- Email: [email protected]
Allelomimesis as escape strategy of pedestrians in two-exit confinements
Physica A 388 (2009) 2469–2475
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Allelomimesis as escape strategy of pedestrians in two-exit confinements Gay Jane Perez, Caesar Saloma ∗ National Institute of Physics, University of the Philippines Diliman, Quezon City 1101, Philippines
article
info
Article history: Received 22 December 2008 Available online 6 March 2009 Keywords: Structures and organization in complex systems Random processes Self-organized systems Computer modeling and simulation
a b s t r a c t We study the efficacy of allelomimesis as an escape strategy of mobile agents (pedestrians) that aim to leave a two-exit room within the shortest possible time. Allelomimesis is the act of copying one’s kindred neighbors. To escape, an agent employs one of the following strategies: (1) It chooses its own route independently (non-copying, α = 0), (2) It imitates the actions of its neighbors at all times (blind copying, α = 1), or (3) It either copies or acts independently according to a certain probability that is set by the copying parameter α (0 < α < 1). Not more than one agent could occupy a given room location. An agent’s knowledge of the two exit locations is set by its information content β (0 ≤ β ≤ 1). When left alone, an agent with complete knowledge of the exit whereabouts (β = 1) always takes the shortest possible path to an exit. We obtain plots of the (group) evacuation time T and exit throughput Q as functions of α and β for cases where the two exits are near (on same room side) and far (on opposite sides of room) from each other. For an isolated agent, T is inversely proportional to β . The chances of escape for an isolated agent with β ≤ 0.2 are higher with adjacent exits. However, for an agent with β > 0.4 the chance is better with opposite exits. In a highly occupied room (occupancy rate R = 80%) with adjacent exits, agents with β > 0.8 escape more quickly if they employ a mixed strategy of copying and non-copying (0.4 < α < 0.6). On the other hand, blind copying (α ≈ 1) gives agents with β < 0.1 a better chance of escaping from the same room. For the same α and R values, opposite exits allow faster evacuation for agents with β < 0.1 due to the likelihood of streaming and the lower probability of exit clogging. Streaming indicates an efficient utilization of an exit and it happens when the arcs that are formed are smaller and arch interference is less likely. Allelomimesis provides a simple yet versatile mechanism for studying the egress behavior of confined crowds in a multi-exit room. © 2009 Elsevier B.V. All rights reserved.
1. Introduction The ability of confined pedestrians to make sound decisions during (unassisted) emergency evacuations is usually hampered by an apparent narrowing of attention and a (real or perceived) reduction in the number of available escape options [1]. The situation is paralyzing and leads to a fatal slowing of the egress rate with disastrous consequences. To prevent if not minimize serious damage, strict regulations have been drawn mandating the availability of properly designed emergency escape routes in theaters, convention centers, sports stadiums and other similar infrastructures [2]. The conduct of evacuation drills could also lead to more efficient evacuation but such training exercises are generally costly and difficult to conduct in large crowds of pedestrians who are not familiar with each other.
∗
Corresponding author. Tel.: +63 2 9247392; fax: +63 2 9247674. E-mail address: [email protected] (C. Saloma).
0378-4371/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2009.02.037
2470
G.J. Perez, C. Saloma / Physica A 388 (2009) 2469–2475
The number of casualties that resulted in recent emergency escape incidents has remained unacceptably high to civil society and safety engineers continue to work with architects and structural engineers to design and construct more effective escape routes that achieve an optimal balance between form and function as well as structural integrity and economic cost [2,3]. The dire consequences of emergency escapes would also be abated if we can improve our understanding of crowd dynamics — a goal that is not simple to reach since crowds are complex social systems. Crowds exhibit several forms of collective behavior that are different and not deducible from the individual performance of their constituent members [2–6]. Understanding accurately the emergence of collective behavior in crowds is difficult since even as individuals, human beings can respond differently to the same external stimuli [7,8]. The development of computationally efficient and versatile models of real world crowd systems is also essential but not easy to accomplish even for computer scientists due to the interdisciplinary nature of the crowd phenomenon [3,8,9]. In addition, the reliability of the simulation results is also difficult to test due to serious ethical and legal concerns that govern the use of innocent civilians in scientific experiments. A number of interesting manifestations of collective behavior have been revealed by computer simulations such as clogging at exits, arch formation around an exit, disruptive interference between arcs, herding and self-organized queuing among others [9–16]. Collective behavior plays a critical role in defining the egress characteristics of confined pedestrians. Different kinds of animal systems have been utilized to test the simulation results and to probe the range of possible collective behavior of real crowds without violating ethical and legal standards [3,6,17–20]. Here, we study the efficacy of allelomimesis as an escape strategy of mobile agents (pedestrians) that aim to leave a twoexit room within the shortest possible time. Allelomimesis is the act of copying one’s kindred neighbors [21]. To escape, an agent employs one of the following strategies: (1) It chooses its own route independently (non-copying, α = 0), (2) It imitates the actions of its neighbors at all times (blind copying, α = 1), or (3) It either copies or acts independently according to a certain probability that is set by the copying parameter (imitation tendency) α (0 ≤ α ≤ 1). Not more than one agent could occupy a given room location. An agent’s knowledge of the two exit locations is set by its information content (or degree of available information) β (0 ≤ β ≤ 1). When left alone, an agent with β = 1 always takes the shortest possible path to an exit. We determine the group escape time T and exit throughput Q as functions of α and β for cases where the two exits are adjacent (near) and opposite (far) each other in the room. The plots are calculated under conditions of low and high room occupancy rates. The formation of a wide variety of real-world clusters could be explained with just one underlying clustering mechanism called allelomimesis [22,23]. Allelomimesis is considered normal behavior among social groups and surmised to be an evolutionary trait in human societies [24]. Many real-world cluster systems obey the cluster-size frequency distribution [21, 22]: D(s) ∝ s−τ , where exponent τ determines the relative abundance of the cluster sizes s. It has been previously shown that allelomimetic interactions are of three general types [22]: (1) Blind copying where agents are most likely to copy conspecifics (α ≈ 1; e.g. tuna fish schools, buffalos, marmots), (2) Information use copying (α ≈ τ ; e.g. urban agglomerations, firms) where agents are deliberate in the decisions whether to copy or not to copy, and (3) non-copying (α ≈ 0; e.g. gene families, crystals, galaxies). It is not hard to see that allelomimesis is a plausible mechanism for driving the emergence of herd behavior in crowds and animal groups. In this paper, we study how allelomimetic interactions between the mobile agents affect their ability to escape out of a two-exit room. It can be argued from experience that novice (inexperienced) agents (or those lacking in self-confidence) are likely to copy the actions of their neighbors while those with prior training are more deliberate and tend to act on their own. However, deliberate actions require longer processing times — a luxury that agents could ill-afford during emergency evacuations. Our presentation will proceed as follows: Section 2 describes the characteristics of our agent-based model of confined mobile pedestrians in a room with exits and the simulation results are presented and analyzed in Sections 3 and 4, respectively. 