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The Impact of Sampling Methods on Evacuation Model Convergence and Egress Time Ruggiero Lovreglio, Mathilde Girault, & Michael Spearpoint December, 2018

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The Impact of Sampling Methods on Evacuation Model Convergence and

Egress Time

Article  in  Reliability Engineering [?] System Safety · May 2019

DOI: 10.1016/j.ress.2018.12.015

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The Impact of Sampling Methods on Evacuation Model Convergence and Egress Time

Ruggiero Lovreglio1, Michael Spearpoint2,3, Mathilde Girault4

1 School of Engineering and Advanced Technology, Massey University, New Zealand 2 Olsson Fire & Risk, Manchester, UK. 3 Department of Civil and Environmental Engineering, University of Canterbury, New Zealand 4 Norman Disney & Young, Auckland, New Zealand ABSTRACT Simulating human behaviour in fire is often one of the main challenges in designing complex buildings, structures or sites for the life safety of occupants. In fact, evacuation simulations represent a fundamental input to assess fire safety performance using a risk analysis approach. The variability in evacuee behaviours (e.g. pre-evacuation delays and uncongested walking speed) can be probabilistically simulated in egress models using distribution functions. The application of probabilistic simulations requires the input distributions to be sampled. This paper describes a series of eight repeated trial evacuations that were carried out using a classroom-based scenario. The paper then investigates how four different sampling methods (namely Simple Random, Stratified, Inversed Stratified and Halton) affect the ability of a computational egress tool to reach convergence when determining the total time for occupants to leave the room. The analysis found that the Stratified and the Inverse Stratified sampling approaches require the least number of simulation runs to converge while the Halton sampling approach needs the greatest number of simulation runs. Moreover, the results indicate that the Halton sampling generates the highest variance for the simulated total evacuation time and thus is more effective at examining scenarios that utilise the extreme ends of the distribution functions. Keywords: Evacuation modelling, Sampling methods, Behavioural uncertainty, Evacuation experiments Published as: Lovreglio R., Spearpoint M., and Girault M, 2019, The Impact of Sampling Methods on Evacuation Model Convergence and Egress Time, Reliability Engineering and System Safety

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1. INTRODUCTION

The design and safety assessment of complex buildings, structures or sites (such as nuclear power plants, chemical plants and hazardous waste facilities) often needs to consider the safety of occupants should an accident occur [1–3]. Part of such a safety assessment can include calculations to determine how long it might take for occupants to reach a place of safety an well as the time available to clear the area threaten by the disaster [4,5]. In the case of fire safety this time is generally referred to as required safe egress time (RSET) and it should be less than the available safe egress time (ASET), often with some safety factor included, to demonstrate the occupants will reach a place of safety without adverse effects. To date, several modelling approaches (e.g. Gaussian models, Box models, Computational Fluid Dynamics models) have been developed to estimate the spread of smoke and fire [6,7]. Various methods are also available to estimate RSET and these range from relatively simple equations (see for instance the SFPE hydraulic model [8]) that can be programmed into a spreadsheet to complex algorithms implemented in computational software tools (e.g. continuous models, discrete models, hybrid models) as reviewed in the literature [9,10]. Traditionally these RSET calculations have been deterministic but the use of probabilistic methods as part of the safety design process under the effect of uncertain factors is gaining prominence especially for complex building and sites [11–13]. In fact, the probabilistic approach is fundamental to simulating behavioural uncertainty through egress models [14–16] and also provides a means to investigate what might be appropriate safety factors for deterministic analyses. As such, evacuation models are a key tool for probabilistic safety assessment of a wide range of occupancies [17–19]. To date, several probabilistic evacuation models have been proposed in the literature to evaluate the performance and risk analysis of complex buildings etc. [20]. Those models provide a suitable representation of human behaviour as they use stochastic or probabilistic methods to simulate pre-evacuation responses, route or exit choices, movement. The randomness of movement speed and navigation is one of the most common feature in existing evacuation software tools implementing the social force model or floor field cellular automaton model [1,21,22]. The randomness of pre-evacuation responses has been considered by most evacuation models by allowing the users to select a pre-evacuation time distribution considering those available in the literature [23]. Finally, the probabilistic features of route selection are available in only a limited number of evacuation models as reviewed in [20]. The treatment of the randomness can be handled using several approaches. However, the Monte Carlo approach represents one of the most popular in the field [2,20,24]. This approach relies on repeated random sampling to obtain numerical results. The Monte Carlo approach has been integrated in several evacuation models such as: Evacuationz [25], CRISP [26], Pedgo [27] and CUrisk [28]. The application of probabilistic simulations based on the Monte Carlo approach requires the selection of values for the input parameters used in the computational tools [9,10]. This selection is generally achieved by sampling from one or more distributions. Typically, the probabilistic simulations are run to a point they reach a defined level of convergence [29,30]. This can be done using broad class of computational algorithms called sampling methods. A particular sampling method may incur a greater or lesser computational cost [14]. The computational cost may manifest itself in the number of simulations required to reach convergence but may also be related to the computational effort required to do the sampling. There are a range of sampling methods of varying complexity that could be implemented by computational tools but several concerns have been raised in the literature [31,32] regarding which sampling technique is appropriate to confirm the precision of the results. However, the authors of this paper are not aware of any studies investigating the impact of different sampling

