types of information and knowl-
edge about the environment, from path planning to the regulation planning [2]: the achieved results were somewhere between
of distance from other pedestrians and obstacles present in the basic path planning exclusively based on a “shortest path”
environment. Pedestrians, moreover, are not robots: although heuristics and a globally optimal solution. Later results [3]
empirical observations show that they consider congestion when allowed us to further enrich the model, on one hand to
planning, there are evidences that their decisions are not al- embed an imitation mechanism for which a change in the
ways optimal (even in normal situations). We present a model
integrating and improving consolidated results mitigating the initially planned line of action can be perceived by nearby
optimization effects of congestion aware path planning by making agents and it can trigger analogous decisions. On the other,
commonsense estimations of the effects of perceivable congestion, it supported a calibration of the significance of the different
also embedding an imitation mechanism stimulating changes in components of the wayfinding model, that are path length,
planned decisions whenever another nearby pedestrian did the perceived congestion and perceivable recent changes in the
same. The model is formally described and experimented both
in a validation scenario as well as in a real-world situation: an intentions of nearby pedestrians. In the present paper we
interesting counterintuitive result, in which reducing available improve and extend the above work by providing a deeper
choices and exits actually reduces overall egress time, is also discussion and a more thorough evaluation of the implications
presented and discussed. of the commonsense evaluation of the effects of perceivable
Index Terms—agent-based simulation, pedestrian simulation, congestion on path planning and evaluating the effects of these
wayfinding modeling choices in a real-world scenario. New results show
that the present model represents a step in the direction of
I. I NTRODUCTION a more plausible, although farther from optimality, overall
Pedestrian and crowd simulation is a consolidated research simulation results.
and application context, in which results that lead to technol- The following Section will better set this work in the
ogy transfer (off-the-shelf available commercial tools are daily relevant literature, while Section III will formally describe the
used by designers and planners) co-exist with open challenges proposed modeling approach, that essentially incorporates an
for researchers in different fields and disciplines, to improve extension of the floor field [4] pedestrian modeling approach
model expressiveness (i.e. simplifying the modeling activity in which agents are provided with a representation of the
or introducing the possibility of representing phenomena that environment in which they are situated, that they can employ
were still not considered) and efficiency of the simulators to construct plans of action considering the above mentioned
based on those approaches. Trying to extend the range of factors. Results of the model are provided in the form of
pedestrian agents’ decision making activities represented in calibration tests, provided in Section IV, and exploration of
a computational model represents a serious challenge, though: implications of the chosen path planning model in a real-world
different types of decisions are taken at different levels of scenario, in Section V. Conclusions and future developments
abstraction, employing different types of information and will end the paper.
knowledge about the environment, from path planning (tactical
level decision [1]) to the regulation of distance from other II. L ITERATURE R EVIEW
pedestrians and obstacles present in the environment (opera- The modeling of pedestrians’ wayfinding decisions, es-
tional level decision). Moreover, the measure of success and pecially considering the inevitable and empirically observed
validity of a model is definitely not the optimality with respect trade off between trajectory length and estimated travel time
to some cost function, as (for instance) in robotics, but the (considering the perceived congestion in alternative choices),
plausibility, the adherence of the simulation results to data that represent an open challenge for pedestrian simulation re-
can be acquired by means of observations or experiments. search. Despite the topic has been considered by different
1
disciplines studying spatial cognition processes for a long generally a small minority of the simulated population and
time, as testified by [5], research trying to provide empirical they are often not considered to evaluate the performance of
evidences supporting modeling and simulation efforts is still the design in withstanding a certain type of demand.
lively: for instance, [6] used a questionnaire to ask pedestrians
what are the most relevant factor influencing their choices, III. S IMULATING WALKING AND ROUTE CHOICE OF
whereas [7] actually performed both a simplified walking P EDESTRIANS
experiment involving wayfinding and also asked the involved
participants to draw a trajectory on a map, in an outdoor A. The Representation of the Environment and the Knowledge
setting; [8] describes observed trajectories followed by pedes- of Agents
trians attending a festival; [3] performed an experiment to The adopted agent environment [14] is discrete and modeled
observe actual wayfinding choices in a very simple situation, in with a grid of 40 cm sided square cells: the size considers
which pedestrians had to choose between a short but congested the average area occupied by a pedestrian [15], and it allows
path and a longer but faster one. All of these works show representing reasonable densities usually observed in real
that wayfinding decisions are not exclusively determined by scenarios. The cells have a state informing nearby agents
the length of the path or the expected travel time, and they about their movement possibilities: they can be vacant or
highlight that pedestrians actually do not choose optimal paths occupied by an obstacle or at most two pedestrians, so as
even in normal, non stressful situations. Despite these results, to be able to manage locally high density situations [16].