2. Methodology We implement an agent-based model to simulate the rapid exit of pedestrians out of a two-exit room. The pedestrians are viewed as discrete mobile agents in a two-dimensional lattice-like neighborhood. The agents move into adjacent lattice sites in discrete steps. The room is divided into M × N cells and each cell is either empty or occupied by not more than one agent at any given time. The total number K of agents inside the room at initial time i = 0, is determined by the room occupancy rate R(%) = 100K /MN. The k-th agent Ak is assigned specific values for the imitation parameter α , information content β and the panic threshold φ where: 0 ≤ φ ≤ 10 [12,14,20]. An agent with φ = 0, is calm and prefers to remain in its present position while that with a large φ value, has a high tendency to move to other available lattice sites. An agent with a high β value represents a pedestrian with prior knowledge of the exit locations — a condition that is realized when the exit locations are labeled with vivid exit signs or if the pedestrian has received previous training. Agent-based modeling is an attractive tool for studying the pedestrian dynamics during emergency evacuations because of its computational simplicity and versatility [25]. It allows us to incorporate the discrete nature of pedestrians and the possible diversity of their interactions with each other [21,22]. Failure to account for discreteness in population models could lead to erroneous results [26]. Agent-based modeling produces results (at least the mean values) that are robust against
G.J. Perez, C. Saloma / Physica A 388 (2009) 2469–2475
2471
Fig. 1. Single agent escaping from a two-exit room (R = 0.396%). Evacuation time T as a function of information content β (20 trials per data point) in range: 0.1 ≤ β ≤ 1. Agent with complete knowledge of the two exit locations is assigned with β = 1. Agent is placed randomly inside the room. Solid curves: T = 10.23β −0.76 for adjacent exits (blue) and T = 5.66β −1.17 for opposite exits (red). Inset plots show T (β) values in full range: 0 ≤ β ≤ 1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
the number of agent population — an important advantage since emergency cases exist where the number of pedestrians involved is not statistically high. Initially (i = 0), the agents are distributed randomly inside the room. We assume that an agent can be found with equal probability in any of the available lattice points. At a later time i > 0, Ak occupies a particular cell except when it has already escaped out of the room. At (i + 1), Ak can occupy any of the eight vacant cells in its Moore neighborhood. Given more than one available neighboring cell to occupy, an Ak with α = 1, chooses that cell which has been recently vacated by its neighbor. If there is more than one such cell then the final choice is randomly selected by Ak . On the other hand, when 0 ≤ α < 1, Ak has a probability (1 − α ) of moving to a cell defined by β . Agent Ak prefers to move to a cell that could bring it closer to the nearest exit when β = 1 and α = 0. On the other hand, Ak moves to any randomly selected vacant cell in its neighborhood when β = 0 or α = 0. Values of α and β between 0 and 1 imply a probabilistic choice between the two extreme strategies. Agent Ak can move into another chosen cell if: l + r < b + φ , where l, r, and b are the number of occupied cells on the left, right and back of Ak , respectively [12,14]. A high value of the panic threshold φ indicates an Ak with a strong urge to move from its current location while an Ak with φ ≈ 0 is passive and prefers to remain in its present location. For the simulations shown here, φ was arbitrarily set to a mid-level value of φ = 5. For computational simplicity, we assume that all the K agents in the room have same values for φ , α and β that could also be interpreted as mean values for the entire group. Two exit configurations were considered in the simulations: (1) Two exits are positioned beside each other on the same side of the room (exit separation = one cell unit), and (2) Two exits positioned on opposite sides of the room and are farthest from each other. The exit width is set to one cell unit so that only one pedestrian can pass through the exit at a time [12,14,20]. The rules for movement and agent interaction described above are based on observations about emergency evacuations. During emergency evacuations, pedestrians have a tendency to escape through the same door that they had previously used as entrance [16]. And when orientation and visibility is poor, such as in smoke-filled rooms or overcrowded areas, only the local information is accessible to each pedestrian. The situation encourages pedestrians to base their decisions on what they know, thus copying the actions of their immediate neighbors, which may result to herding [5,10,14]. Confined pedestrians have also been observed to become disoriented during emergency evacuations and accordingly move in a haphazard way [8]. At i = 0, the room is occupied by K = RMN agents and the group escape time T is the time duration that is needed to evacuate all the agents present. On the other hand, to determine the throughput Q which is the number of agents that has passed through an exit per unit time, we maintain the room occupancy rate R constant at all times by adding (in next time step i + 1) a new agent at a randomly-chosen available lattice site location for every agent that escapes at time step i. 3. Simulation results and discussion First, we determine how the information content β value affects the escape time T of a single agent that is randomly located in a two-exit room (R = 100/MN = 100/252 = 0.396%; 20 trials). The agent has no neighbor to imitate and all cells in the Moore neighborhood are available for occupation. Fig. 1 shows that T is inversely proportional with β i.e., T = Γ β −γ , where: Γ = 10.23 and γ = 0.76 for adjacent exits, and Γ = 5.66 and γ = 1.17 for opposite exits. The plots are independent of the copying parameter α value. When an isolated agent does not have good information of the exit whereabouts (β ≤ 0.2), it tends to move into a randomly chosen lattice site per unit time-step and escape is faster if the exits are positioned near each other. On the other hand, an agent with a good (prior) awareness of the exit locations, can leave more quickly if the exits are farthest from
2472
G.J. Perez, C. Saloma / Physica A 388 (2009) 2469–2475
Fig. 2. Two-dimensional plot of Ω = (1 − α)β , where 0 (non-copying agent) ≤ α ≤ 1 (blind copying agent) and 0 ≤ β ≤ 1. Maximum Ω value indicate a condition that supports a faster escape and it happens when an agent is non-copying (α = 0) and fully knowledgeable of the exit locations (β = 1).