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methods on the output of evacuation simulation tools. As such, investigating the impact of different sampling methods is fundamental as the generated probabilistic evacuation results could have a significant impact on safety assessments especially when focusing on extreme events. This paper investigates how sampling methods affect (1) the convergence of the evacuation simulations and (2) variance of the total evacuation time of those simulations. In this paper, we compare four sampling methods, namely Simple Random, Stratified, Inversed Stratified and Halton. The convergence criteria used in this work are those based on the functional operators introduced in refs. [14,30]. These criteria are defined using experimental data as proposed by Lovreglio et al. [14]. As such, eight trial evacuations of the same scenario were carried out in a lecture room at the University of Canterbury (New Zealand) and are presented in this paper.

2. METHOD

2.1 Sampling of distributions

To illustrate the sampling methods investigated in this paper we can consider an exemplar distribution shown as a probability density function in Figure 1 (a). In the case of the Simple Random method, a sample point is randomly selected from the distribution as illustrated in Figure 1 (b). For a small number of samples there may not be a representative range of values obtained that adequately encompasses the distribution. An alternative approach is to stratify the distribution into regions and take the same number of samples from each region so that values are selected across the distribution [33]. Within a given region samples can be selected using a uniform sub-distribution function between the upper and lower bounds. The upper and lower bounds of the regions can be weighted using the distribution itself so that relatively more or less samples are taken from different parts of the distribution. In the case of the Stratified Method used in this paper the area of each region under the probability density curves are equal so that a greater number of samples are taken where the probability is higher [33]. Thus, for illustrative purposes the exemplar distribution has been stratified into five arbitrary regions in Figure 1(c) and one sample taken from each of those regions. The result is that more samples come from the distribution in which the probability is at its greatest. For the Inverse Stratified method, illustrated in Figure 1(d), the sampling regions are configured so that the weighting of the upper and lower bounds results in more samples come from regions of the distribution in which the probability is proportionally less. By breaking a distribution into regions there also needs to be a technique for selecting the region from which a sample is to be taken. The selection of the region for the next sample could be done randomly or in sequential order, as in this study. An alternative to using a random number generating function is to use a deterministic quasi-random number sequence and in this paper the Halton sequence [34] has been investigated. The Halton sequence uses coprime numbers as the base for each dimension, where a dimension is the sampling distribution associated with a particular simulation input variable. In this work the simulations selected random prime numbers in the range 2 to 7,919 (i.e. the first 1,000 prime numbers) for each dimension. A given prime number was only applied to one dimension for each run of the simulation model but prime numbers were reassigned to different dimensions each run. This ensured the same prime number was not used for a given dimension each simulation run.

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(a)

(b)

(c)

(d)

Figure 1. (a) Exemplar distribution; (b) Simple Random sampling; (c) Stratified Method sampling; (d) Inverse Stratified Method sampling.

The previous discussion considers that the selection of a random number is truly random. However, many computational modelling environments are not able to generate truly random numbers but samples are generated using a pseudo-random number generator. It is not the purpose of this paper to discuss how pseudo-random numbers are generated or used other than there are commonly used techniques to select seed values that assist in giving different sequences of numbers. The computational modelling employed the linear congruential pseudo-random number generator from the standard C++ rand() function. To ensure a different series of pseudo-random numbers was used for each simulation the seeds were selected using an approach in which the date and time that a given simulation run was initiated. 2.2. Convergence Criteria and Functional Operators

The convergence measures used for the study of behavioural uncertainty are based on functional approach proposed in ref. [30]. Functional analysis is here intended as the branch of mathematics used for the study of vector spaces through operators. Functional analysis operators are used to investigate the uncertainty of the occupant-evacuation curves, which are studied as vectors where the components are ordered times. Three concepts of functional analysis can be used to analyse these curves, namely the Euclidean Relative Difference (ERD), the Euclidean Projection Coefficient (EPC) and

