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On performance measures for evacuation systems
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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH
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ELSEVIER
European Journal of Operational Research 85 (1995) 352-367
Theory and Methodology
On performance measures for evacuation systems Gunnar G. I~v~s Department of Mathematics, Universityof Oslo, P.O. Box 1053, N-0316 Oslo, Norway Received July 1993; revised December 1993
Abstract
How can a large, complex building with many occupants be evacuated safely? Before this question can be addressed it is necessary to define what 'safely' means, and to define some relevant measures which can be used to characterize the safety level. This paper studies different performance measures for evacuation systems. The primary measures are related to accident effects, e.g. the expected number of fatalities, and to evacuation time, e.g. the expected time needed to evacuate k persons out of the building. The paper presents a model of the evacuation system and shows how queueing network theory and simulation methods can be used to calculate the performance measures. The paper includes an example and a brief description of a real case study. Keywords: Evacuation; Modelling; Simulation; Queueing network; Human behaviour
I. Introduction
Hazardous situations may occur during a building's life, situations where it is necessary to evacuate the building population in order to take care of its safety. Several accidents over the last decades have m a d e it clear that evacuation systems are not always able to supply the required 'level of safety'. The goal of this p a p e r is to define, describe and discuss some relevant measures of evacuation system performance. The measures are used to describe how well the evacuation system performs, relative to its goal of 'safe' evacuation. A n evacuation system (ES) is a complex system of many 'components': The physical environment in which the population circulates is called a building. The building is composed of elements (rooms, corridors, stairs) which will be called rooms. T h e layout of the building is determined
by the way in which these rooms are interconnected by doom. The population within a building also deserves closer attention. Differences in personal characteristics such as sex, age, physical ability, familiarity with the building, with safety equipment and with evacuation procedures, etc. may be very important. Potential hazards within or close to the building determine how likely it is that a rapid evacuation will be needed. Management decisions are important in the planning of an emergency evacuation system, and managers may also play an important role as 'on-line' decision makers in a real accident situation. Detection systems are required so dangers may be observed as early as possible, initiating alarm warnings which can be understood by the building population. ESs have to satisfy certain safety requirements of which there are typically three different categories:
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G.G. Levis/European Journal of Operational Research 85 (1995) 352-367
1. Qualitative ideal goals, e.g. evacuation should be safe. 2. Quantitative requirements on system level, e.g. the total evacuation time should be below 10 minutes. 3. Detailed requirements to ' c o m p o n e n t solutions', e.g. doors should be wider than 1.2 meters. The first category requirements are too imprecise to make them useful for evaluation purposes. The third category requirements, frequently used (in the past) to evaluate ESs, are easily evaluated, but one does not have any guarantee that good components necessarily compose a good system. Based on these observations, there is a growing interest in the second category requirements, which will hereafter be called functional requirements. To evaluate if systems cope with functional requirements it is necessary to measure the performance of the system. Performance measures of ESs describe the likely behavior of the evacuation process, i.e. how the ES is able to function. Therefore, typical and important examples of such measures are • expected n u m b e r of fatalities; • expected n u m b e r of safe evacuees at time t; • expected time needed to evacuate k persons. These measures will be related to a specific set of scenarios, describing the accident, the population distribution, etc. In this paper, we will study such measures further, and give them precise mathematical definitions. In the literature, little has been written about measures of ES performance. No deep discussions about different measures with their advantages and limitations are known to this author. However, several sources have discussed different ' m e a s u r e m e n t methods', such as network flow models, either deterministic (e.g. [1,2,4,10]) or stochastic (e.g. [12,24-26]), or simulation methods (e.g. [3,6,13,23]). The cited references are concerned with a few performance measures only, and some of them are mainly oriented towards optimization of the ES with respect to some small subset of the performance measures. This p a p e r has the following organization: Section 2 presents a mathematical model of the ES, and stochastic variables related to system perfor-
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mance are defined. In Section 3, several performance measures are defined, and, in Section 4, these measures are discussed with respect to their usefulness. This p a p e r does not present detailed methods on how to calculate the performance measures analytically, but some ideas are presented in Section 5. Examples are given in Sections 6 and 7 to illustrate some of the measures presented in the paper.
2. The model
2.1. Parameters A building composed of N rooms and M doors will be modelled as the network G(V, E), consisting of the nodes u 1. . . . . u N in the set V and M links in the set E. We write eij = (vi, v ) ~ E if and only if there is a link between nodes ui and vi. The nodes represent rooms, and the links represent doors. The nodes are divided into three subsets: Source nodes V1, transportation nodes V 2 and exit (destination) nodes V 3. The nodes in V3 are absorbing, i.e. persons will not move from a n o d e i n V 3 t o a n o d e i n V 1 U V 2. The population of finite size K is modeled as K separate individuals moving around in the evacuation network G, from nodes in V1 to nodes in V3. Their movement is influenced by human decisions (initial response and route choice logic) and walking ability (see L~v~s [19]). Many parameters may be used to describe the persons and their behaviour, but it is outside the scope of this p a p e r to present them here. The persons are located in the building, at time t = 0, with probability Pu, according to the rule K u = ( K l u ..... K N u ) , where Kiu is the initial number of persons in node i. The index u is a m e m b e r in the set ~ of indices for all possible location rules. We syncronize our clocks so that at time t = 0, the hazardous situation starts, e.g. when a flammable gas is ignited (syncronization is important, but the choice of time-origin may be redefined). Assume that it is possible to define a reasonable time-horizon, T ~ (0, oo], for the evacuation process, so that there is high probability that the process has terminated before T. We
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study the evacuation process in the time-interval [0, T] only. It is natural to use a continuous rather than discrete time model, since the evacuation process is of a continuous nature; it is hard to define a suitable time step if a discrete approach should be applied. However, this discussion does not really relate to the performance measures defined in the paper, but it relates to the choice of solution methods. Obviously, certain - but not all - solution methods use numerical methods which apply discretization, such that the modelling gain is in fact only a formal one. The accident (or hazard) is represented as a set of possible scenarios. The accident follows scenario Hm(t) = (nxm(t) . . . . . nNm(t)) with probability qm, in which case the accident effects in node i at time t can be expressed a s Him(t). For the index m we have m ~.¢t', where Jr" is the set of indices for all possible accident scenarios. With inspiration from multistate reliability theory [22], we require that H/re(t) e {0, 1, 2 . . . . . MH}, where Him(t) = 0 iff no accident effects are present, and Him(t) = M H iff the accident will kill anyone present in node i at time t. Let d(i, j) be the shortest distance between (the centers of) node v i and node vj. If no connection exist, d(i, j ) = oo. Let us define d i as the shortest distance from node v; to some exit v i
v3, d i=
min d(i, j). j:v~e V3
The above definition of d i is based on the underlying assumption that persons walk the shortest path to the closest exit. However, it is easy to redefine d i if each person is believed to walk the shortest path to one specific exit (e.g. exit l, according to an evacuation plan) or that each person has to follow a specified path of movement (e.g. visit nodes i, Jl, J2 . . . . . Jn, 1). In general, we may also consider the distances to depend on changes in the environment, e.g. accident effects, and write di(t).