each other where the mean exit distance from the agent’s initial (i = 0) position does not vary significantly with the initial position coordinates. The evacuation time T decreases rapidly with β implying that a small improvement in its knowledge of the exit whereabouts translates to a much faster egress. For a room with opposite exits, a reduction of more than 50% in the T value is achieved when β increases from 0.2 to 0.4. On the other hand, the T value does not vary significantly in the range: (0.8 ≤ β ≤ 1) — complete knowledge of the exit locations (β = 1) is not necessary to achieve the quickest exit possible when the exits are opposite each other. The same observation does not hold for the case of adjacent exits. Fig. 2 plots the product Ω = (1 − α)β , for 0 ≤ α ≤ 1 and 0 ≤ β ≤ 1. The product Ω achieves maximum of unity when β = 1 and α = 0, which corresponds to the case of an escaping agent that knows the exit locations exactly and acts independently of its neighbors. Such an agent attempts to go to the nearest exit at all times. On the other hand, Ω becomes zero when β = 0 or α = 0, which is the case for an agent that has no knowledge of the exit locations and whose next steps are either copied from its neighbors or merely taken at random. Even when left alone in the room, such an agent requires a long time to get out of the room. The Ω values are approximately symmetric about the α = β diagonal line implying that the same Ω value is produced by varying α while keeping β constant and vice-versa. We proceed to investigate the egress characteristics of a group of interacting agents that aims to leave a two-exit room in the shortest possible time. Fig. 3(a)–(d) show the dependence of the group evacuation time T (in color gradations) with α and β for R = 11.9% (3(a) and (c)) and R = 80% (3(b) and (d)) for a two-exit room (area: MN = 252 cells). Both cases of adjacent (Fig. 3(a) and (b)) and opposite (3(c) and (d)) are considered at low (R = 11.9%, 30 agents) and high (R = 80%, 200 agents) room occupancy rates. For given R, α and β values, the group egress is quickest (slowest) when the T (α, β) values are colored black (white). In general, group evacuation is faster (low T values) when agents have accurate knowledge of the exit locations (high β values) and do not imitate the actions of their neighbors (low α values). The dark regions in T (α, β) plots are larger when the exits are opposite each other which imply less stringent requirements for high exit information content and good independent decision-making capability for agents to achieve faster egress. Total lack of knowledge about the exit locations (β ≈ 0) and the tendency to decide independently (α ≈ 0) is a fatal combination for agents particularly when the exits are near each other. The combination results in random selections of escape directions among agents that give rise to relatively long evacuation times. In a highly-occupied room with adjacent exits (Fig. 3(b)), faster egress is achieved when agents with a good knowledge of the exit locations (β > 0.5), employ a mixed strategy of copying and non-copying (0.4 ≤ α ≤ 0.6) their neighbors during the evacuation process. Finally we study the egress throughput Q of each exit as a function of α and β . Throughput Q is the number of agents that has escaped via a specific exit per unit time. Since at most only one agent can pass through an exit at any given time, one obtains bursts of different sizes from the Q profile which is a time-series of 1 and 0 (no escapee) values. We plot the frequency distribution D(s) of the burst sizes where burst size s is defined as the total number of agents that has escaped through a specific exit successively in time. A burst of size s is bounded by two time instants, i and [i + (s + 1)], that no agent is able to leave from the same exit. The room occupancy rate R is held constant in time while we monitor the Q profile. Bursts with large s values may be interpreted as manifestations of streaming (or queuing) — a desirable condition that supports a more efficient room evacuation of crowds [12,20]. Fig. 4(a)–(b) present the D(s) plots that were obtained with α = 0 (blue bars), 0.5 (green) and α = 1 (red) and when the two exits are adjacent and opposite each other, respectively. In all cases, the room occupancy rates are high (R = 80%) and the agents have no knowledge of the exit locations (β = 0). For each α value, the corresponding Q profile has a time length of 20,000 iteration steps.
G.J. Perez, C. Saloma / Physica A 388 (2009) 2469–2475
2473
Fig. 3. Group evacuation time T (in log scale) versus imitation tendency α and information content β (grid resolution: 0.66) for R = 11.9% (upper row) and R = 80% (lower row) for a two-exit room (area: 18 × 14 cells). Rooms with adjacent (first column) and opposite (second column) exits are considered. Each grid value represents the average of 20 trials. Exits are located on shorter side(s) of the room. Black color represents conditions for fastest group exit (lowest possible T value).
For both exit arrangements, increasing α from α = 0 (non-copying) to α = 1 (blind copying) leads to the tailing of D(s) that is characterized by the emergence of larger burst sizes. When agents have no knowledge of the exit locations, copying the actions of one’s neighbor results in agents streaming out of an exit. Comparing the impact of exit arrangements, Fig. 4 reveals that having the two exits positioned far from each other results in a higher number of large burst sizes implying a more efficient egress of agents. 4. Summary We have investigated, via agent-based modeling, the efficacy of copying one’s neighbor as a strategy during emergency evacuations where the goal is to get out of a two-exit room in the shortest possible time. Three possible types of allelomimetic behavior were studied for the agents: (1) non-copying, α = 0), (2) blind copying (α = 1), and (3) Mixed response where an agent imitates or decide independently according to a certain probability (0 < α < 1). Allelomimesis is the act of copying one’s kindred neighbors [23]. It is considered normal behavior among social groups and surmised to be an evolutionary trait in human societies [24]. We both considered the case when the two exits are located (near each other) on the same side or in the opposite sides of the room. The agents possess a certain degree of knowledge of the exit locations that is set by the information content β (0 ≤ β ≤ 1). When left alone in a random room location, an agent with β = 1, always takes the shortest possible path to an exit. The room evacuation time T of an isolated agent is inversely proportional to β . We found that for an isolated agent with poor knowledge of the exit whereabouts (β ≤ 0.2), (accidental) escape is more likely when the exits are located near each other. For a knowledgeable agent however, escape is generally faster with opposite exits. When the two exits are on the same side of the room, we also found that agents with good knowledge of the exit whereabouts (0.8 < β < 1.0) are able to leave more quickly (smaller T values) in a highly occupied room (R = 80%) if they employ a mixed strategy of copying and non-copying (0.4 < α < 0.6). On the other hand, blind copying (α ≈ 1) gives agents with no knowledge of the exit locations (β < 0.1), a relatively greater chance of escaping out of the same room. For the same α and R values, opposite exits allow faster evacuation for agents with β < 0.1 due to the likelihood of streaming and the lower probability of exit clogging. Streaming indicates an efficient utilization of an exit and it happens when the arcs that are formed are smaller and arch interference is less probable.