5

the Secant Cosine (SC) (see the Appendix of this paper). The mathematical formulations for these functional operators are described in detail by Ronchi et al. [30] and Galea et al. [35]. Ronchi et al. [30] also introduce five convergence measures for the study of behavioural uncertainty, which are presented in Equations 1–5. These criteria assess the convergence towards the average of consecutive aggregate occupant-evacuation curves. Equation 1 and 2 investigate the variation of Total Evacuation Time (TETi), namely the highest time of an occupant-evacuation curve. In fact, Equation 1 and 2 investigate respectively the consecutive average (𝑇𝑇𝑇𝑇𝑇𝑇������𝑖𝑖) and standard deviations (SDi) of total evacuation times from different curves. Similarly, convergence measures ERDconv i, EPCconv i and SCconv i are produced for the functional analysis operators ERDi, EPCi and SCi.

where Txx are the thresholds for each criterion. Those criteria need to be accepted for a consecutive number of runs to verify that the convergence measures are stable under the specified thresholds. 2.3 Comparison Procedure

Three steps are proposed for the procedure used to compare the impact of the four sampling methods on the evacuation results (see Figure 2). The first step deals with the assessment of the thresholds introduced in Equations 1-5. This is done using running consecutive trials for the same evacuation scenario generating a set of occupant-evacuation curves whose uncertainty can be measured using the convergence parameters described in Section 2.2. Starting from these measures, the definition of threshold values in Equations 1-5 for the five convergence criteria can be performed as suggested in ref. [14]. The second step focuses on the analysis of evacuation simulation results. For each sampling method introduced in Section 2.1, the general approach consists of running M simulations of the same evacuation scenario and verifying whether the convergence criteria in Equation 1-5 are met. If the convergences are not met, then an additional M simulations need to be run and the convergence criteria reassessed. This iterative process needs to be repeated until each convergence criterion is met. This process is iterated for N duplications and thus it is possible to define a N-size vector 𝑹𝑹 ={𝑅𝑅1,𝑅𝑅2, … ,𝑅𝑅𝑖𝑖 , … ,𝑅𝑅𝑁𝑁}, where 𝑅𝑅𝑖𝑖 is the number of runs required to reach the convergence for the ith duplication. Moreover, for each iteration, it is possible to collect a vector 𝑻𝑻𝑻𝑻𝑻𝑻𝒊𝒊 ={𝑇𝑇𝑇𝑇𝑇𝑇1,𝑇𝑇𝑇𝑇𝑇𝑇2, … ,𝑇𝑇𝑇𝑇𝑇𝑇𝑀𝑀} including all the total evacuation times for the ith duplication.

Criterion 1: 𝑇𝑇𝑇𝑇𝑇𝑇𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑖𝑖 = �𝑇𝑇𝑇𝑇𝑇𝑇������𝑖𝑖−𝑇𝑇𝑇𝑇𝑇𝑇������𝑖𝑖−1

𝑇𝑇𝑇𝑇𝑇𝑇������𝑖𝑖� < 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 Equation 1

Criterion 2: 𝑆𝑆𝑆𝑆𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑖𝑖 = �𝑆𝑆𝑆𝑆𝑖𝑖−𝑆𝑆𝑆𝑆𝑖𝑖−1𝑆𝑆𝑆𝑆𝑖𝑖

� < 𝑇𝑇𝑆𝑆𝑆𝑆 Equation 2

Criterion 3: 𝑇𝑇𝑅𝑅𝑆𝑆𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑖𝑖 = |𝑇𝑇𝑅𝑅𝑆𝑆𝑖𝑖 − 𝑇𝑇𝑅𝑅𝑆𝑆𝑖𝑖−1| < 𝑇𝑇𝑇𝑇𝐸𝐸𝑆𝑆 Equation 3

Criterion 4: 𝑇𝑇𝐸𝐸𝐸𝐸𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑖𝑖 = |𝑇𝑇𝐸𝐸𝐸𝐸𝑖𝑖 − 𝑇𝑇𝐸𝐸𝐸𝐸𝑖𝑖−1| < 𝑇𝑇𝑇𝑇𝐸𝐸𝐸𝐸 Equation 4

Criterion 5: 𝑆𝑆𝐸𝐸𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑖𝑖 = |𝑆𝑆𝐸𝐸𝑖𝑖 − 𝑆𝑆𝐸𝐸𝑖𝑖−1| < 𝑇𝑇𝑆𝑆𝐸𝐸 Equation 5

6

Figure 2. Schematic flow chart of the comparison procedure.