2.2. Stochastic variables In the following, several stochastic variables are defined. They will later be used as the basis
for the definition of relevant performance measures. Many of the stochastic variables are interdependent in a way that might make traditional statistical solution methods unsuitable, since they normally require that variables are independent, or at most associated. However, the dependencies among the stochastic variables do not influence the coming definition of the performance measures. The dependencies of the stochastic variables have to be clarified before a solution method is chosen and applied. The number of persons in node i at time t will be noted X~(t). Let the aggregate system state X(t) = (Xl(t) .... , X u ( t ) ) be an element in the finite space X of all possible system states. Define Ai(t) to be the number of arrivals at node i before time t. Define Zi(t)= X~(O)+Ai(t), as the n u m b e r of visits to node i within time t. Let Zij(t) be the number of persons using link eij within time t. It will be shown in the rest of this section that many of the most interesting performance variables (and measures) are related, more or less directly, to the process {X(t), t ~ [0, T]}. Hence, knowledge about the stochastic process X(t) will often be sufficient.
2.2.1. Accident effects Let H(t) = (Hx(t) . . . . . HN(t)) describe the state of the accident. From the set of all possible accident scenarios, a specific scenario is 'drawn' according to the probabilities qm; if scenario m is selected, then H(t) = Hm(t). A person will be exposed to accident effects depending on the route he follows and the length of time he spends in the different nodes. Let Kk(t) be an indicator taking value 1 if person k is alive at time t and value zero otherwise. Define K(t) = E~ Kk(t) as the total number of persons being alive.
2.2.2. Evacuation time Define M(t) as the n u m b e r of persons who evacuated within time t,
M(t)= E x/(t). i:oi~ V3
G.G. LCvSs/ European Journal of OperationalResearch 85 (1995) 352-367 The evacuation time for one person is defined as the total time he needs to get out of the building (i.e. to arrive at a node in V3), including reaction time, walking time and waiting time in queues. The individual evacuation time of exiting person n u m b e r k is Tk. We define T ~ = T if person k does not evacuate before T. Hence, the times are ordered so that T 1 < T 2 <_ . . . <_ TK. Observe that Tk = inf{t e [0, T ] : M ( t )
> k}.
2.2.3. Queueing and waiting The last time when s persons or more are present in node i is Li(s) = sup{t ~ [0, T]:Xi(t) >s}. Let L ( s ) = m a x Li(s) , the maximization taken over all nodes in V1 U V2. Node i is busy with s persons or more for Y/(s) time units, and the total time spent in node i when at least s persons are present is Ii(s), hence defined as
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n u m b e r k, k = 1. . . . . Zi(T). The average time a visitor spends in node i is called W/, easily seen to be
li(O) W/
Zi(T) "
The maximum waiting time will be called W/max . Let Qij(t) be the number of persons waiting to use link eij at time t. This variable should be interpreted as the size of the " q u e u e " in front of a door. As we defined ~ and W/, based on Xi(t), we can define similar variables based on Qij(t).
2.2.4. Network distances The total remaining route length D(t) and the average remaining route length Da(t) are defined as N
D(t) = ~_~X i ( t ) d i ( t ) , i=l
Yi(s) = fo~I(Xi(t) > s) dt, Da(t)-
O(t) K-M(t)
Ii(s ) = foTSi( t ) I( Xi( t ) > S) dr, where I ( . ) equals one if (-) is true, zero otherwise. The average population size is obviously closely related to a subjective choice of time interval over which the n u m b e r of persons is averaged. Let y ( s ) be the length of the chosen time interval, and let y ( s ) equal Y/(s), t i ( s ) o r L(S), corresponding to three different types of averages. The following definition of the average population size X,(y, s) of node i is only meaningful if y ( s ) > O,
li(s) s) =
With s = 0, this expression corresponds to a standard time average over [0, T]. However, we get more interesting results with s = 1; therefore, if s = 1, we may simplify the notation by omitting the p a r a m e t e r - for example, L i = Z i ( 1 ) and X~(y) = Xi(y, 1). The maximum population size is
Si max= suPtSi(t ). Let W/k be the time spent in node i by visit
We define No(d, t) as the number of persons with a longer minimum remaining distance than d to walk at time t, N
ND(d, t) = ~, X i ( t ) l ( d i ( t ) >_d). i-1 2. 2. 5. Network redundancy Define V~ as all nodes in V1 w V2 which are neighbours to a node in V3. A viVa-path is a sequence of neighbouring nodes from vi to a node in V~. A viVa-path is safe at level s at time t iff H j ( t ) < s for all nodes vj in the path (in particular, a path is safe if it is safe at level 0). Two viVa-paths are independent iff all their common nodes are in ui I,_JV3. Let fi(t, s) be the n u m b e r of independent viVa-paths that are safe at level s at time t. The average redundancy number F(t, s) is defined as
F ( t , s) =
Y:i:v, ev, uv2Xi( t) fi( t, s) K-M(t)
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3. Performance measures
q~sa = Probability distribution of D(t). C1)5b =
In order to give a good and valuable description of the ES it is necessary that we consider the whole range of possible values of the variables we have defined. Therefore, we will typically define the relevant performance measures as the expected value of a stochastic variable, its distribution, quantiles, etc. The measures defined in this section are related to accident effects, evacuation times, queueing, network distances and network redundancy. A discussion about the usefulness of the different measures can be found in the next section. The performance measures, n a m e d (~la, (~lb, (~2a, (~2b . . . . . are listed below in six different categories whose use is discussed in Section 4. 1. These measures are related to K(t), the n u m b e r of u n t r a p p e d individuals, defined in Section 2.2.1. ~ l a -~- Probability distribution of K(t).
t~lb = EK(t). 2. These measures are related to M(t), the n u m b e r of safe evacuees at time t, defined in Section 2.2.2. ~2a = Probability distribution of M(t). (1)2b =
EM( t ).
~2c = Quantiles (upper) of M(t). 3. These measures are related to the evacuation time Tk, defined in Section 2.2.2. q~3a = Probability distribution of T k. (I)3b =
E T k.
4. These measures are directly related to Xi(t), defined in Section 2.2. ¢~4a = Probability distribution of Xi(t). (~')4b =
EXi( t )"
ED( t ).
~5¢ = Probability distribution of Da(t).
CI)sd = EDa( t ). cI)se = ENo( d, t ). 6. These measures are related to the average redundancy n u m b e r F(t, s), defined in Section 2.2.5. ~6a = Probability distribution of F(t, s). t~6b ~
EF(t, s).
4. Discussion
The measures carry information about the ES, hopefully, and some of the measures are better than the others, because they are easy to understand or easy to calculate. Since many elements in a complex ES are subject to random fluctuations, it is often a problem that the stochastic variables, e.g. T k or M(t), have very high variances, reducing the 'information value' of the performance measures. One way to overcome this problem is to measure the performance of the ES under given 'initial conditions', i.e. that a specific population localization K , and an accident scenario H m is known. The performance measures may be studied for all interesting combinations of u ~ ~' and m ~ ¢ ' . In this way it is possible to identify the 'critical components' in the ES, whether it is the accident development, the initial population distribution or the evacuation movement process itself. This simplification, with fixed u and m, is applied in Sections 5.1 and 6 (even though it is not required).
~4c = Quantiles (upper) of Xi(t). (~ 4d = CI)4e =
EYi( s )" ELi(s). EI~s). EXi(y, s).
¢I)4f = ¢I)4g = tTJ54h = E X i max. (~4i =
EZ__L(t )"
(~4j = E W i .
(/)4k = EW/max. ¢~41 =
EQij(t)"
5. These measures are related to the total remaining distance D(t), defined in Section 2.2.4.
4.1. Measures of category 1 The safety of the building population is our main concern. Therefore, m e a s u r e s ~)l" are two of the most important measures, especially when evaluated at t = T. However, m e a s u r e (~lb will often be a poor measure since the expected value is not a good measure of centrality for a skew distribution. Using qbla to calculate the probability that more than k persons are dead, is much more informative.