2474
G.J. Perez, C. Saloma / Physica A 388 (2009) 2469–2475
Fig. 4. Burst-size frequency distributions D(s) for escaping agents with α = 0 (non-copying, blue), α = 0.5 (green) and α = 1 (blind copying, red) and β = 0 (no knowledge of exit whereabouts) in a two-exit room (area: 18 × 14 cells) with 80% occupancy with: (a) Adjacent exits on the same side, and (b) Exits on opposite (shorter) sides of the room. Agents do not know where the exit locations are (β = 0). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
We have shown that allelomimesis provides a simple yet versatile mechanism for studying the egress behavior of confined crowds including the value of exit arrangements in a room. Interesting new results have been presented regarding the allelomimesis-driven emergence of streaming (queuing) and the relationship between information content and the degree of allelomimetic interactions between agents. Acknowledgments We have benefited from past discussions with Dr May Lim, Dr Cynthia Palmes-Saloma and Dr Giovanni Tapang regarding the development of computer models for studying the emergency escape of confined pedestrians. The project is supported by a faculty grant (for CS) and an OVCRD dissertation (for GJP) grant from the University of the Philippines. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
G. Proulx, Cool under fire, Fire Protection Engineering 16 (Fall) (2002) 33. J. Bryan, A selected historical review of human behavior in fire, Fire Protection Eng. 16 (Fall) (2002) 4–10. See articles in: N. Waldau, P. Gatterman, H. Knoflacher, M. Schreckenberg (Eds.), Pedestrian and Evacuation Dynamics 2005, Springer, Berlin, 2007. S. Nolfi, Behavior as a complex adaptive system: On the role of self-organization in the development of individual and collective behavior, Complexus 2 (2004–2005) 195–203. D. Helbing, I. Farkas, T. Vicsek, Dynamical features of escape panic, Nature 407 (2000) 487–490. P. Ball, The physical modelling of human social systems, Complexus 1 (2003) 190–206. A. Baddeley, G. Hitch, in: G. Bower (Ed.), The Psychology of Learning and Motivation: Advances in Research and Theory, Vol. 8, Academic Press, New York, 1974, pp. 47–89. M. Bierlaire, G. Antonini, M. Weber, Behavior dynamics of pedestrians, in: K. Axhausen (Ed.), Moving Through Nets: The Physical and Social Dimensions of Travel, Elsevier, New York, 2003, pp. 1–22. V. Blue, J. Adler, Cellular automata microsimulation of bi-directional pedestrian flows, J. Transport. Res. Board (2000) 1678. C. Burstedde, K. Klauck, A. Schadschneider, J. Zittarts, Simulation of pedestrian dynamics using two-dimensional cellular automata, Physica A 295 (2001) 507–525. D. Helbing, Traffic and related self-driven many-particle systems, Rev. Modern Phys. 73 (2001) 1067–1141. G.J. Perez, G. Tapang, M. Lim, C. Saloma, Streaming, disruptive interference and power-law behavior in the exit dynamics of confined pedestrians, Physica A 312 (2002) 609–618.
G.J. Perez, C. Saloma / Physica A 388 (2009) 2469–2475
2475
[13] D. Helbing, M. Isobe, T. Nagatani, K. Takitomo, Lattice gas simulation of experimentally studied evacuation dynamics, Phys. Rev. E 67 (2003) 067101. [14] C. Saloma, G.J. Perez, Herding in real escape panic, in: N. Waldau, P. Gatterman, H. Knoflacher, M. Schreckenberg (Eds.), Pedestrian and Evacuation Dynamics 2005, Springer, Berlin, 2007. [15] H. Yue, H. Hao, X. Chen, C. Shao, Simulation of pedestrian flow on square lattice based on cellular automata model, Physica A 384 (2007) 567–588. [16] N. Johnson, W.E. Feinberg, The impact of exit instructions and number of exits in fire emergencies: A computer simulation investigation, J. Environ. Psych. 17 (1997) 123–133. [17] D. Sumpter, The principles of collective animal behaviour, Philos. Trans. R. Soc. Lond. B 361 (2006) 5–22. [18] E. Altshuler, O. Ramos, Y. Nuñez, J. Fernandez, A.J. Batista-Leyva, C. Noda, Symmetry breaking in escaping ants, Amer. Naturalist 166 (2005) 643–649. [19] D. Helbing, A. Johansson, H.Z. Al-Abideen, Dynamics of crowd disasters: An empirical study, Phys. Rev. E 75 (2007) 046109. [20] C. Saloma, G.J. Perez, G. Tapang, M. Lim, C. Palmes-Saloma, Self-organized queueing and scale-free behavior in real escape panic, PNAS USA 100 (2003) 11947–11952. [21] J. Parrish, L. Edelstein-Keshet, Complexity, pattern, and evolutionary trade-offs in animal aggregation, Science 284 (1999) 99–101. [22] D.E. Juanico, C. Monterola, C. Saloma, Allelomimesis as generic clustering mechanism for interacting agents, Physica A 320C (2003) 590–600. [23] D.E. Juanico, C. Monterola, C. Saloma, Cluster formation by allelomimesis in real-world complex adaptive systems, Phys. Rev. E 17 (2005) 041905. [24] P.G. Higgs, The Mimetic Transition: A simulation study of the evolution of learning by imitation, Proc. Roy. Soc. Lond. Ser. B 276 (2000) 1355–1361. [25] S. Wolfram, A New Kind of Science, Wolfram Media Inc, Champaign, IL, 2002. [26] N. Shnerb, Y. Lousoun, E. Bettelheim, S. Solomon, The importance of being discrete: Life always wins on the surface, PNAS USA 97 (2000) 10322–10324.