The third step is the actual comparison of the sampling methods. In this work we compare those methods to investigate (1) which sample method is the fastest in meeting the convergence criteria in Equations 1-5 by comparing the 𝑹𝑹 vectors generated for each sampling method and (2) which sample method generate the highest variance of for the simulated total evacuation times by comparing the 𝑻𝑻𝑻𝑻𝑻𝑻𝒋𝒋 vectors for each sampling method.

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3. CASE STUDY 3.1 Experimental Data

Eight repeat trial evacuations were carried out in a classroom at the University of Canterbury. The geometry and layout of the room is represented in Figure 3. Thirteen master’s degree students attending one of the fire safety course at the University agreed to participate to this experiment. Previous similar research is described in the literature where students have been used as part of trial evacuations. For instance, the work by Kimura and Sime [36] in which 43 participants and 77 participants were involved in two different scenarios; the work by Zhang et al. [37] in which 60 participants; the work by Purser [38] where participant numbers ranged between 17 and 18; and various trial evacuations used in the work by Spearpoint and Xiang [25] where numbers ranged between 8 and 34 participants. Clearly the number of participants in this study is at the lower end of those reported in the literature and it is likely that having more participants would have been advantageous. However, in this work we have repeated the trial evacuations to assess variability whereas the previous studies have only carried out single trials.

Figure 3. Geometry of the classroom and location of the participants at the beginning of the trials.

Prior to the trial evacuations, the students involved in the study were asked to fill a survey to determine their demographic profile. Although the age of the participants and BMI (23.5 ± 2.7 kg/m2) were recorded these were only applied to a deterministic simulation exercise that is not reported in

8

this paper and the uncongested walking speed of the participants was not measured. Each student was provided with an ID number and a fixed pre-defined pre- evacuation time as indicated in Table 1. Since the students were aware of their participation in the study it was not viable to assess their pre-evacuation time as though it was an unannounced trial. Thus, their pre-evacuation times were pseudo-randomly selected from a triangular distribution with a minimum of 0 s, most likely of 8 s and a maximum of 20 s. This distribution was selected based on the previous work on unannounced classroom evacuations [39] in which representative total pre-evacuation times of 14 s and 22 s were measured. The students were distributed throughout the room, as indicated in Figure 3. For each trial students started from the same location from a sitting position and the approximate travel distance to the door was measured.

Table 1. Participant demographics and pre-evacuation time.

Student ID Gender Age Assigned pre-evacuation time (s)

Approximate distance from the door (m)

1 M 29 5 9.0 2 M 29 5 8.0 3 F 31 9 9.0 4 M 23 14 8.0 5 F 22 11 7.5 6 M 25 6 4.5 7 M 33 13 13.0 8 M 33 11 12.0 9 M 23 17 10.5

10 F 30 7 7.0 11 F 22 9 13.0 12 M 25 8 12.5 13 M 47 16 10.0

The participants were asked to evacuate the room using a single 0.84 m wide door located at the approximate centre of one of the walls. The door had a self-closer but it was propped permanently open during all the trials. At the beginning of each trial an instructor in the room started a timer and the students started evacuating when their pre-evacuation time had elapsed. The time to leave the room for each student was estimated based on the videos taken during the trials using a camera located close to the exit. During the analysis of those videos, a participant was considered out of the room when their shoulders had passed the door frame. Participants were instructed to move away from the door once they had exited to not impede those people behind them. The evacuation curves of the eight trials are shown in Figure 4 and the times in Table 2. Figure 4 also illustrates the average evacuation curve and the 0.15 and 99.85 percentile curves (i.e. average ±3 standard deviation). As such the probability of having an evacuation curve between the upper and lower curves is 99.7 %. Finally, it possible to observe that the average total evacuation time was 25.6 ± 0.5 s. Since the door from the classroom was propped open it was observed that there was little delay on the people travelling out of the classroom when traversing the door opening.

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Figure 4. Evacuation curves of the eight trials (lines are used to join the data points to aid with

visualisation). The full data is reported in Table 2.

Table 2. Evacuation times (in seconds) of the 13 participants for the 8 runs. Average values rounded to the nearest integer.