G.G. Levers/European Journal of Operational Research 85 (1995) 352-367
In practice, the measures q~. are used as 'filters' in the sense that their values give an idea whether the ES performs bad or well. Even if the ES seems to perform well, one should study the other performance measures to see if the performance is really OK, or if it could easily be improved. This is in line with the idea that the risk for the building occupants should be 'as low as reasonably practicable' (ALARP), an idea influencing much of m o d e r n risk analysis. 4.2. Measures o f category 2
Measures related to time are very important because most hazards evolve over time so that there is limited time available for safe escape. Let TA be the time available to escape the building. Obviously, TA is a stochastic variable, and its distribution will not be known to the building occupants (or anyone else). Nevertheless, based on knowledge about accidents and their potential development, it is customary to determine TA as a deterministic parameter. It is also possible to interpret TA as the time accepted by management, e.g. one functional requirement may be that 98% of the population should have evacuated within time TA. Measures q~2. tell quite a lot about the behaviour of the ES, they are easy to interpret, and they are a good basis for an evaluation of the ES's efficiency. Measures ~2b and ~2c carry valuable information over the whole time-span [0, T], especially in the first part of the interval, while (J~)za is the best measure for high t-values, e.g. t = TA. Observe also that E M ( t ) ~ E K ( t ) as t ~ oo; hence, there is a strong connection between measures q~l. and q~2.4.3. Measures o f category 3
The expected evacuation time E T k will often be a poor measure because of the following reason: If T is finite and Pr(T k < T ) < 1, then the evacuation process is truncated and E T k will be difficult to interpret. If T = oo and P r ( T k < ~) < 1, then we also get E T k = ~, which is not really very informative. This corresponds to an accident scenario where
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there is a positive probability that ( K - k + 1) persons will be trapped (dead or unconscious). Since there will always be a positive probability that at least some persons may be trapped otherwise it was not necessary to evacuate - we know that E T K = oo. One way to overcome this difficulty is to study the expected average evacuation time for the group of 'survivors'. However, as this may be a little bit confusing, it is difficult to interpret the information carried by measure 1~3b, and its use is not recommended. The probability that all the K persons have evacuated within the available time is an important number. This probability may be of interest at other times and other values of k too, making t~)aa an interesting measure. 4.4. Measures o f category 4
Even if queueing may seem to be of minor importance, several sources claim that queueing is in itself dangerous, maybe causing people to crush each other, giving rise to panic, etc. Therefore, we choose to describe an evacuation system also by some queueing measures. The measures of this category may be related to the concept of 'level of service', as defined and used by Fruin [5] and Mori [21]. These authors give a method for measuring performance quality, related to the comfort and safety of the moving crowd; basically, the density of people is the main determinant for the service level. When the capacity C i of node i is known, it is easy to use the measures based on Xg(t) to find the expected level of service (at each time t) and its probability distribution. The evacuation process {Xi(t), t > 0 } tells much about the conditions in node i. The probability distribution of Xg(t) and its expectation and quantiles are relevant measures (q~4a, ~4b and ~4c), but it will often be sufficient to consider measures ~4d to ~4g, which are all 'extracts' of information included in ~4~(~4d and t~4e are measures of the length of time when a node is used by s or m o r e persons. A high value to these measures means that the node is important in the sense that its 'failure' is likely to have consequences for the persons. For most
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applications, measure ~4d is more informative than measure t~4e. The expected total time spent in a node (~4f) measures the 'popularity' of the node, and may suggest something about the importance of the node. The average population size in node i (measure ~4g) is related both to different levels s of population size and to different time periods Y~, Z i and L. For example, .~i(Y/, s) is the average number of persons in node i given that there are at least s persons in the node. Another example, Xi(Li, 0) is the average number of persons in node i over the whole time period [0, T]. If it is likely that persons are hurt in overcrowded situations, it is interesting for us to know E S i max (measure (~)4h), indicating if the conditions are untolerable or not. Measure gai indicates if a node is visited by many persons, and measure ~4j give us the average time spent in the node by each visit; these are helpful measures when one tries to identify 'bottlenecks' in the system.
4.5. Measures of category 5 These measures, related to the remaining distance D(t), are interesting measures of the evacuation system performance, the complexity of the evacuation network, and the queueing problems present in the system. Define Dm(t) as the lower limit for D(t) given that the ES performs perfectly. Starting with Din(0) = D(0), and assuming that all persons walk immediately and with unimpeded speed to the nearest exit, it is easy to calculate Dm(t) for all t ~ [0, T]. If ED(t) is close to the lower limit, then the evacuation process is performing well. The expected value of No(d, t) tells something about the size of both the building and the population, and hence about the potential for fatal outcome of an accident.
4.6. Measures of category 6 For the persons in the building at time important that there exist safe paths out building. Measure ~6b gives information the average number of safe independent
t, it is of the about paths
for the persons in the building. This number will normally fall as t grows, i.e. the situation will be worse for the persons, since the accident develops and make some evacuation routes 'less attractive'. If EF(t, 0) equals zero, then the E ( K - M ( t ) ) persons in the building have a serious problem, since there are no more safe routes to follow. Then, it is interesting to know EF(t, s) to see if there are any paths which are safe at level s. Measures 4 6 . are well suited to tell whether the ES has high enough redundancy or not. This was noticed by [17] who used similar measures to describe some buildings in the USA.
5. Calculation
of measures
Many of the measures are hard to calculate, and no attempt will be made in this paper to arrive at nice analytical solutions. With a large building ( N large) and many building occupants ( K high), any calculation principle based on realistic assumptions is likely to run into computational problems. In Section 5.1 it is shown how an analytical solution may be derived - although no simple answers are given - and in Section 5.2 a simulation model is described.
5.1. An analytical approach A short sketch of an analytical solution method, based on ideas from queueing network theory (see [14, Section 4.8] or [8, Chapter 4]), is included in the following. An overview over other analytical solution methods can be found in [18]. A queueing system is a system where 'customers' arrive, spend some time (usually, but not necessarily, to get some kind of service), and then leave. Each node in G is modelled as a queueing system, inspired by Jackson [11]. Obviously it can be useful to describe a system as a queueing system even if 'queueing' in the traditional sense - waiting in lines - is not a (big) problem. Actually, if queueing at a station is certain (i.e. someone is always waiting), the departure process from the station is not related to the input process, and the apparatus of queueing theory is not really
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needed. Therefore, the presented stochastic approach is of particular interest when N is large and K is relatively small, corresponding to a large but not overcrowded building. In the opposite case, when a building is heavily overcrowded, traditional 'hydraulic' flow models will often suffice as a method for obtaining the performance measures. As noted before, let the aggregate system state X(t) = (Xl(t) ..... XN(t)) be an element in the finite space X of all possible system states. Note that X(t) does not necessarily contain all information about the actual state of the system. The number of distinguishable states of X(t) is equal to the number of ways in which one can place K persons among N nodes, known to be the binomial coefficient N+K-1 K-1
× P r ( m o v e m e n t from / t o
=p(k,t) N
+ ~ ~ P(kij, t)P(i, t, h, k)r(i, j, k). i=lj=l
j~i
(1) Rearranging in (1), taking the limit as h ~ O, changing the order of summation and limit, we find
Op( k, t) at
N ~ (limP(i,t,h,k)/h}p(k,t) i=1 N
N
+ E E(limP(i,t,h,k)/h}
p(k, t) as below:
p(k, t)
t) = k l , . . . , X u ( t ) = kN)
= Pr(X(t) =k). The system state may change in one way only, namely when a person moves from one node to another. We assume that in a very short time-interval only one such change may take place. For the rest of this subsection, let us assume that the process is Markovian. Let P(i, t, h, k) be the probability that a person leaves node v i in the short time-interval [t, t + h ) given the present state k of the evacuation process. Assume that the moving person selects his new node vj according to the probability r(i, j, k). We require that )Zk~ = K and define k~j = k except that its i-th component is k~ + 1 and its j-th component is k j - 1. Then, the forward equation, describing the change in the population process from t to t + h, equals
p(k, t +h) =p(k, t ) P r ( n o change in [t, t+h)) N
E P(kiy, t)
i-lj=l
j¢i
~h~0
j~i
Define the state-probability
+ E
t+h))
i=1 N
i=1 j=l
N
j in It,
1- ~P(i,t,h,k)
)"
= Pr(X,(t)
359
Xr(i, j, k)p(kij , t).