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Allelomimesis as escape strategy of pedestrians in two-exit confinements Gay Jane Perez, Caesar Saloma ∗ National Institute of Physics, University of the Philippines Diliman, Quezon City 1101, Philippines
article
info
Article history: Received 22 December 2008 Available online 6 March 2009 Keywords: Structures and organization in complex systems Random processes Self-organized systems Computer modeling and simulation
a b s t r a c t We study the efficacy of allelomimesis as an escape strategy of mobile agents (pedestrians) that aim to leave a two-exit room within the shortest possible time. Allelomimesis is the act of copying one’s kindred neighbors. To escape, an agent employs one of the following strategies: (1) It chooses its own route independently (non-copying, α = 0), (2) It imitates the actions of its neighbors at all times (blind copying, α = 1), or (3) It either copies or acts independently according to a certain probability that is set by the copying parameter α (0 < α < 1). Not more than one agent could occupy a given room location. An agent’s knowledge of the two exit locations is set by its information content β (0 ≤ β ≤ 1). When left alone, an agent with complete knowledge of the exit whereabouts (β = 1) always takes the shortest possible path to an exit. We obtain plots of the (group) evacuation time T and exit throughput Q as functions of α and β for cases where the two exits are near (on same room side) and far (on opposite sides of room) from each other. For an isolated agent, T is inversely proportional to β . The chances of escape for an isolated agent with β ≤ 0.2 are higher with adjacent exits. However, for an agent with β > 0.4 the chance is better with opposite exits. In a highly occupied room (occupancy rate R = 80%) with adjacent exits, agents with β > 0.8 escape more quickly if they employ a mixed strategy of copying and non-copying (0.4 < α < 0.6). On the other hand, blind copying (α ≈ 1) gives agents with β < 0.1 a better chance of escaping from the same room. For the same α and R values, opposite exits allow faster evacuation for agents with β < 0.1 due to the likelihood of streaming and the lower probability of exit clogging. Streaming indicates an efficient utilization of an exit and it happens when the arcs that are formed are smaller and arch interference is less likely. Allelomimesis provides a simple yet versatile mechanism for studying the egress behavior of confined crowds in a multi-exit room. © 2009 Elsevier B.V. All rights reserved.
1. Introduction The ability of confined pedestrians to make sound decisions during (unassisted) emergency evacuations is usually hampered by an apparent narrowing of attention and a (real or perceived) reduction in the number of available escape options [1]. The situation is paralyzing and leads to a fatal slowing of the egress rate with disastrous consequences. To prevent if not minimize serious damage, strict regulations have been drawn mandating the availability of properly designed emergency escape routes in theaters, convention centers, sports stadiums and other similar infrastructures [2]. The conduct of evacuation drills could also lead to more efficient evacuation but such training exercises are generally costly and difficult to conduct in large crowds of pedestrians who are not familiar with each other.
∗
Corresponding author. Tel.: +63 2 9247392; fax: +63 2 9247674. E-mail address: [email protected] (C. Saloma).
0378-4371/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2009.02.037
2470
G.J. Perez, C. Saloma / Physica A 388 (2009) 2469–2475
The number of casualties that resulted in recent emergency escape incidents has remained unacceptably high to civil society and safety engineers continue to work with architects and structural engineers to design and construct more effective escape routes that achieve an optimal balance between form and function as well as structural integrity and economic cost [2,3]. The dire consequences of emergency escapes would also be abated if we can improve our understanding of crowd dynamics — a goal that is not simple to reach since crowds are complex social systems. Crowds exhibit several forms of collective behavior that are different and not deducible from the individual performance of their constituent members [2–6]. Understanding accurately the emergence of collective behavior in crowds is difficult since even as individuals, human beings can respond differently to the same external stimuli [7,8]. The development of computationally efficient and versatile models of real world crowd systems is also essential but not easy to accomplish even for computer scientists due to the interdisciplinary nature of the crowd phenomenon [3,8,9]. In addition, the reliability of the simulation results is also difficult to test due to serious ethical and legal concerns that govern the use of innocent civilians in scientific experiments. A number of interesting manifestations of collective behavior have been revealed by computer simulations such as clogging at exits, arch formation around an exit, disruptive interference between arcs, herding and self-organized queuing among others [9–16]. Collective behavior plays a critical role in defining the egress characteristics of confined pedestrians. Different kinds of animal systems have been utilized to test the simulation results and to probe the range of possible collective behavior of real crowds without violating ethical and legal standards [3,6,17–20]. Here, we study the efficacy of allelomimesis as an escape strategy of mobile agents (pedestrians) that aim to leave a twoexit room within the shortest possible time. Allelomimesis is the act of copying one’s kindred neighbors [21]. To escape, an agent employs one of the following strategies: (1) It chooses its own route independently (non-copying, α = 0), (2) It imitates the actions of its neighbors at all times (blind copying, α = 1), or (3) It either copies or acts independently according to a certain probability that is set by the copying parameter (imitation tendency) α (0 ≤ α ≤ 1). Not more than one agent could occupy a given room location. An agent’s knowledge of the two exit locations is set by its information content (or degree of available information) β (0 ≤ β ≤ 1). When left alone, an agent with β = 1 always takes the shortest possible path to an exit. We determine the group escape time T and exit throughput Q as functions of α and β for cases where the two exits are adjacent (near) and opposite (far) each other in the room. The plots are calculated under conditions of low and high room occupancy rates. The formation of a wide variety of real-world clusters could be explained with just one underlying clustering mechanism called allelomimesis [22,23]. Allelomimesis is considered normal behavior among social groups and surmised to be an evolutionary trait in human societies [24]. Many real-world cluster systems obey the cluster-size frequency distribution [21, 22]: D(s) ∝ s−τ , where exponent τ determines the relative abundance of the cluster sizes s. It has been previously shown that allelomimetic interactions are of three general types [22]: (1) Blind copying where agents are most likely to copy conspecifics (α ≈ 1; e.g. tuna fish schools, buffalos, marmots), (2) Information use copying (α ≈ τ ; e.g. urban agglomerations, firms) where agents are deliberate in the decisions whether to copy or not to copy, and (3) non-copying (α ≈ 0; e.g. gene families, crystals, galaxies). It is not hard to see that allelomimesis is a plausible mechanism for driving the emergence of herd behavior in crowds and animal groups. In this paper, we study how allelomimetic interactions between the mobile agents affect their ability to escape out of a two-exit room. It can be argued from experience that novice (inexperienced) agents (or those lacking in self-confidence) are likely to copy the actions of their neighbors while those with prior training are more deliberate and tend to act on their own. However, deliberate actions require longer processing times — a luxury that agents could ill-afford during emergency evacuations. Our presentation will proceed as follows: Section 2 describes the characteristics of our agent-based model of confined mobile pedestrians in a room with exits and the simulation results are presented and analyzed in Sections 3 and 4, respectively. 