Run number

Number of evacuees who have left the room 1 2 3 4 5 6 7 8 9 10 11 12 13

1 11 12 13 15 16 18 18 19 21 22 23 24 26 2 11 12 13 14 17 18 18 19 21 22 23 24 26 3 11 12 13 14 16 17 18 19 20 22 23 24 25 4 11 12 13 14 16 16 17 18 21 22 23 24 25 5 11 12 13 14 16 17 18 19 22 22 23 24 26 6 11 12 13 15 16 17 19 20 21 22 23 24 26 7 11 12 13 15 16 17 18 19 22 23 24 24 26 8 11 12 13 15 17 18 19 20 21 22 23 24 25

Average 11 12 13 15 16 17 18 19 21 22 23 24 26

10

12

14

16

18

20

22

24

26

28

1 2 3 4 5 6 7 8 9 10 11 12 13

Evac

uatio

n tim

e (s

)

Number of evacuees who have left the room

Trial 1

Trial 2

Trial 3

Trial 4

Trial 5

Trial 6

Trial 7

Trial 8

Average

0.15Percentile

99.85Percentile

10

The experimental behavioural uncertainty observed in the evacuation trials was estimated using the functional operators (𝑇𝑇𝑇𝑇𝑇𝑇𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑇𝑇 , 𝑆𝑆𝑆𝑆𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑇𝑇 , 𝑇𝑇𝑅𝑅𝑆𝑆𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑇𝑇 , 𝑇𝑇𝐸𝐸𝐸𝐸𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑇𝑇 and 𝑆𝑆𝐸𝐸𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑇𝑇 ) using the ‘Convergence Calculator’ package [14]. The calculated values are illustrated in Figure 5 showing how the functional operator decreases with the increment of the evacuation trials1. For the calculation of 𝑆𝑆𝐸𝐸𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑇𝑇 , the skip s is set equal to 1, as suggested in reference [35] where reasonable values of the s/n ratio are between 0.03 and 0.05, with n being the number of agents, 13, and s being an integer value greater than or equal to one. As explained in [14], the calculation of all of the functional operators requires at least three trials in the dataset, thus Figure 5 necessarily starts at Trial 3.

(a) (b)

Figure 5. Estimate of experimental behavioural uncertainty using the functional operators introduced in Section 2.2; (a) 𝑇𝑇𝑇𝑇𝑇𝑇𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑇𝑇 , 𝑆𝑆𝑆𝑆𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑇𝑇 , 𝑇𝑇𝑅𝑅𝑆𝑆𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑇𝑇 , 𝑇𝑇𝐸𝐸𝐸𝐸𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑇𝑇 and (b) 𝑆𝑆𝐸𝐸𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑇𝑇 .

The behavioural uncertainty associated with simulation results (i.e. the simulated behavioural uncertainty) should be at least at the same order of magnitude as the experimental behavioural uncertainty [14]. To achieve this goal, it is necessary to define some thresholds for the five functional operators and to make sure that simulated behavioural uncertainty, measured by 𝑇𝑇𝑇𝑇𝑇𝑇𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆 , 𝑆𝑆𝑆𝑆𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆 , 𝑇𝑇𝑅𝑅𝑆𝑆𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆 , 𝑇𝑇𝐸𝐸𝐸𝐸𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆 and 𝑆𝑆𝐸𝐸𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆 , are less the selected thresholds. These thresholds are defined by observing the trends in 𝑇𝑇𝑇𝑇𝑇𝑇𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑇𝑇 , 𝑆𝑆𝑆𝑆𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑇𝑇 , 𝑇𝑇𝑅𝑅𝑆𝑆𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑇𝑇 , 𝑇𝑇𝐸𝐸𝐸𝐸𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑇𝑇 and 𝑆𝑆𝐸𝐸𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑇𝑇 in Figure 5. The thresholds shown in Equation 6 are chosen as it is possible to see that all the functional operators are consistently stable below those thresholds after four evacuation trials.

TRTET = 0.01, TRSD = 0.1, TRERD = 0.01, TREPC = 0.01, TRSC = 0.01 Equation 6 3.2 Evacuation Simulations

The simulations described in this paper were carried out using the Evacuationz [40] agent-based probabilistic computational software tool. The tool uses a node-based approach to represent a building geometry with paths of user-defined lengths to connect nodes. Paths include the effect of constrictions, such as doors, on the flow of agents as well as the influence of stairs on agent movement speeds through the application of equations given in Gwynne and Rosenbaum [8]. Agents are individually represented in the software where each agent can be assigned their own pre-evacuation time, uncongested walking speed, age and Body Mass Index (BMI). The effect of occupant density,