(2)
The initial distribution, p(k, 0), is already known since p(k, O)= E, ~.~I(K u = k) Pu. Given p(k, 0) and (2) with P ( - ) and r ( ' ) specified, it is possible to study, at least by numerical methods, the transient development of p(k, t) in the timeinterval [0, T]. Grassmann [7] and Gross and Miller [9] solved the problem under the assumption that P(i, t, h, k) = t~ih and r(i, j, k) = 7rij. Assume now that p(k, t) is known for all vectors (k, t). Then it is easy to see how some of the performance measures can be quantified. Measure q~4b is identified as
EXi(t) = E ki p(k, t) k~x
and, measure tbaa, the distribution of be found directly as
Xi(t), can
Pr(Xi(t ) <_k)= E I(ki<-k)p(k, t). k~ X
Similarly, we can find measure q02b, the expected number of safe evacuees at time t, by
E M ( t ) = ~ ( ~, ki)P(k , t). kEX i:viEV 3
Several of the other measures can be related to
p(k, t) in the same fashion.
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5.2. Simulation methods Several simulation tools have been developed to evaluate the effectiveness of ESs. These tools differ in complexity, input requirements, underlying assumptions, and specific application areas. An overview of some existing methods has been given by Watts [27], and a discussion about different models has been presented by l_~vfis [18]. E V A C S I M is an evacuation simulation program under development in Norway and France. A short introduction to E V A C S I M can be found in [3], and a detailed description of the 'pedestrian flow' is given in [19]. It is not within the scope of this p a p e r to go into details about any simulation program, but a short description of E V A C S I M will be given. E V A C S I M simulates the evacuation process on a microscopic level, in the sense that each person is treated as a separate 'flow object'. Each person is characterized by a set of attributes, e.g. initial position and characteristic walking speed. The building is modelled as the graph G, with nodes and links, also with their own attributes, e.g. capacity of a node and width of a link. The accident scenario is represented as a dynamic scenario, specifying time and type of changes in accident effects in each node. E V A C S I M performs a process-oriented discrete-event simulation [16], using the QNAP2-tool (Queueing Network Analyzing Package). This means that each person is followed closely and that critical events in his movement process are identified. These events are then scheduled into an event calender, and the program continues by handling the most imminent event on the event calender, corresponding to an event (activity/movement) for another person. E V A C S I M performs R replications of the simulation from the same initial state (i.e. the same building topology, the same personnel localization, the same accident scenario). On each replication, values to the personal attributes are sampled from the appropriate distributions, and K individual m o v e m e n t processes are 'started'. Each movement process is stochastic, e.g. because the persons are allowed to make ' r a n d o m ' choices among alternative routes [20]. Process k
interacts with the other K - 1 processes: when 'it' evaluates the actual walking speed of person k; when the route-choice-probabilities are established; when a link is used (to control that only one person uses the same link at the same time); when a person tries to enter a new node (to see if there is more free capacity in the node); etc. Each replication is finished when all of the K movement processes have stopped or when the maximum time T is reached. Statistical results are collected during the simulation [15], and estimates to the performance measures are presented as the main results. In addition, E V A C SIM produces data which can be used to animate the evacuation process on a computer screen.
6. Example To get more familiar with the different performance measures, an example will be presented. The building network G is shown in Fig. 1; the nodes are numbered i = 1 , . . . , 25 as indicated in the figure. Each link has length d(i, j ) = 10 meters. We study a population of K = 100 persons who evacuate from room 1 to room 25 (which is considered safe), i.e. V1 = {vl}, V2 = {u2,...,u24 } and V3 = {v25}. In order to simplify, no accident scenario is specified, meaning that all persons will succeed to evacuate sooner or later. We assume, in this example, that all persons start to evacuate immediately at t = 0, and we follow the evacuation process until every person has exited, i.e. T = ~. The walking speed of the persons is as-
25 20 13
15 10
1
2
3
4
5
Fig. 1. Network of nodes (dots) and links (lines).
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361
No. of p e r s o n s 10 0
I.
~ - - - ' , .......... ,...........................................
-
Z.; /
100
200
T i m e t in seconds. Fig. 2. Measures @2b and @2c: The average value of M(t), with its observed 2% lower and upper quantiles.
sumed to be lognormally distributed with mean 1.5 m / s and standard deviation 0.3 m/s. We assume that the persons are familiar with the building, being able to select paths leading them effectively to the destination v25. The simulation program EVACSIM has been used to simulate the evacuation process. In this example, R = 50 replications have been performed. Several interesting results will be presented in the following. However, since no accident sce-
@~.
nario is specified, measures and @6. are of no interest. Many of the performance measures are functions of time, and well suited for graphical presentations. Measures @2.: Fig. 2 shows the average of M(t), with its observed upper and lower 2% quantiles. From this curve it can be seen that it is expected that approximately 90 evacuees are safe (at node 25) after 160 seconds. The distribution of M(t) varies with time, and Fig. 3 shows the
Probability O.5
....
....
/~M(80)
/I
/. y( o> y"i j "',,, j" ! M ( l O 0 )
.... o,
/
1
/
lo
20
N u m b e r of persons. Fig. 3. Measure @2a: Empirical distribution of M(t) at t = 80, 90, 100.
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362
Cum. prob. (%)
Ts0
~.50
2OO
25O
T i m e Tk in s e c o n d s . Fig. 4. Measure
t~b3a: Empirical cumulative distribution of evacuation times, from left to right, 7"5o, Tso , T90 , T95 and
distribution of M(t) at t = 80, 90, 100. These curves can be used to obtain an estimate of measure 42a. For example, less than 26 persons are safe after 100 seconds with probability 0.98. Measures 43.: The distribution of evacuation time Tk is shown in Fig. 4 for k - - 5 0 , 80, 90, 95, 100. It is interesting to observe how variable the evacuation time of the last person is (T100) compared to the median evacuation time (Ts0). Measures 44.: Fig. 5 shows the average of
X13(/) together with its observed upper and lower 2% quantiles. For the nodes on the right-hand side of Fig. 1, the average of Xi(t) is presented in Fig. 6. Similar results are collected for all nodes. These curves give information about the conditions in the node, the node usage, the average density, etc. It can, for example, be seen that no persons are present in the center node (i = 13) after 160 seconds. Some examples of the measures 44d to 44S are shown in Table 1. One could
No. o f p e r s o n s lo
yi,Jv ,, ,, g \
., ,
,"2_A_JiJ.........
0 0
Tloo.
loo
Time t in seconds. Fig. 5. Measures ~4b and ~4c: Average value of Xz3(t) with its observed lower and upper 2% quantiles.