2. Methodology We implement an agent-based model to simulate the rapid exit of pedestrians out of a two-exit room. The pedestrians are viewed as discrete mobile agents in a two-dimensional lattice-like neighborhood. The agents move into adjacent lattice sites in discrete steps. The room is divided into M × N cells and each cell is either empty or occupied by not more than one agent at any given time. The total number K of agents inside the room at initial time i = 0, is determined by the room occupancy rate R(%) = 100K /MN. The k-th agent Ak is assigned specific values for the imitation parameter α , information content β and the panic threshold φ where: 0 ≤ φ ≤ 10 [12,14,20]. An agent with φ = 0, is calm and prefers to remain in its present position while that with a large φ value, has a high tendency to move to other available lattice sites. An agent with a high β value represents a pedestrian with prior knowledge of the exit locations — a condition that is realized when the exit locations are labeled with vivid exit signs or if the pedestrian has received previous training. Agent-based modeling is an attractive tool for studying the pedestrian dynamics during emergency evacuations because of its computational simplicity and versatility [25]. It allows us to incorporate the discrete nature of pedestrians and the possible diversity of their interactions with each other [21,22]. Failure to account for discreteness in population models could lead to erroneous results [26]. Agent-based modeling produces results (at least the mean values) that are robust against
G.J. Perez, C. Saloma / Physica A 388 (2009) 2469–2475
2471
Fig. 1. Single agent escaping from a two-exit room (R = 0.396%). Evacuation time T as a function of information content β (20 trials per data point) in range: 0.1 ≤ β ≤ 1. Agent with complete knowledge of the two exit locations is assigned with β = 1. Agent is placed randomly inside the room. Solid curves: T = 10.23β −0.76 for adjacent exits (blue) and T = 5.66β −1.17 for opposite exits (red). Inset plots show T (β) values in full range: 0 ≤ β ≤ 1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
the number of agent population — an important advantage since emergency cases exist where the number of pedestrians involved is not statistically high. Initially (i = 0), the agents are distributed randomly inside the room. We assume that an agent can be found with equal probability in any of the available lattice points. At a later time i > 0, Ak occupies a particular cell except when it has already escaped out of the room. At (i + 1), Ak can occupy any of the eight vacant cells in its Moore neighborhood. Given more than one available neighboring cell to occupy, an Ak with α = 1, chooses that cell which has been recently vacated by its neighbor. If there is more than one such cell then the final choice is randomly selected by Ak . On the other hand, when 0 ≤ α < 1, Ak has a probability (1 − α ) of moving to a cell defined by β . Agent Ak prefers to move to a cell that could bring it closer to the nearest exit when β = 1 and α = 0. On the other hand, Ak moves to any randomly selected vacant cell in its neighborhood when β = 0 or α = 0. Values of α and β between 0 and 1 imply a probabilistic choice between the two extreme strategies. Agent Ak can move into another chosen cell if: l + r < b + φ , where l, r, and b are the number of occupied cells on the left, right and back of Ak , respectively [12,14]. A high value of the panic threshold φ indicates an Ak with a strong urge to move from its current location while an Ak with φ ≈ 0 is passive and prefers to remain in its present location. For the simulations shown here, φ was arbitrarily set to a mid-level value of φ = 5. For computational simplicity, we assume that all the K agents in the room have same values for φ , α and β that could also be interpreted as mean values for the entire group. Two exit configurations were considered in the simulations: (1) Two exits are positioned beside each other on the same side of the room (exit separation = one cell unit), and (2) Two exits positioned on opposite sides of the room and are farthest from each other. The exit width is set to one cell unit so that only one pedestrian can pass through the exit at a time [12,14,20]. The rules for movement and agent interaction described above are based on observations about emergency evacuations. During emergency evacuations, pedestrians have a tendency to escape through the same door that they had previously used as entrance [16]. And when orientation and visibility is poor, such as in smoke-filled rooms or overcrowded areas, only the local information is accessible to each pedestrian. The situation encourages pedestrians to base their decisions on what they know, thus copying the actions of their immediate neighbors, which may result to herding [5,10,14]. Confined pedestrians have also been observed to become disoriented during emergency evacuations and accordingly move in a haphazard way [8]. At i = 0, the room is occupied by K = RMN agents and the group escape time T is the time duration that is needed to evacuate all the agents present. On the other hand, to determine the throughput Q which is the number of agents that has passed through an exit per unit time, we maintain the room occupancy rate R constant at all times by adding (in next time step i + 1) a new agent at a randomly-chosen available lattice site location for every agent that escapes at time step i. 3. Simulation results and discussion First, we determine how the information content β value affects the escape time T of a single agent that is randomly located in a two-exit room (R = 100/MN = 100/252 = 0.396%; 20 trials). The agent has no neighbor to imitate and all cells in the Moore neighborhood are available for occupation. Fig. 1 shows that T is inversely proportional with β i.e., T = Γ β −γ , where: Γ = 10.23 and γ = 0.76 for adjacent exits, and Γ = 5.66 and γ = 1.17 for opposite exits. The plots are independent of the copying parameter α value. When an isolated agent does not have good information of the exit whereabouts (β ≤ 0.2), it tends to move into a randomly chosen lattice site per unit time-step and escape is faster if the exits are positioned near each other. On the other hand, an agent with a good (prior) awareness of the exit locations, can leave more quickly if the exits are farthest from
2472
G.J. Perez, C. Saloma / Physica A 388 (2009) 2469–2475
Fig. 2. Two-dimensional plot of Ω = (1 − α)β , where 0 (non-copying agent) ≤ α ≤ 1 (blind copying agent) and 0 ≤ β ≤ 1. Maximum Ω value indicate a condition that supports a faster escape and it happens when an agent is non-copying (α = 0) and fully knowledgeable of the exit locations (β = 1).