1 Such a property is a direct consequence of the Central Limit Theorem as argued in ref. [30]

0.000

0.005

0.010

0.015

0.020

0.025

3 4 5 6 7 8

TETE co

nv,,

ERDE co

nv, E

PCE co

nv,S

CE conv

Number of evacuation trials

TETconv ERDconvEPCconv SCconv

0.00.10.20.30.40.50.60.70.80.91.0

3 4 5 6 7 8

SDE co

nv

Number of evacuation trials

SDconv

11

age, BMI are accounted for using the data published in refs. [41,42]. Input values for the simulation parameters can be specified as constants or through the use of probability distribution functions. The trial evacuation configuration is modelled using a 12 m by 12 m node to represent the classroom with a 0.84 m wide door-type connection to an exit node. Each agent is given an initial starting distance, taken from Table 1, to represent their position relative to the door. In order to apply the sampling methods, agents are assigned an uncongested walking speed by a normal distribution with a mean speed of 2.1 m/s and standard deviation of 0.25 m/s [43]. A more complex approach using the age, BMI and gender as presented in [35] could have been used although the relatively narrow range of ages and BMI values of the participants means there would have been little additional benefit in doing this. Similarly, the sampling methods are applied to agent pre-evacuation times using the same triangular distribution discussed in Section 3.1. Since the people in the trial evacuations started from a sitting position then the simulations have included an additional random response time to account for the time it took to stand up before commencing movement. Following the work of Yamada and Demura [44] the response time has been characterised by a normal distribution with a mean of 3.07 s and standard deviation of 0.35 s. Simulations have been carried out using a 0.5 s time step with each agent processed in a random order during each time step.

(a) (b)

Figure 6. Representation of the evacuation scenario in Evacuationz [40]; (a) node network; (b) snapshot from a Smokeview [45] animation.

For each of the four sampling methods 90 duplications (i.e. N = 90 as shown in Figure 2) of the evacuation scenario were run in which each duplicate consisted of a batch of 200 simulations (i.e. M = 200 as shown in Figure 2). The number of duplications was determined using the G*Power software tool [46]. This tool assesses the impact on the sampling methods for the number of simulations required to reach the convergence criteria using an ANOVA test. The effect size (f) was found to be from small (f=10) to medium (f=25), substantiating that 90 duplications provided an acceptably low level of impact. The number of simulations within each batch of 200 that were necessary to reach the convergence criteria in Equations 1–5 were recorded. It was not necessary to run any further batches of simulations for each duplication since all the convergence criteria were met prior to the 200th simulation (see Section 3.3.1). Thus, 90 × 200 = 18,000 simulations per sampling method and a total of 360 duplicates (90 × 4 sampling methods) were carried out.

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3.3 Sampling Method Comparison

3.3.1 Convergence Comparison

The four sampling methods have been assessed to determine whether there is any difference in terms of the number of simulation runs necessary to meet the convergence criteria given in Equations 1–5. To make this assessment the analysis determines which sampling method requires the least number of runs to reach the criteria. Figure 7 illustrates box plots of the mean, 25-, 50- and 75-percentile number of runs (with outliers indicated by the dots) required to meet the five convergence criteria. In addition, descriptive statistics of the simulation runs are given in Table 3. An ANOVA test has been carried out with the hypothesis that a p-val. < 0.05 indicates a significant difference between sampling methods and these results are also given in Table 3. For the Equation 1 convergence criterion the ANOVA test indicates that there is a significant difference between sampling methods and Figure 7(a) indicates that Stratified sampling method requires the least mean number of runs to reach the convergence criterion while the Halton sampling approach requires the greatest number. Table 3. Descriptive statistics for convergence criteria for the four sampling methods.

Criterion Sampling method

Number of runs to reach convergence ANOVA p-val. Mean Std.

Dev. Minimum Maximum

Equation 1 TRTET = 0.01

Simple Random 137.0 18.3 84 187

0.000 Stratified 122.8 16.0 70 156 Inverse Stratified 130.1 14.5 85 160 Halton 145.9 21.0 84 192

Equation 2 TRSD = 0.1

Simple Random 107.0 25.8 63 180

0.000 Stratified 106.4 23.1 67 170

Inverse Stratified 105.2 20.2 66 143

Halton 121.6 35.8 64 194 Equation 3 TRERD = 0.01

Simple Random 65.9 9.6 44 91

0.034 Stratified 63.5 10.7 41 90

Inverse Stratified 61.5 10.7 42 94

Halton 64.4 10.6 44 90 Equation 4 TREPC = 0.01

Simple Random 107.0 25.8 63 180

0.001 Stratified 106.4 23.1 67 170

Inverse Stratified 105.2 20.2 66 143

Halton 121.6 35.8 64 194 Equation 5 TRSC = 0.01

Simple random 38.3 3.9 31 51

0.453 Stratified 38.2 3.9 31 48

Inverse Stratified 38.8 3.5 31 50

Halton 37.9 4.0 30 52

Similarly, Figure 7(b) illustrates the box plots of the number of runs required to meet the Equation 2 convergence criterion which clearly indicates that the Halton sampling approach requires the most simulation runs to reach convergence. In Table 3 the ANOVA test indicates that there is a significant difference between sampling methods since p-val. = 0.000. A second ANOVA test is used to compare

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the Simple Random, Stratified and Inverse Stratified results. This second test indicates that there were no statistical differences between the remaining sampling methods (p-val. = 0.859 > 0.05).