however, the support to the modeling of this kind of decision This modification to the basic floor field approach is based
making activities is still in relatively early stages. on observations described by [17] that highlight the fact
Most works in the area of pedestrian and crowd simulation that in some situations (especially counter-flows) pedestrians
investigate wayfinding from the perspective of including in actually adjust their spatial orientation to temporarily accept a
the model the necessary elements to perform the wayfinding reduction of their personal space to allow a smoother flow.
operation from the first execution of the model. The approach To allow the configuration of a pedestrian simulation sce-
described in the previously mentioned work by [8] considers nario, several markers are defined with different purposes in
spatial cognition aspects, in particular combining allocentric addition to the basic map of the area describing obstacles and
and egocentric contributions to overall pedestrian navigation walkable space. This set of objects has been introduced to
in an integrative approach considering different heuristics. allow the movement at the operational level and the reasoning
Results of the approach have been proposed showing a good at the tactical level, identifying intermediate and final targets
accordance with empirical observations in outdoor situations, for agents’ plans: (i) start areas, generating agents in the simu-
the kind of scenario in which the approach seems more lation; (ii) openings, sets of cells that (together with obstacles)
plausible. [9] explored the implications of different strategies divide the environment into regions; (iii) final destinations of
for the management of route choice operations, through the agents, generally implying their removal from the simulation.
combination of applying the shortest or quickest path, with An example of environment annotated with this set of
local (i.e., minimize time to vacate the room) or global (i.e., markers is proposed in Fig. 1(b): the above model implies the
minimize overall travel time) strategies. [10] proposed the fact that the environment is divided in regions that, as well as
modification of the floor-field Cellular Automata [4] approach other markers, are associated to additional relevant metadata
for enabling pedestrian choices also considering the impact of used to characterize the associated element (e.g. define the
congestion on the expected travel time in evacuation situations. demand associated to a start area).
[11] proposed a pedestrian model to simulate route choice in This model uses an extended version of the floor fields
case of evacuation and they were able to reproduce the ob- approach [4] for supporting agents’ navigation at locomo-
served data of an experiment. [12] also considered evacuation tion/operational level, using the agents’ environment as a con-
situations, discussing the results of an experiment about the tainer of information for the management of the interactions
evacuation of a classroom with two exits, proposing a cellular between entities. In this particular model, discrete potentials
automata model, whereas [13] studied the evacuation of a two are spread from cells of obstacles and destinations, informing
exits classroom, proposing a differentiation between rational about distances to these objects. The two types of floor fields
behavior, mainly aimed at optimizing the own travel time, and are denoted as path field, spread from openings and final
irrational one, attracted by the choices of other people and destinations (one per destination object), and obstacle field,
leading to higher evacuation times. Although the results of the a unique field spread from all the cells marked as obstacle.
above cited works are an interesting starting point for further In addition, a dynamic floor field that has been denoted as
studies, they are not conclusive. proxemic field is used to reproduce a proxemic behavior [18]
It is important to highlight that all of the above cited in a repulsive sense, allowing agents to preserve acceptable
approaches imply that modeled pedestrians are provided with distances from other agents. The overall approach generates
a complete map of the environment in which they are situated. a plausible navigation of the environment as well as an
Whereas this seems a rather implausible assumption, we must anthropologically founded means of regulating interpersonal
consider that these research efforts are generally set within distances among pedestrians.
or very close to the transportation research area: within a This framework, on one hand, enables the agents to have a
train station, it is almost inevitable to find first time visitors position in the discrete environment and to perform movement
of such an environment, nonetheless their presence represents towards a user configured final destination. On the other hand,
2
(a) (b)
Fig. 2. The complete life-cycle of the agent.