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363
No. of p e r s o n s 8
6 2O
4
0
2O0
100
T i m e t in seconds. Fig. 6. Measure qbab: Average of Xi(t) for transportation nodes on the right hand side of Fig. 1.
for example see that the persons spend more time in node 20 than in node 2, even though the nodes have the same number of visits; this is due to queueing phenomena, restricting the flow out of node 20 (only one link, 20-25, is used) more than the flow out of node 2 (two links, 2-3 and 2-7, are used). Measures qOs.: The average total remaining distance D(t) is shown in Fig. 7 together with its lower limit Din(t). We see that the persons need a minimum of 54 seconds to evacuate (corresponding to the lower limit Din(t)). In the average simulated case, however, it is seen that the population needs about 175 seconds to walk all the total distance. Hence, the population requires
about 3 times as much time to leave the building in 'reality' as in the 'idealized uncongested world'. This is due to two factors: The links impose restrictions on the flow, leading to queues, and the slow walkers need more time to walk 80 meters than the 'average walker'.
7. Case study A real case study for a Norwegian offshore oil production platform will be briefly reported here. The actual oil platform is a very large 'building' with about four kilometers of walk-ways, with a population ranging from 150 to 300 persons.
Table 1 Node i
1 2 3 4 5 10 13 15 20
Performance measures
t~4f
t~ad
t~4e
t~4g
t~4g
t~4i
(J~aj
~4h
Eli
EY~
EL,
E.~,(Y~)
EXi(L i)
EZi(T)
E~
EXimax
maxR X/max
5534 451 219 117 64 171 342 344 603
103.7 92.4 86.3 68.9 48.0 82.2 97.3 99.8 109.6
103.7 108.6 114.5 117.5 121.4 140.8 138.5 158.9 173.2
53.4 4.9 2.5 1.7 1.3 2.1 3.5 3.4 5.5
53.4 4.2 1.9 1.0 0.5 1.2 2.5 2.2 3.5
100.0 50.5 25.1 13.5 7.4 19.4 37.8 35.4 50.7
55.3 8.9 8.7 8.6 8.7 9.0 9.1 9.7 11.9
100.0 7.0 5.2 3.8 2.8 5.0 10.3 7.3 7.8
100 8 7 6 6 7 12 10 15
364
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Total distance \
\
loo
200
Time t in seconds. Fig. 7. Measure ~Sb: The average total remaining distance D(t) and its lower limit Din(t).
The platform is operated throughout the hours (12 hour shifts), but the majority of the crew work at daytime. Most of the persons are located within the living quarters (LQ). The rest of the workforce is spread around the platform in many different modules. Since large areas of the platform are not continuously manned, it was decided to locate the personnel in the network model according to a specially given probability distribution. The platform was modelled as a walkway network with 219 nodes, 314 links and 6 exits, each corresponding to a lifeboat with capacity to carry 50 persons. The purpose of the study has been to analyse the effectiveness of two different evacuation strategies, explained below: Strategy 1. All personnel move from their present location to a temporary safe area within the LQ, waiting for further messages. When the last person arrives, they all move to their respective (preassigned) lifeboat positions. This strategy will not be followed in the unlikely case that a serious accident occurs within the LQ.
Strategy 2. All personell move directly from their present location to their respective (preassigned) lifeboat positions.
The second strategy is obviously quicker, and the goal of the study was to find 'how much'. However, the first strategy is more manageable, making communication between the leaders and the personnel more reliable, which is of course important in an emergency situation. 'Gas and oil blowout in the wellhead area' is the single accident scenario which contibutes most to the total risk level. A blowout may - if it is ignited - lead to a pool fire at sea. Based on knowledge of the weather conditions, the well pressure, the sea surface water flow, etc. it can be found that - in this case - the lifeboats can be safely launched within the first 4 - 8 minutes. After this time, there is a large risk that there will be a fire below the boat. Performance measures related to time are therefore very important. The differences between strategies 1 and 2 can be measured even without a model of the accident. In the case study, the strategies were considered both for the blowout scenario and under 'normal' evacuation conditions. The latter will be described here. First, it is necessary to point out that the initial response patterns of the 'building occupants' are extremely important for the outcome of the evacuation process. This is in contrast to the previous, rather academic, example. In the case study it was assumed that each individual had his re-
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365
No. of p e r s o n s
Strategy 2
//'/ o
Strategy 1
50o
1000
T i m e t in seconds. Fig. 8. Average number of persons at lifeboats at different times for the two different strategies.
sponse time drawn from a lognormal distribution, where the mean and standard deviation varied between the different areas of the platform. Response from offices was assumed quicker than response from bedrooms or noisy workplaces. The persons were also assumed to communicate with each other, e.g. meaning that persons in the same node influence each others response times. For each strategy, 100 replicated simulations were performed with EVACSIM, assuming a con-
stant population size of 200 persons. A short summary of the main results are shown in Table 2. These numbers clearly indicate that the strategies are significantly different with respect to performance measures ~3.. The curves in Fig. 8 even better visualize the differences between the strategies, showing measure ~2b for both strategies. At an offshore platform, the time until the last person has reached the lifeboats is very important, because it determines how long time
Probability density rategy 2
60O
BOO
I000
T i m e t in seconds. Fig. 9. Probability distribution for the time needed to evacuate all persons under the two strategies.
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G.G. Levis/European Journal of Operational Research 85 (1995) 352-367
Table 2 Main results (All times in seconds) Person Average Confidenceinterval (95%) number evac.-time of average evac.-time Strategy 1 First 521 (489, 553) Median 632 (599, 665) Last 741 (708, 774) Strategy 2 First 44 (42, 46) Median 174 (173, 175) Last 447 (417, 477)
many persons have to wait before a lifeboat can be launched. Fig. 9 shows measure ~3a, the probability distribution of this time for the two strategies. The bimodal shape of the density curves is due to the fact that there is a r a n d o m n u m b e r of persons, normally one or none, in the most remote region of the platform. The final conclusion from the case study t e a m was that the effectiveness of the standard evacuation strategy (number 1) is significantly lower than that of strategy 2. Based on this, the oil company started an internal discussion about the validity of the results and about the possibility to draw conclusions of relevance to the offshore installation management. At the time of the writing of this article, the m a n a g e m e n t conclusions and decisions are not known.
8. Concluding remarks W h e n we study evacuation systems, we do so because we want to see if the building can be evacuated in an acceptable way. Hence, the performance measures form a basis for all quantitative studies of evacuation systems. If the performance measures indicate that the system will (or may) perform unacceptably, then some corrective actions are needed to 'speed up' the evacuation process (or 'slow down' the accident development). The performance of the evacuation system may be improved by one of the following actions: • Reduce reaction time of personnel (e.g. better detector- and alarm systems). • Reduce interpretation time (e.g. better messages to building occupants).
• Reduce walking time by improving the escapeway system: - shorter routes (e.g. new connections, new doors); - higher flow capacities (e.g. wider doors); - easy wayfinding (e.g. better marking and orientation help). • Train personnel (e.g. react correctly, walk faster, follow planned routes, etc.). • Improve organization of the evacuation process (e.g. 'optimal' routing of evacuees). • Delay accident development (e.g. sprinkler systems, fire-doors). • Minimize accident effects (e.g. use materials producing non-toxic combustion products).
Acknowledgements Many thanks to Professors Terje Aven and Bent Natvig for helpful discussions over the presented topic; their experience with reliability thepry, especially that of multistate systems and network flows, has been very inspiring. Thanks also to Jo Wiklund, Quasar Consultants, for useful ideas, critical comments, and for his kind permission to let me present the case-study, in which he was responsible for the modelling and simulations. Many thanks to a referee for valuable comments on a previous version of the manuscript. The author gratefully acknowledges the financial support of this work by the Royal Norwegian Council for Scientific and Industrial Research.