each other where the mean exit distance from the agent’s initial (i = 0) position does not vary significantly with the initial position coordinates. The evacuation time T decreases rapidly with β implying that a small improvement in its knowledge of the exit whereabouts translates to a much faster egress. For a room with opposite exits, a reduction of more than 50% in the T value is achieved when β increases from 0.2 to 0.4. On the other hand, the T value does not vary significantly in the range: (0.8 ≤ β ≤ 1) — complete knowledge of the exit locations (β = 1) is not necessary to achieve the quickest exit possible when the exits are opposite each other. The same observation does not hold for the case of adjacent exits. Fig. 2 plots the product Ω = (1 − α)β , for 0 ≤ α ≤ 1 and 0 ≤ β ≤ 1. The product Ω achieves maximum of unity when β = 1 and α = 0, which corresponds to the case of an escaping agent that knows the exit locations exactly and acts independently of its neighbors. Such an agent attempts to go to the nearest exit at all times. On the other hand, Ω becomes zero when β = 0 or α = 0, which is the case for an agent that has no knowledge of the exit locations and whose next steps are either copied from its neighbors or merely taken at random. Even when left alone in the room, such an agent requires a long time to get out of the room. The Ω values are approximately symmetric about the α = β diagonal line implying that the same Ω value is produced by varying α while keeping β constant and vice-versa. We proceed to investigate the egress characteristics of a group of interacting agents that aims to leave a two-exit room in the shortest possible time. Fig. 3(a)–(d) show the dependence of the group evacuation time T (in color gradations) with α and β for R = 11.9% (3(a) and (c)) and R = 80% (3(b) and (d)) for a two-exit room (area: MN = 252 cells). Both cases of adjacent (Fig. 3(a) and (b)) and opposite (3(c) and (d)) are considered at low (R = 11.9%, 30 agents) and high (R = 80%, 200 agents) room occupancy rates. For given R, α and β values, the group egress is quickest (slowest) when the T (α, β) values are colored black (white). In general, group evacuation is faster (low T values) when agents have accurate knowledge of the exit locations (high β values) and do not imitate the actions of their neighbors (low α values). The dark regions in T (α, β) plots are larger when the exits are opposite each other which imply less stringent requirements for high exit information content and good independent decision-making capability for agents to achieve faster egress. Total lack of knowledge about the exit locations (β ≈ 0) and the tendency to decide independently (α ≈ 0) is a fatal combination for agents particularly when the exits are near each other. The combination results in random selections of escape directions among agents that give rise to relatively long evacuation times. In a highly-occupied room with adjacent exits (Fig. 3(b)), faster egress is achieved when agents with a good knowledge of the exit locations (β > 0.5), employ a mixed strategy of copying and non-copying (0.4 ≤ α ≤ 0.6) their neighbors during the evacuation process. Finally we study the egress throughput Q of each exit as a function of α and β . Throughput Q is the number of agents that has escaped via a specific exit per unit time. Since at most only one agent can pass through an exit at any given time, one obtains bursts of different sizes from the Q profile which is a time-series of 1 and 0 (no escapee) values. We plot the frequency distribution D(s) of the burst sizes where burst size s is defined as the total number of agents that has escaped through a specific exit successively in time. A burst of size s is bounded by two time instants, i and [i + (s + 1)], that no agent is able to leave from the same exit. The room occupancy rate R is held constant in time while we monitor the Q profile. Bursts with large s values may be interpreted as manifestations of streaming (or queuing) — a desirable condition that supports a more efficient room evacuation of crowds [12,20]. Fig. 4(a)–(b) present the D(s) plots that were obtained with α = 0 (blue bars), 0.5 (green) and α = 1 (red) and when the two exits are adjacent and opposite each other, respectively. In all cases, the room occupancy rates are high (R = 80%) and the agents have no knowledge of the exit locations (β = 0). For each α value, the corresponding Q profile has a time length of 20,000 iteration steps.
G.J. Perez, C. Saloma / Physica A 388 (2009) 2469–2475
2473
Fig. 3. Group evacuation time T (in log scale) versus imitation tendency α and information content β (grid resolution: 0.66) for R = 11.9% (upper row) and R = 80% (lower row) for a two-exit room (area: 18 × 14 cells). Rooms with adjacent (first column) and opposite (second column) exits are considered. Each grid value represents the average of 20 trials. Exits are located on shorter side(s) of the room. Black color represents conditions for fastest group exit (lowest possible T value).
For both exit arrangements, increasing α from α = 0 (non-copying) to α = 1 (blind copying) leads to the tailing of D(s) that is characterized by the emergence of larger burst sizes. When agents have no knowledge of the exit locations, copying the actions of one’s neighbor results in agents streaming out of an exit. Comparing the impact of exit arrangements, Fig. 4 reveals that having the two exits positioned far from each other results in a higher number of large burst sizes implying a more efficient egress of agents. 4. Summary We have investigated, via agent-based modeling, the efficacy of copying one’s neighbor as a strategy during emergency evacuations where the goal is to get out of a two-exit room in the shortest possible time. Three possible types of allelomimetic behavior were studied for the agents: (1) non-copying, α = 0), (2) blind copying (α = 1), and (3) Mixed response where an agent imitates or decide independently according to a certain probability (0 < α < 1). Allelomimesis is the act of copying one’s kindred neighbors [23]. It is considered normal behavior among social groups and surmised to be an evolutionary trait in human societies [24]. We both considered the case when the two exits are located (near each other) on the same side or in the opposite sides of the room. The agents possess a certain degree of knowledge of the exit locations that is set by the information content β (0 ≤ β ≤ 1). When left alone in a random room location, an agent with β = 1, always takes the shortest possible path to an exit. The room evacuation time T of an isolated agent is inversely proportional to β . We found that for an isolated agent with poor knowledge of the exit whereabouts (β ≤ 0.2), (accidental) escape is more likely when the exits are located near each other. For a knowledgeable agent however, escape is generally faster with opposite exits. When the two exits are on the same side of the room, we also found that agents with good knowledge of the exit whereabouts (0.8 < β < 1.0) are able to leave more quickly (smaller T values) in a highly occupied room (R = 80%) if they employ a mixed strategy of copying and non-copying (0.4 < α < 0.6). On the other hand, blind copying (α ≈ 1) gives agents with no knowledge of the exit locations (β < 0.1), a relatively greater chance of escaping out of the same room. For the same α and R values, opposite exits allow faster evacuation for agents with β < 0.1 due to the likelihood of streaming and the lower probability of exit clogging. Streaming indicates an efficient utilization of an exit and it happens when the arcs that are formed are smaller and arch interference is less probable.