(a)

(b)

(c)

(d)

(e)

Figure 7. Number of runs required to meet the convergence criteria given in Equation 6 for the four sampling methods: (a) TRTET = 0.01; (b) TRSD = 0.1; (c) TRERD = 0.01; (d) TREPC = 0.01 and

(e) TRSC = 0.01. The ANOVA tests for the Equation 3 convergence criteria indicates that there is a significant difference between sampling methods (p-val. < 0.05). Table 3 and Figure 7(c) indicate that the Inverse Stratified sampling approach took the least number of runs to reach convergence. A second ANOVA test is used to compare the Simple Random, Stratified and Halton results. This second test indicates that there were no statistical differences between the remaining sampling methods (p-val. = 0.272 > 0.05). The ANOVA tests for the Equation 4 convergence criteria indicates that there is a significant difference between sampling methods (p-val. < 0.05). Table 3 and Figure 7(d) indicate that the Inverse Stratified

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sampling approach took the least number of runs to reach convergence while the Halton sampling approach took the greatest. For the Equation 5 convergence criterion the ANOVA test shows there is not a significant difference between sampling methods since p-val. = 0.453. A summary of the proposed results is shown in Table 4 where overall the Stratified and Inverse Stratified sampling methods require the least number of simulation runs to converge while the Halton sampling method requires the greatest number. An alternative approach could be comparing the four sampling methods using the five criteria at the same time. However, this approach has not implemented in this paper as it can be strongly biased by the most constraining criterion. For instance, considering the proposed case study, it is possible to observe that such analysis will coincide with the results obtained for the first criterion as it is the one requiring a larger number of simulations runs to converge (see Figure 7).

Table 4. Comparison of the four sampling approaches to identify the least (L) and greatest (G) number of simulation runs or the absence of a difference (=) and ‘n’ indicates neither least nor

greatest number of simulation runs.

Criterion Simple Random Stratified Inverse Stratified Halton Equation 1 n L n G Equation 2 L L L G Equation 3 G G L G Equation 4 n n L G Equation 5 = = = =

3.3.2 Variance of the Total Evacuation Time

The four sampling methods are compared in this section to assess if there is any difference in terms of the 18,000 simulated total evacuation times for each sampling method. Figure 8 illustrates the box plots of the total evacuation times while the descriptive statistics of those times and the F-test is shown in Table 5. The F-test is used to compare the variance of the Stratified, Inverse Stratified and Halton sampling methods against the variance of the Simple Random method. The results indicate that the Halton sampling method has the highest variance while the Stratified sampling method has the lowest variance while there is no difference between the variance of the Simple Random and Inverse Stratified (p-val. > 0.01) sampling methods. This indicates that the Halton method is more effective at obtaining sample values from the extreme limits of the distributions.

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Figure 8. Simulated total evacuation times for the four sampling methods.

Table 5. Simulated total evacuation times for the four sampling methods (n/a = not applicable)

Sampling method Total evacuation times (s)

F-test p-val. Mean Std. Dev. Variance

Simple Random 37.25 2.57 6.58 n/a Stratified 37.18 2.35 5.52 0.000* Inverse Stratified 37.72 2.60 6.76 0.052* Halton 38.39 2.68 7.17 0.000*

*Comparison with the variance from the Simple Random sampling method.

When comparing the average total evacuation time from the trial evacuations against the simulation it has been found that Evacuationz over-predicts by a factor of around 1.4. The difference between the predicted total evacuation times and the experimental times is related to the modelling assumption that the door was on a self-closer during the trial evacuations. For the simulations an initially closed door which is on a self-closer adds an additional delay to the transfer of agents to the exit node. The purpose of this paper is not to provide a direct benchmarking comparison between the model and the experiments so that the difference in total evacuation times is not a critical element in terms of the analysis of the sampling methods. 4. CONCLUSION