References [1] Chalmet, L.G., Francis, R.L., and Saunders, P.B. "Network models for building Evacuation", Management Science 28/1 (1982) 86-105. [2] Choi, W., Hamacher, H.W., and Tufekci, S., "Modeling of building evacuation problems with side constraints", European Journal of Operational Research 35 (1988) 98110. [3] Drager, K.H., LOves, G.G., Wiklund, J., et al. "EVACSIM. A comprehensive evacuation simulation tool", in: J. Sullivan (ed.), Proc. of the 1992 Int. Emergency Management and Engineering Conference, Florida, April 1992,
The Society for Computer Simulation (SCS), 1992, 101108. [4] Francis, R.L., "A uniformity principle' for evacuation
G.G. LOt,as/European Journal of Operational Research 85 (1995) 352-367 route allocation", Journal of Research of the National Bureau of Standards 86/5 (1981) 509-513. [5] Fruin, J.J., Pedestrian Planning and Design, Metropolitan Association of Urban Designers and Environmental Planners Inc., New York, 1971. [6] Glowacki, A., "Using simulation to prepare for emergency mine fire evacuations", in: J. Sullivan (ed.), Proc. of the 1992 Int. Emergency Management and Engineering Conference, April 1992, The Society for Computer Simulation, 1992, 79-83. [7] Grassmann, W.K., "Transient solutions in Markovian queueing systems", Computers and Operations Research 4 (1977) 47-53. [8] Gross, D., and Harris, C.M., Fundamentals of Queueing Theory, 2nd ed., Wiley, New York, 1985. [9] Gross, D., and Miller, D.R., "The randomization technique as a modeling tool and simulation procedure for transient Markov processes", Operations Research 32/2 (1984) 341-361. [10] Hamacher, H.W., and Tufekci, S., "On the use of lexicographic min cost flows in evacuation modeling", Naval Research Logistics 34 (1987) 487-503. [11] Jackson, J.R., "Jobshop-like queueing systems", Management Science 10/1 (1963) 131-142. [12] Karbowicz, C.J., and Smith, J.M., "A K-shortest paths routing heuristic for stochastic network evacuation models", Engineering Optimization 7 (1984) 253-280. [13] Kisko, T.M., and Francis, R.L., "EVACNET+: A network model of building evacuation", in: Computer Simulation in Emergency Planning, The Society for Computer Simulation (SCS), 1981. [14] Kleinrock, L., Queueing Systems: Volume I: Theory, Wiley, New York, 1975. [15] Law, A.M., "Statistical analysis of the output data from terminating simulations", Naval Research Logistics Quarterly 27 (1980) 131-143. [16] Law, A.M., and Kelton, W.D., Simulation Modeling and Analysis, McGraw-Hill, New York, 1991.
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[17] Levin, B.M., "A modeling procedure for analyzing the effect of design on emergency escape potential", Part of the Proceedings from a Seminar on Human Behavior in Fire Emergencies, National Bureau of Standards, NBSIR 80-2070, 1980, 13-41. [18] LCv~s, G.G., Wiklund, J., and Drager, K.H., "Evacuation models and objectives", in: J. Sullivan (ed.), Proc. of the International Emergency Management and Engineering Conference, March 1993, Arlington, VA, The Society for Computer Simulation, 1993. [19] Levis, G.G., "Modeling and simulation of pedestrian traffic flow", Transportation Research B 28B/6 (1994) 429-443. [20] Levis, G.G., "Wayfinding in a complex labyrinth", to appear in European Journal of Operational Research. [21] Mori, M., and Tsukaguchi, H., "A new method for evaluation of level of service in pedestrian facilities", Transportation Research. A 21A/3 (1987) 223-234. [22] Natvig, B., "Multistate coherent systems", in: N.L. Johnson and S. Kotz (eds.), Encyclopedia of Statistical Sciences, Vol. 5, Wiley, New York, 1985, 732-735. [23] Ozel, F., "Simulation modeling of human behavior in buildings", Simulation 58/6 (1992) 377-384. [24] Smith, J.M., "State-dependent queueing models in emergency evacuation networks", Transportation Research. B 25B/6 (1991) 373-389. [25] Smith, J.M., "Multi-objective routing in stochastic evacuation networks", in: J. Sullivan (ed.), Proc. of the 1992 lnt. Emergency Management and Engineering Conference, Florida, April 1992, The Society for Computer Simulation, 1992, 113-117. [26] Talebi, K., and Smith, J.M., "Stochastic network evacuation models", Computers & Operations Research 12/6 (1985) 559-577. [27] Watts, J.M., "Computer models for evacuation analysis", Fire Safety Journal 12 (1987) 237-245.
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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH
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ELSEVIER
European Journal of Operational Research 85 (1995) 352-367
Theory and Methodology
On performance measures for evacuation systems Gunnar G. I~v~s Department of Mathematics, Universityof Oslo, P.O. Box 1053, N-0316 Oslo, Norway Received July 1993; revised December 1993
Abstract
How can a large, complex building with many occupants be evacuated safely? Before this question can be addressed it is necessary to define what 'safely' means, and to define some relevant measures which can be used to characterize the safety level. This paper studies different performance measures for evacuation systems. The primary measures are related to accident effects, e.g. the expected number of fatalities, and to evacuation time, e.g. the expected time needed to evacuate k persons out of the building. The paper presents a model of the evacuation system and shows how queueing network theory and simulation methods can be used to calculate the performance measures. The paper includes an example and a brief description of a real case study. Keywords: Evacuation; Modelling; Simulation; Queueing network; Human behaviour
I. Introduction
Hazardous situations may occur during a building's life, situations where it is necessary to evacuate the building population in order to take care of its safety. Several accidents over the last decades have m a d e it clear that evacuation systems are not always able to supply the required 'level of safety'. The goal of this p a p e r is to define, describe and discuss some relevant measures of evacuation system performance. The measures are used to describe how well the evacuation system performs, relative to its goal of 'safe' evacuation. A n evacuation system (ES) is a complex system of many 'components': The physical environment in which the population circulates is called a building. The building is composed of elements (rooms, corridors, stairs) which will be called rooms. T h e layout of the building is determined
by the way in which these rooms are interconnected by doom. The population within a building also deserves closer attention. Differences in personal characteristics such as sex, age, physical ability, familiarity with the building, with safety equipment and with evacuation procedures, etc. may be very important. Potential hazards within or close to the building determine how likely it is that a rapid evacuation will be needed. Management decisions are important in the planning of an emergency evacuation system, and managers may also play an important role as 'on-line' decision makers in a real accident situation. Detection systems are required so dangers may be observed as early as possible, initiating alarm warnings which can be understood by the building population. ESs have to satisfy certain safety requirements of which there are typically three different categories:
0377-2217/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0377-2217(94)00054-G
G.G. Levis/European Journal of Operational Research 85 (1995) 352-367
1. Qualitative ideal goals, e.g. evacuation should be safe. 2. Quantitative requirements on system level, e.g. the total evacuation time should be below 10 minutes. 3. Detailed requirements to ' c o m p o n e n t solutions', e.g. doors should be wider than 1.2 meters. The first category requirements are too imprecise to make them useful for evaluation purposes. The third category requirements, frequently used (in the past) to evaluate ESs, are easily evaluated, but one does not have any guarantee that good components necessarily compose a good system. Based on these observations, there is a growing interest in the second category requirements, which will hereafter be called functional requirements. To evaluate if systems cope with functional requirements it is necessary to measure the performance of the system. Performance measures of ESs describe the likely behavior of the evacuation process, i.e. how the ES is able to function. Therefore, typical and important examples of such measures are • expected n u m b e r of fatalities; • expected n u m b e r of safe evacuees at time t; • expected time needed to evacuate k persons. These measures will be related to a specific set of scenarios, describing the accident, the population distribution, etc. In this paper, we will study such measures further, and give them precise mathematical definitions. In the literature, little has been written about measures of ES performance. No deep discussions about different measures with their advantages and limitations are known to this author. However, several sources have discussed different ' m e a s u r e m e n t methods', such as network flow models, either deterministic (e.g. [1,2,4,10]) or stochastic (e.g. [12,24-26]), or simulation methods (e.g. [3,6,13,23]). The cited references are concerned with a few performance measures only, and some of them are mainly oriented towards optimization of the ES with respect to some small subset of the performance measures. This p a p e r has the following organization: Section 2 presents a mathematical model of the ES, and stochastic variables related to system perfor-
353
mance are defined. In Section 3, several performance measures are defined, and, in Section 4, these measures are discussed with respect to their usefulness. This p a p e r does not present detailed methods on how to calculate the performance measures analytically, but some ideas are presented in Section 5. Examples are given in Sections 6 and 7 to illustrate some of the measures presented in the paper.