(c) (d) mapped to targets in the environment and each edge refers to
Fig. 1. (a) An example of plausible (continuous line) and implausible (dashed) a particular path between two targets. The root of the tree is a
paths in a simple environment. (b) An experimental scenario (two open areas final destination, while the underlying nodes are only mapped
connected by means of a constrained corridor separated in two sections to openings connected or reachable from it. To complete the
connected by three openings - o1 , o2 , and o3 ,) with the considered annotation
tools and its respective cognitive map (c) and the shortest path tree leading information, each node n is labeled with the free flow expected
to the southern exit (d). travel time (i.e. without encountering any congestion in the
path) associated to the path starting from the center of the
opening associated to n and passing through the center of all
the presence of intermediate targets supports choices at the openings mapped by the parent nodes of n, until the final
tactical level of the agent, with the computation of a graph- destination.
like representation of the walkable space, based on the concept For the choice of their path, agents access the information
of cognitive map [19]. The method for the computation of this of a Paths Tree generated from a final destination End
environment abstraction has been defined in [20] and it uses with the function Paths(R, End ). Given the region R of
the information of the scenario configuration, together with the agent, the function returns a set of couples {(Pi , tt i )}.
the floor fields associated to openings and final destinations. Pi = {Ωk , . . . , End } is the ordered set describing paths which
In this way a data structure for a complete knowledge of the start from Ωk , belonging to Openings(R), and lead to End .
environment is pre-computed. The cognitive map identifies tti is the associated free flow travel time.
regions (e.g. a room) as nodes of the labeled graph and
openings as edges. An example of the data structure associated B. The Route Choice Model of Agents
to the sample scenario is illustrated in Fig. 1(c). Overall the This aspect of the model is inspired by previously discussed
cognitive map allows the agents to identify their position in empirical evidences suggesting the following stylized facts
the environment and it constitutes a basis for the generation of about pedestrian tactical level decisions: (i) they are able to
an additional knowledge base, which will enable the reasoning consider perceivable congestion when planning their paths; (ii)
for the route calculation. their reasoning is inevitably imprecise, both due to the limited
This additional data structure has been called Paths Tree time spent for the decision as well as to the imprecise estimates
and it contains the information about plausible paths towards a results of individual perception; (iii) they are influenced by
final destination, starting from each region of the environment. nearby pedestrians also through imitation mechanisms, appar-
The concept of plausibility of a path is encoded in the ently conflicting with the general avoidance tendency.
algorithm for the computation of the tree, which is discussed By considering these aspects, the proposed approach enables
in [2] and only briefly described here. The procedure starts agents to choose their path considering distances as well as the
by considering the destination as the root of a tree that evolution of the overall simulation dynamics, especially con-
is recursively expanded, adding child nodes mapped to an sidering visible changes in decisions of preceding agents. At
intermediate destination reachable in the region. Nodes are the same time, the model must provide a sufficient variability
added if the constraints describing the plausibility of a path of the results (i.e. of the paths choices) and the possibility to
are satisfied: in particular, paths that imply cycles or a not be calibrated to reflect observed empirical data.
reasonable usage of the space (e.g. passing inside a room to The workflow of the agent is again reported in Figure 2, to
reach the exit of a corridor, as illustrated in Fig. 1(a)) are allow the understanding of the tasks related to route choice.
simply avoided. First of all, the agent performs a perception of its surround-
The results of the computation is a tree whose nodes are ing situation, considering its knowledge of the environment,
3
aimed at understanding its position and the markers perceiv- of emerging leaders). All the three functions provide values
able from its region (e.g. intermediate targets). At the very normalized within the range [0, 1], thus the value of U (P ) is
beginning of its life, the agent does not have any information included in the range [−κq , κtt + κf ].