2474
G.J. Perez, C. Saloma / Physica A 388 (2009) 2469–2475
Fig. 4. Burst-size frequency distributions D(s) for escaping agents with α = 0 (non-copying, blue), α = 0.5 (green) and α = 1 (blind copying, red) and β = 0 (no knowledge of exit whereabouts) in a two-exit room (area: 18 × 14 cells) with 80% occupancy with: (a) Adjacent exits on the same side, and (b) Exits on opposite (shorter) sides of the room. Agents do not know where the exit locations are (β = 0). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
We have shown that allelomimesis provides a simple yet versatile mechanism for studying the egress behavior of confined crowds including the value of exit arrangements in a room. Interesting new results have been presented regarding the allelomimesis-driven emergence of streaming (queuing) and the relationship between information content and the degree of allelomimetic interactions between agents. Acknowledgments We have benefited from past discussions with Dr May Lim, Dr Cynthia Palmes-Saloma and Dr Giovanni Tapang regarding the development of computer models for studying the emergency escape of confined pedestrians. The project is supported by a faculty grant (for CS) and an OVCRD dissertation (for GJP) grant from the University of the Philippines. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
G. Proulx, Cool under fire, Fire Protection Engineering 16 (Fall) (2002) 33. J. Bryan, A selected historical review of human behavior in fire, Fire Protection Eng. 16 (Fall) (2002) 4–10. See articles in: N. Waldau, P. Gatterman, H. Knoflacher, M. Schreckenberg (Eds.), Pedestrian and Evacuation Dynamics 2005, Springer, Berlin, 2007. S. Nolfi, Behavior as a complex adaptive system: On the role of self-organization in the development of individual and collective behavior, Complexus 2 (2004–2005) 195–203. D. Helbing, I. Farkas, T. Vicsek, Dynamical features of escape panic, Nature 407 (2000) 487–490. P. Ball, The physical modelling of human social systems, Complexus 1 (2003) 190–206. A. Baddeley, G. Hitch, in: G. Bower (Ed.), The Psychology of Learning and Motivation: Advances in Research and Theory, Vol. 8, Academic Press, New York, 1974, pp. 47–89. M. Bierlaire, G. Antonini, M. Weber, Behavior dynamics of pedestrians, in: K. Axhausen (Ed.), Moving Through Nets: The Physical and Social Dimensions of Travel, Elsevier, New York, 2003, pp. 1–22. V. Blue, J. Adler, Cellular automata microsimulation of bi-directional pedestrian flows, J. Transport. Res. Board (2000) 1678. C. Burstedde, K. Klauck, A. Schadschneider, J. Zittarts, Simulation of pedestrian dynamics using two-dimensional cellular automata, Physica A 295 (2001) 507–525. D. Helbing, Traffic and related self-driven many-particle systems, Rev. Modern Phys. 73 (2001) 1067–1141. G.J. Perez, G. Tapang, M. Lim, C. Saloma, Streaming, disruptive interference and power-law behavior in the exit dynamics of confined pedestrians, Physica A 312 (2002) 609–618.
G.J. Perez, C. Saloma / Physica A 388 (2009) 2469–2475
2475
[13] D. Helbing, M. Isobe, T. Nagatani, K. Takitomo, Lattice gas simulation of experimentally studied evacuation dynamics, Phys. Rev. E 67 (2003) 067101. [14] C. Saloma, G.J. Perez, Herding in real escape panic, in: N. Waldau, P. Gatterman, H. Knoflacher, M. Schreckenberg (Eds.), Pedestrian and Evacuation Dynamics 2005, Springer, Berlin, 2007. [15] H. Yue, H. Hao, X. Chen, C. Shao, Simulation of pedestrian flow on square lattice based on cellular automata model, Physica A 384 (2007) 567–588. [16] N. Johnson, W.E. Feinberg, The impact of exit instructions and number of exits in fire emergencies: A computer simulation investigation, J. Environ. Psych. 17 (1997) 123–133. [17] D. Sumpter, The principles of collective animal behaviour, Philos. Trans. R. Soc. Lond. B 361 (2006) 5–22. [18] E. Altshuler, O. Ramos, Y. Nuñez, J. Fernandez, A.J. Batista-Leyva, C. Noda, Symmetry breaking in escaping ants, Amer. Naturalist 166 (2005) 643–649. [19] D. Helbing, A. Johansson, H.Z. Al-Abideen, Dynamics of crowd disasters: An empirical study, Phys. Rev. E 75 (2007) 046109. [20] C. Saloma, G.J. Perez, G. Tapang, M. Lim, C. Palmes-Saloma, Self-organized queueing and scale-free behavior in real escape panic, PNAS USA 100 (2003) 11947–11952. [21] J. Parrish, L. Edelstein-Keshet, Complexity, pattern, and evolutionary trade-offs in animal aggregation, Science 284 (1999) 99–101. [22] D.E. Juanico, C. Monterola, C. Saloma, Allelomimesis as generic clustering mechanism for interacting agents, Physica A 320C (2003) 590–600. [23] D.E. Juanico, C. Monterola, C. Saloma, Cluster formation by allelomimesis in real-world complex adaptive systems, Phys. Rev. E 17 (2005) 041905. [24] P.G. Higgs, The Mimetic Transition: A simulation study of the evolution of learning by imitation, Proc. Roy. Soc. Lond. Ser. B 276 (2000) 1355–1361. [25] S. Wolfram, A New Kind of Science, Wolfram Media Inc, Champaign, IL, 2002. [26] N. Shnerb, Y. Lousoun, E. Bettelheim, S. Solomon, The importance of being discrete: Life always wins on the surface, PNAS USA 97 (2000) 10322–10324.