2. The model
2.1. Parameters A building composed of N rooms and M doors will be modelled as the network G(V, E), consisting of the nodes u 1. . . . . u N in the set V and M links in the set E. We write eij = (vi, v ) ~ E if and only if there is a link between nodes ui and vi. The nodes represent rooms, and the links represent doors. The nodes are divided into three subsets: Source nodes V1, transportation nodes V 2 and exit (destination) nodes V 3. The nodes in V3 are absorbing, i.e. persons will not move from a n o d e i n V 3 t o a n o d e i n V 1 U V 2. The population of finite size K is modeled as K separate individuals moving around in the evacuation network G, from nodes in V1 to nodes in V3. Their movement is influenced by human decisions (initial response and route choice logic) and walking ability (see L~v~s [19]). Many parameters may be used to describe the persons and their behaviour, but it is outside the scope of this p a p e r to present them here. The persons are located in the building, at time t = 0, with probability Pu, according to the rule K u = ( K l u ..... K N u ) , where Kiu is the initial number of persons in node i. The index u is a m e m b e r in the set ~ of indices for all possible location rules. We syncronize our clocks so that at time t = 0, the hazardous situation starts, e.g. when a flammable gas is ignited (syncronization is important, but the choice of time-origin may be redefined). Assume that it is possible to define a reasonable time-horizon, T ~ (0, oo], for the evacuation process, so that there is high probability that the process has terminated before T. We
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354
study the evacuation process in the time-interval [0, T] only. It is natural to use a continuous rather than discrete time model, since the evacuation process is of a continuous nature; it is hard to define a suitable time step if a discrete approach should be applied. However, this discussion does not really relate to the performance measures defined in the paper, but it relates to the choice of solution methods. Obviously, certain - but not all - solution methods use numerical methods which apply discretization, such that the modelling gain is in fact only a formal one. The accident (or hazard) is represented as a set of possible scenarios. The accident follows scenario Hm(t) = (nxm(t) . . . . . nNm(t)) with probability qm, in which case the accident effects in node i at time t can be expressed a s Him(t). For the index m we have m ~.¢t', where Jr" is the set of indices for all possible accident scenarios. With inspiration from multistate reliability theory [22], we require that H/re(t) e {0, 1, 2 . . . . . MH}, where Him(t) = 0 iff no accident effects are present, and Him(t) = M H iff the accident will kill anyone present in node i at time t. Let d(i, j) be the shortest distance between (the centers of) node v i and node vj. If no connection exist, d(i, j ) = oo. Let us define d i as the shortest distance from node v; to some exit v i
v3, d i=
min d(i, j). j:v~e V3
The above definition of d i is based on the underlying assumption that persons walk the shortest path to the closest exit. However, it is easy to redefine d i if each person is believed to walk the shortest path to one specific exit (e.g. exit l, according to an evacuation plan) or that each person has to follow a specified path of movement (e.g. visit nodes i, Jl, J2 . . . . . Jn, 1). In general, we may also consider the distances to depend on changes in the environment, e.g. accident effects, and write di(t).
2.2. Stochastic variables In the following, several stochastic variables are defined. They will later be used as the basis
for the definition of relevant performance measures. Many of the stochastic variables are interdependent in a way that might make traditional statistical solution methods unsuitable, since they normally require that variables are independent, or at most associated. However, the dependencies among the stochastic variables do not influence the coming definition of the performance measures. The dependencies of the stochastic variables have to be clarified before a solution method is chosen and applied. The number of persons in node i at time t will be noted X~(t). Let the aggregate system state X(t) = (Xl(t) .... , X u ( t ) ) be an element in the finite space X of all possible system states. Define Ai(t) to be the number of arrivals at node i before time t. Define Zi(t)= X~(O)+Ai(t), as the n u m b e r of visits to node i within time t. Let Zij(t) be the number of persons using link eij within time t. It will be shown in the rest of this section that many of the most interesting performance variables (and measures) are related, more or less directly, to the process {X(t), t ~ [0, T]}. Hence, knowledge about the stochastic process X(t) will often be sufficient.
2.2.1. Accident effects Let H(t) = (Hx(t) . . . . . HN(t)) describe the state of the accident. From the set of all possible accident scenarios, a specific scenario is 'drawn' according to the probabilities qm; if scenario m is selected, then H(t) = Hm(t). A person will be exposed to accident effects depending on the route he follows and the length of time he spends in the different nodes. Let Kk(t) be an indicator taking value 1 if person k is alive at time t and value zero otherwise. Define K(t) = E~ Kk(t) as the total number of persons being alive.
2.2.2. Evacuation time Define M(t) as the n u m b e r of persons who evacuated within time t,
M(t)= E x/(t). i:oi~ V3
G.G. LCvSs/ European Journal of OperationalResearch 85 (1995) 352-367 The evacuation time for one person is defined as the total time he needs to get out of the building (i.e. to arrive at a node in V3), including reaction time, walking time and waiting time in queues. The individual evacuation time of exiting person n u m b e r k is Tk. We define T ~ = T if person k does not evacuate before T. Hence, the times are ordered so that T 1 < T 2 <_ . . . <_ TK. Observe that Tk = inf{t e [0, T ] : M ( t )
> k}.
2.2.3. Queueing and waiting The last time when s persons or more are present in node i is Li(s) = sup{t ~ [0, T]:Xi(t) >s}. Let L ( s ) = m a x Li(s) , the maximization taken over all nodes in V1 U V2. Node i is busy with s persons or more for Y/(s) time units, and the total time spent in node i when at least s persons are present is Ii(s), hence defined as
355
n u m b e r k, k = 1. . . . . Zi(T). The average time a visitor spends in node i is called W/, easily seen to be
li(O) W/
Zi(T) "
The maximum waiting time will be called W/max . Let Qij(t) be the number of persons waiting to use link eij at time t. This variable should be interpreted as the size of the " q u e u e " in front of a door. As we defined ~ and W/, based on Xi(t), we can define similar variables based on Qij(t).