about its location, thus the first assignment to execute is a 2) The Evaluation of Traveling Times: The evaluation of
self localization: it basically implies to perceive the values traveling times is a crucial element of the model: even if
of floor fields in its physical position and infers the location it is not the only considered factor, it still represents an
in the Cognitive Map. Once the agent is aware of the region extremely significant element for routing decisions. First of all,
where it is situated, it loads the Paths Tree and evaluates the the information about the travel time tti of a path Pi is derived
alternatives leading to its final destination. from the relevant Paths Tree. In particular, P aths(R, End) is
Figure 2 also emphasizes that the evaluation of the possible used, where End is the agent’s final destination (used to select
paths and the re-consideration of the plan do not only occur the appropriate Paths Tree), and R is the region in which the
when the agent is created in the simulation or when it passes agent is situated (it is used to select the relevant path Pi in the
from a region to another (i.e. when new elements influencing Paths Tree structure). This information is integrated with the
the choice can be perceived). The evaluation is also performed free flow travel time to reach the first opening Ωk described
at specific intervals, according to a timer that can assume by each path:
two possible values: (i) a value defining a short interval, set
right after the agent performs a change of its current plan,
to evaluate its adequacy (the new plan, in fact, could lead to P FΩk (x, y)
TravelTime(Pi ) = tti + (2)
acquire new information about the state of the environment Speedd
potentially indicating that the new path leads to a worst
congestion than the one avoided); as a result of the evaluation
where P FΩk (x, y) is the value of the path field associated
associated to the short interval timer, (ii) a higher value is
to Ωk in the position (x, y) of the agent and Speedd is the
set when the agent confirms the current choice of path, or
desired velocity of the agent, that can be an arbitrary value.
if it changes back to the previous choice employed after the
The value of the traveling time is then evaluated by means of
short interval. This timer-based mechanism is introduced to
the following function:
limit the natural non-determinism of the probabilistic approach
employed in this model and to avoid excessively frequent
changes in the adopted plan. min (TravelTime(Pi ))
Pi ∈Paths(r)
The evaluation of a potential plan is designed through the Eval tt (P ) = Ntt · (3)
TravelTime(P )
concept of path utility, assigned to each alternative: just like
for the selection of the next cell at operational level, also the
choice of the overall plan of actions (i.e. set of intermediate
where Ntt is the normalization factor, i.e., 1 over the sum
markers leading from the current region to the desired exit)
of TravelTime(P ) for all paths. By using the minimum value
is in fact based on a probabilistic decision. The result of this
of the list of possible paths leading the agent towards its own
process generates a new intermediate target of the agent, used
destination from the current region, the range of the function
to update the reference to the floor field to be followed at the
is set to (0,1], being 1 for the path with minimum travel
operational layer.
time and decreasing as the difference with the other paths
1) The Utility and Choice of Paths: The function that
increases. This modeling choice, makes this function describe
defines the probability of choosing a path is exponential with
the utility of the route in terms of travel times, instead of its
respect to the utility value associated to it. This is essentially
cost, but the most important consideration is that it allows
analogous to the choice of movement at the operational layer:
performing a normalization employing the minimum travel
P rob(P ) = N · eU (P )
time instead of the maximum. This improves the robustness
The usage of the exponential function for the computation
of the function with respect to the presence of outliers, few
of the probability of adopting a path P is a good solution to
paths (even just one) characterized by very high travel times
emphasize the differences in the perceived utility values of
that would essentially flatten the differences among cost values
paths, limiting the choice of relatively bad solutions, such as
of other reasonable choices after the normalization, reducing
those associated to much longer paths. More precisely, U (P )
its discriminating power.
comprises the three observed components influencing the route
choice decision, which are aggregated with a weighted sum: 3) The Evaluation of Congestion: The behavior modeled
in the agent in this model considers congestion as a negative
element for the evaluation of the path. However, by acting
U (P ) = κtt Evaltt (P ) − κq Evalq (P ) + κf Evalf (P ) (1)
on the calibration of the parameter κq it is possible to define
different classes of agents with customized (and potentially
where the first element evaluates the expected travel times; dynamic) behaviors, also considering attraction to congested
the second provides a commonsense evaluation of the queuing paths with the configuration of a negative value to generate
(crowding) conditions through the considered path and the last mere following or herding behaviors.
one introduces a positive influence of perceived choices of For the evaluation of this component of the route decision
nearby agents to pursue the associated path P (i.e. imitation making activity associated to a path P , a function is first
4
introduced for denoting agents that precede the evaluating
agent a in the route towards the opening Ω of a path P :
Forward (Ω, a) =
|{a0 ∈ Ag\{a} : Dest(a0 ) = Ω ∧ (4)
PF Ω (Pos(a0 )) < PF Ω (Pos(a))}|
where P os and Dest indicate respectively the position and
current destination of the agent; the fact that PF Ω (Pos(a0 )) <
PF Ω (Pos(a)) assures that a0 is closer to Ω than a, due to the (a)
nature of floor fields. Each agent is therefore able to perceive
the main direction of the others (its current destination). This
kind of perception is plausible considering that only preceding
agents are counted, but we want to restrict its application
when agents are sufficiently close to the next passage (i.e.