This paper compares the impact of sampling methods on evacuation simulations. It investigates how several sampling methods affect: (1) the convergence of the evacuation simulations and (2) variance of the total evacuation time of those simulations. To answer those points, this paper uses the five convergence criteria introduced in refs. [14,30]. A methodology to compare sampling methods is proposed and applied to four sampling methods, namely Simple Random, Stratified, Inverse Stratified and Halton. Those four sampling methods have been compared using the evacuation scenario of a university classroom including 13 participants and the sampling methods were used to generate the reaction time and the travel speed of agents in a simulation tool. The simulation results indicate that the Stratified and Inverse Stratified sampling methods require the least number of runs to meet the convergence criteria while the Halton requires

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a greater number. However, from the analysis of the variance of the total evacuation times it is possible to observe that the Halton distribution is the one having the highest variance and is thus more effective at sampling extreme values from distributions than the other sampling methods. The results are fundamental for evacuation model development showing the benefit of integrating several sampling methods in existing and new evacuation models as they can generate different convergence results and different values of variance. However, future studies are required to verify if similar trends are observed in different types of evacuation scenarios, building geometries and evacuation procedures. This paper presents the results from eight repeated trial evacuations from the same space using the same group of participants. Having data from repeated trial evacuations is rarely reported in the literature but necessary for evacuation model benchmarking [14]. However, the proposed scenario is limited by the relatively low number of participants and the simplicity of the room geometry. The data was collected during a post-graduate teaching course and so the number of participants was a function of the number of enrolled students. As such, this was a variable that could not be controlled in the experiments. Further studies are needed providing results of repeated trial evacuations in other buildings, including more evacuees. The results show that the use of different sampling methods affected the number of simulations required to reach convergence a comparison between the measured. However, the simulated total evacuation times shows that Evacuationz consistently over-estimates by a factor of 1.4 on average if a door is simulated as being on a self-closer. The results of this paper aids users of evacuation simulation tools, such as risk analysts, in their potential selection of the most appropriate sampling methods depending on the simulation goals whether that be in reaching a result that requires fewer runs or having a sampling technique that is more likely to select values at the extremes of distributions. This work provides some useful reference guidance for future generations of probabilistic safety assessments of complex buildings, structures or sites such as such as nuclear power plants, chemical plants and hazardous waste facilities. ACKNOWLEDGEMENTS

The authors would like to thank the participants for agreeing to be involved in this study and for the useful comments on a draft of the paper made by Kevin Frank from BRANZ. REFERENCES

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APPENDIX different occupant-evacuation time curves, then the Euclidean Relative Difference (ERD), the Euclidean Projection Coefficient (EPC) and the Secant Cosine (SC) of these two vectors are shown in Equations A.1, A.2 and A.3.

The pi parameter is the index of the xi component if pi is referred to x while s is the skip parameter.

𝑇𝑇𝑅𝑅𝑆𝑆(𝒙𝒙,𝒚𝒚) = �∑ (𝑥𝑥𝑖𝑖 − 𝑦𝑦𝑖𝑖)2𝑐𝑐𝑖𝑖=1∑ (𝑦𝑦𝑖𝑖)2𝑐𝑐𝑖𝑖=1

Equation A.1

𝑇𝑇𝐸𝐸𝐸𝐸(𝒙𝒙,𝒚𝒚) =∑ (𝑥𝑥𝑖𝑖𝑦𝑦𝑖𝑖)𝑐𝑐𝑖𝑖=1∑ 𝑦𝑦𝑖𝑖2𝑐𝑐𝑖𝑖=1

Equation A.2

𝑆𝑆𝐸𝐸(𝒙𝒙,𝒚𝒚) =∑ (𝑥𝑥𝑖𝑖 − 𝑥𝑥𝑖𝑖−𝑠𝑠)(𝑦𝑦𝑖𝑖 − 𝑦𝑦𝑖𝑖−𝑠𝑠)

𝑠𝑠2(𝑝𝑝𝑖𝑖 − 𝑝𝑝𝑖𝑖−1) 𝑐𝑐𝑖𝑖=𝑠𝑠+1

�∑ (𝑥𝑥𝑖𝑖 − 𝑥𝑥𝑖𝑖−𝑠𝑠)2𝑠𝑠2(𝑝𝑝𝑖𝑖 − 𝑝𝑝𝑖𝑖−1) 𝑐𝑐

𝑖𝑖=𝑠𝑠+1 ∑ (𝑦𝑦𝑖𝑖 − 𝑦𝑦𝑖𝑖−𝑠𝑠)2𝑠𝑠2(𝑝𝑝𝑖𝑖 − 𝑝𝑝𝑖𝑖−1) 𝑐𝑐

𝑖𝑖=𝑠𝑠+1

Equation A.3

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