they perceive as important the choice of continuing to pursue
that path or change it). A schema providing a sample situation
describing the above defined functions is shown in Figure 3
(a): in particular, agent A1 has five other agents that should
reach passage Ω1 before it, according to current intentions and
state of the environment (agent A2 is actually farther from (b)
Ω1 ), whereas there are six other agents that should arrive to
Fig. 3. Example situations describing Forward function (a) and ChoiceField
Ω2 before it, should it change its plan (something that seems (in particular, for ρc = 3) (b).
implausible given this state of the system). To introduce a
way to calibrate this perception, the following function and
an additional parameter γ are introduced:
needed by the values to decay. The diffusion of values from
( an agent a, choosing a new target Ω0 , is performed in the cells
Forward (Ω, a), if PF Ω (Pos(a)) < γ c of the grid with Dist(Pos(a), c) ≤ ρ with the following
PerceiveForward (Ω, a) = c
0, otherwise function:
(5)
(
The function Eval q is finally defined with the normalization 1/Dist(Pos(a), c) if Pos(a) 6= c
Diffuse(c, a) = (7)
of values of PerceiveForward for all the openings connecting 1 otherwise
the region of the agent:
The diffused values persist in the ChoiceField grid for
PerceiveForward (FirstEl (P ), myself ) τc simulation steps, then they are simply discarded. The
Eval q (P ) = N · (6)
width(FirstEl (P )) index of the target Ω0 is stored together with the diffusion
values, thus the grid contains in each cell a vector of couples
where FirstEl returns the first opening of a path, myself {(Ωm , diff Ωm ), . . . , (Ωn , diff Ωn )}, describing the values of
denotes the evaluating agent and width scales the evaluation influence associated to each opening of the region where the
over the width of the door (larger doors sustain higher flows). cell is situated. While multiple neighbor agents changes their
It must be emphasized that this modeling choice represents choices towards the opening Ω0 , the values of the diffusion are
a deliberately imprecise estimation of the expected increase summed up in the respective diff Ω0 . In addition, after having
in the travel time towards the next intermediate goal, a form changed its decision, an agent spreads the gradient in the grid
of commonsense reasoning, in the vein of [21], unlike what for a configurable amount of time steps represented by an
happens in a previous modeling effort [2]. additional parameter τa . In this way it influences the choices of
4) Propagation of Choices - Following Behavior: This its neighbors for a certain amount of time. Figure 3 (b) shows
component of the decision making model aims at representing a sample situation in which agent A1 , as a consequence of
the effect of an additional stimulus perceived by the agents changing its next intermediate target from Ω1 to Ω2 , spreads
associated to sudden decision changes of other persons that a ChoiceField (with range ρc = 3) that can be perceived by
might have an influence. An additional grid has been in- the agent immediately following it.
troduced to model this kind of event, whose functioning is The existence of values diff Ωk > 0 for some opening Ωk
similar to the one of a dynamic floor field. The grid, called implies that the agent is influenced in the evaluation phase
ChoiceField, is used to spread a gradient from the positions by one of these openings, but the probability for which this
of agents that, at a given time-step, change their plan due to influence is effective is, after all, regulated by the utility weight
the perception of congestion. κf . In case of having multiple diff Ωk > 0 in the same cell,
The functioning of this field is described by two parameters a individual influence is chosen with a simple probability
ρc and τc , which defines the diffusion radius and the time function based on the normalized weights diff associated to
5
Procedure 2 Patha Pathb Pathc
the cell. Hence, for an evaluation performed by an agent a at Experiment 23.2 22.8 0
time-step t, the utility component Evalf can be equal to 1 wayfinding based on shortest path 30.1 14.9 0
wayfinding based on [2] 40.8 5.2 0
only for one path P , between the paths having diff Ωk > 0 in Present model 23.9 22.1 0
the position of a. Procedure 3 Patha Pathb Pathc
Experiment 28 0 18
wayfinding based on shortest path 46 0 0
IV. VALIDATION OF THE MODEL wayfinding based on [2] 44.9 0 1.1
Present model 28.6 0 17.4
While operational level aspects of pedestrian modeling Procedure 4 Patha Pathb Pathc
Experiment 20.8 18 7.2
and simulation have a reasonably stable set of results that wayfinding based on shortest path 30.1 14.9 0
a plausible simulation model should produce (to the point wayfinding based on [2] 40.8 5.2 0
Present model 19.3 17.8 8.9
that there is even a technical note by the National Institute TABLE I
of Standards and Technology on this point [22]), a similar AVERAGE CHOSEN PATHS ( OBSERVED AND SIMULATED ) OF PEDESTRIANS
IN THE EXPERIMENTAL SCENARIO DESCRIBED IN [3].
type of standard validation process for tactical level decisions
and wayfinding is still not feasible due to lack of knowledge
and data. For these reasons, an experiment involving human
participants in a controlled setting has been performed in 2015
and its results have inspired and have been used for the design, a pure floor-field approach and with the model proposed in [2],
calibration and initial validation of the wayfinding model. which describes a model of wayfinding considering in a more
The experiment has been configured to achieve evidences analytically precise the effects on the perceivable level of con-
regarding the influence of crowding conditions on the route gestion, without considering imitative effects. Finally, a simple
choice. 46 students participated to the experiment, and the modification of the analyzed spatial configuration is evaluated
setting was designed to describe an elementary choice: it was with additional simulations, identifying how counterintuitive
characterized by a rectangular environment divided in two results – similar to well-known paradoxes in the transportation
areas of equal size along the long side, each one of 7.2×6 m2 . field [23], and also present in pedestrian dynamics [24] – can
The two sides were connected by three passages, which were be observed in this particular scenario.
creating three paths of different lengths, respectively Patha ,
Decision Parameters Value
Pathb and Pathc in order of length: Figure 1(b) graphically utility parameter κtt 100.0
describes this scenario. The two gates defining longer routes utility parameter κq 25.0
were closed according to four different procedures: (1) only utility parameter κf 5
ChoiceField Parameters
the shortest path was available; (2) Patha and Pathb open; (3) diffusion radius parameter ρc 1.2 m
Patha and Pathc open; (4) all paths available. Each procedure decay parameter τc 0.5 s
has been repeated four times to achieve more consistent data. diffusion time of agent τa 1s
TABLE II
For each procedure, the number of people employing each C ALIBRATION PARAMETERS USED FOR THE WAYFINDING MODEL .
path has been manually counted. More thorough details about
the experiment can be found in [3].
A similar setting has been simulated with the three sim-
The scenario is represented in Fig. 4(a): 4 starting areas
ulation case studies: (i) wayfinding based on shortest path;
(green in the figure) are associated to the bleachers of the
(ii) wayfinding based on quickest path, as defined in [2]; (iii)
stadium and they generate the agents in the simulation, whose
wayfinding with the proposed model. Results are reported in
aim is to reach the outside area indicated with the blue object
Table I. To achieve consistent and reliable results, a set of 50
(i.e. the Northern and Eastern borders of the scenario). Cyan
simulations has been performed for the bottleneck scenario, for
objects are the intermediate targets, generating the alternative
each width of the door. A smaller set containing longer runs
opportunities for agents’ wayfinding decisions. Larger ones
has been configured for the fundamental diagram tests, where
and closer to the start areas represents the corridors connecting
the corridor was configured as toroidal with respect to the long
the bleachers to the atrium, where a total of 11 doors of 1.2
side in order to maintain the same global density. The chosen
m of width provide the way out from the stadium. 250 agents
configuration (κtt , κq , κf ) = (100, 25, 5) of the parameters is
are generated in random positions of the related start area at
effective to reproduce the distribution of chosen paths over
the beginning of the simulation, producing a total of 1000
the simulated pedestrians, leading to much closer results to
pedestrians. The parameters of the model are the same one
the empirical data than with the other two case studies.
employed for the validation tests.
Differences among pedestrians are introduced with respect
V. S IMULATION OF A R EALISTIC S CENARIO to the desired walking speed, defined through a discretization
This section shows the application of the proposed and of a Gaussian distribution described by µ = 1.4 m/s and
overall model in a realistic scenario, simulating a sample σ = 0.2 m/s, to represent an egress situation in normal
egress from a football arena similar to the one described conditions. The maximum speed in the model is set to 1.8
by [9]. The aims are: (i) to allow the reader to understand m/s to cover the majority of values defined by the distribution.
the impact of the chosen modeling purposes on the simulated An example distribution from one simulation run is shown in
dynamics; (ii) to discuss the difference of the results proposed Figure 4(b). The structure of the environment and the nature of
by the current model with a baseline implementation based on the simulated situation limit the impact of the assumption that
6
(a) (b)
Fig. 4. (a) A screenshot from a simulation run describing the environment used for the experiments. Colors of objects define their type as explained in
Section III, while the color of the agents informs about their current target. (b) Configured distribution of desired speeds (blue) and final average speeds (blue)
of agents. Both images refer to one simulation run of the 3rd case study.
of pedestrians causes congestions that are apparent in space
utilization diagrams that will be described later on, and that
also have an influence on the total evacuation time. Case Study
N. 2 represents uses the wayfinding model proposed in [2]
that essentially tries to select the quickest path, employing
an analytically precise estimation of the impact of perceived
congestion surrounding the nearest intermediate targets (that
are supposed to be perceivable from the regions of which they
represent a border). Agents are therefore able to select the
best alternative at the time of planning to the shortest path
in case of a congested environment. Case Studies N. 3 and 4
use the model described in this paper, with parameters set to
κtt = 100, κq = 25, κf = 5. The fourth scenario is config-
ured with a modified version of the environment, achieved by
actually closing the middle gateway connecting the bleachers
to the atrium: this represents a counterintuitive design choice,
since it makes more difficult to exit the bleachers area but, on
the other hand, the atrium turns out to be much less congested,
Fig. 5. Comparison of evacuation times of the whole environment achieved
among the 4 scenarios. The black square represents the average, while the smoothening the flow towards the final exits, reducing the
individual time related to one iteration is plotted with a transparency effect overall egress time.
to describe the distribution.
These above discussed features are clarified with the results
shown in Figure 5, and 6. In particular, Figure 5 shows the
evacuation times of each run of the simulation sets of the
agents are provided with a complete map of the environment: case studies, simply calculated as the time interval starting
first of all, pedestrians are in the process of exiting from at the beginning of the simulation and ending when the last
the environment, which means that they have already seen agent reaches its final destination, vacating the area. The model
a portion of the area at least once; moreover, the map is quite described in [2] is substantially more efficient in terms of
simple, with associated plans that require the passage between travel times of agents, achieving an average evacuation time
just one intermediate target (i.e. the associated path trees are of about 62 s. This is due to the effective strategy of the
surely wide but quite shallow). agents that leads to an extremely well balanced usage of the
Within the illustrated scenario, four case studies have been exit doors, shown in Figure 6(b): this kind of diagram shows
simulated with sets of 50 simulation runs, a number sufficient the evacuation times over the space. The values shown in
to achieve consistent results. Case Study N. 1 aims only at the map are achieved by storing the latest time step τ̂ in
achieving a base line dataset, describing the results achievable which each cell has been occupied by a pedestrian in the
by only using the model at the locomotion layer following simulation, representing essentially how long it takes to vacate
the “shortest path” heuristics, without a dynamic wayfinding: the area associated to a given cell. Adopting the quickest path
the fact that most direct gateways are used by a large number approach the emptying times are extremely well balanced in
7
the available exits, something that does not happen with the to the empirical results and calibration performed in [3], we
case study 3, where the two exit doors at the extremes of the consider the present results as a step in the direction of more
scenario become the most used due to their attractiveness in plausible wayfinding decisions, that are surely more effective
terms of utility (they are the most obvious choices for the than baseline “shortest path” based heuristics, but not as close
agents coming from the top left and bottom right start areas, to optimality as results of a previous approach [2].
due to the short distance). Some pedestrians initially directed Current and future works are aimed at incorporating results
towards those exits, actually change the initial plan and finally on the impact of groups in the simulated pedestrian population
select other nearby and less congested exits, but this does not within the wayfinding decisions; we are also considering the
happen so systematically as for case study 2. opportunity of developing a serious game, also employing
Finally, case study 4 represents a typical “what–if” scenario: video games and virtual reality technologies, to achieve a
in fact, we considered the issue of congestion in the exits more sustainable way of acquiring empirical evidences on
from the atrium and tried to actually reduce the flow from the human wayfinding. This would allow to consider arbitrary
bleachers area by deciding not to use the central of the three environmental structures, also more complicated than the ones
gateways from those areas to the common atrium. While this studied and simulated so far, and to achieve a more thorough
certainly increases the congestion in the remaining passages validation of the proposed model.
(although the maximum measured level of density is lower
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