2.2.4. Network distances The total remaining route length D(t) and the average remaining route length Da(t) are defined as N
D(t) = ~_~X i ( t ) d i ( t ) , i=l
Yi(s) = fo~I(Xi(t) > s) dt, Da(t)-
O(t) K-M(t)
Ii(s ) = foTSi( t ) I( Xi( t ) > S) dr, where I ( . ) equals one if (-) is true, zero otherwise. The average population size is obviously closely related to a subjective choice of time interval over which the n u m b e r of persons is averaged. Let y ( s ) be the length of the chosen time interval, and let y ( s ) equal Y/(s), t i ( s ) o r L(S), corresponding to three different types of averages. The following definition of the average population size X,(y, s) of node i is only meaningful if y ( s ) > O,
li(s) s) =
With s = 0, this expression corresponds to a standard time average over [0, T]. However, we get more interesting results with s = 1; therefore, if s = 1, we may simplify the notation by omitting the p a r a m e t e r - for example, L i = Z i ( 1 ) and X~(y) = Xi(y, 1). The maximum population size is
Si max= suPtSi(t ). Let W/k be the time spent in node i by visit
We define No(d, t) as the number of persons with a longer minimum remaining distance than d to walk at time t, N
ND(d, t) = ~, X i ( t ) l ( d i ( t ) >_d). i-1 2. 2. 5. Network redundancy Define V~ as all nodes in V1 w V2 which are neighbours to a node in V3. A viVa-path is a sequence of neighbouring nodes from vi to a node in V~. A viVa-path is safe at level s at time t iff H j ( t ) < s for all nodes vj in the path (in particular, a path is safe if it is safe at level 0). Two viVa-paths are independent iff all their common nodes are in ui I,_JV3. Let fi(t, s) be the n u m b e r of independent viVa-paths that are safe at level s at time t. The average redundancy number F(t, s) is defined as
F ( t , s) =
Y:i:v, ev, uv2Xi( t) fi( t, s) K-M(t)
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3. Performance measures
q~sa = Probability distribution of D(t). C1)5b =
In order to give a good and valuable description of the ES it is necessary that we consider the whole range of possible values of the variables we have defined. Therefore, we will typically define the relevant performance measures as the expected value of a stochastic variable, its distribution, quantiles, etc. The measures defined in this section are related to accident effects, evacuation times, queueing, network distances and network redundancy. A discussion about the usefulness of the different measures can be found in the next section. The performance measures, n a m e d (~la, (~lb, (~2a, (~2b . . . . . are listed below in six different categories whose use is discussed in Section 4. 1. These measures are related to K(t), the n u m b e r of u n t r a p p e d individuals, defined in Section 2.2.1. ~ l a -~- Probability distribution of K(t).
t~lb = EK(t). 2. These measures are related to M(t), the n u m b e r of safe evacuees at time t, defined in Section 2.2.2. ~2a = Probability distribution of M(t). (1)2b =
EM( t ).
~2c = Quantiles (upper) of M(t). 3. These measures are related to the evacuation time Tk, defined in Section 2.2.2. q~3a = Probability distribution of T k. (I)3b =
E T k.
4. These measures are directly related to Xi(t), defined in Section 2.2. ¢~4a = Probability distribution of Xi(t). (~')4b =
EXi( t )"
ED( t ).
~5¢ = Probability distribution of Da(t).
CI)sd = EDa( t ). cI)se = ENo( d, t ). 6. These measures are related to the average redundancy n u m b e r F(t, s), defined in Section 2.2.5. ~6a = Probability distribution of F(t, s). t~6b ~
EF(t, s).
4. Discussion
The measures carry information about the ES, hopefully, and some of the measures are better than the others, because they are easy to understand or easy to calculate. Since many elements in a complex ES are subject to random fluctuations, it is often a problem that the stochastic variables, e.g. T k or M(t), have very high variances, reducing the 'information value' of the performance measures. One way to overcome this problem is to measure the performance of the ES under given 'initial conditions', i.e. that a specific population localization K , and an accident scenario H m is known. The performance measures may be studied for all interesting combinations of u ~ ~' and m ~ ¢ ' . In this way it is possible to identify the 'critical components' in the ES, whether it is the accident development, the initial population distribution or the evacuation movement process itself. This simplification, with fixed u and m, is applied in Sections 5.1 and 6 (even though it is not required).
~4c = Quantiles (upper) of Xi(t). (~ 4d = CI)4e =
EYi( s )" ELi(s). EI~s). EXi(y, s).
¢I)4f = ¢I)4g = tTJ54h = E X i max. (~4i =
EZ__L(t )"
(~4j = E W i .
(/)4k = EW/max. ¢~41 =
EQij(t)"
5. These measures are related to the total remaining distance D(t), defined in Section 2.2.4.
4.1. Measures of category 1 The safety of the building population is our main concern. Therefore, m e a s u r e s ~)l" are two of the most important measures, especially when evaluated at t = T. However, m e a s u r e (~lb will often be a poor measure since the expected value is not a good measure of centrality for a skew distribution. Using qbla to calculate the probability that more than k persons are dead, is much more informative.
G.G. Levers/European Journal of Operational Research 85 (1995) 352-367
In practice, the measures q~. are used as 'filters' in the sense that their values give an idea whether the ES performs bad or well. Even if the ES seems to perform well, one should study the other performance measures to see if the performance is really OK, or if it could easily be improved. This is in line with the idea that the risk for the building occupants should be 'as low as reasonably practicable' (ALARP), an idea influencing much of m o d e r n risk analysis. 4.2. Measures o f category 2
Measures related to time are very important because most hazards evolve over time so that there is limited time available for safe escape. Let TA be the time available to escape the building. Obviously, TA is a stochastic variable, and its distribution will not be known to the building occupants (or anyone else). Nevertheless, based on knowledge about accidents and their potential development, it is customary to determine TA as a deterministic parameter. It is also possible to interpret TA as the time accepted by management, e.g. one functional requirement may be that 98% of the population should have evacuated within time TA. Measures q~2. tell quite a lot about the behaviour of the ES, they are easy to interpret, and they are a good basis for an evaluation of the ES's efficiency. Measures ~2b and ~2c carry valuable information over the whole time-span [0, T], especially in the first part of the interval, while (J~)za is the best measure for high t-values, e.g. t = TA. Observe also that E M ( t ) ~ E K ( t ) as t ~ oo; hence, there is a strong connection between measures q~l. and q~2.4.3. Measures o f category 3
The expected evacuation time E T k will often be a poor measure because of the following reason: If T is finite and Pr(T k < T ) < 1, then the evacuation process is truncated and E T k will be difficult to interpret. If T = oo and P r ( T k < ~) < 1, then we also get E T k = ~, which is not really very informative. This corresponds to an accident scenario where
357
there is a positive probability that ( K - k + 1) persons will be trapped (dead or unconscious). Since there will always be a positive probability that at least some persons may be trapped otherwise it was not necessary to evacuate - we know that E T K = oo. One way to overcome this difficulty is to study the expected average evacuation time for the group of 'survivors'. However, as this may be a little bit confusing, it is difficult to interpret the information carried by measure 1~3b, and its use is not recommended. The probability that all the K persons have evacuated within the available time is an important number. This probability may be of interest at other times and other values of k too, making t~)aa an interesting measure. 4.4. Measures o f category 4
Even if queueing may seem to be of minor importance, several sources claim that queueing is in itself dangerous, maybe causing people to crush each other, giving rise to panic, etc. Therefore, we choose to describe an evacuation system also by some queueing measures. The measures of this category may be related to the concept of 'level of service', as defined and used by Fruin [5] and Mori [21]. These authors give a method for measuring performance quality, related to the comfort and safety of the moving crowd; basically, the density of people is the main determinant for the service level. When the capacity C i of node i is known, it is easy to use the measures based on Xg(t) to find the expected level of service (at each time t) and its probability distribution. The evacuation process {Xi(t), t > 0 } tells much about the conditions in node i. The probability distribution of Xg(t) and its expectation and quantiles are relevant measures (q~4a, ~4b and ~4c), but it will often be sufficient to consider measures ~4d to ~4g, which are all 'extracts' of information included in ~4~(~4d and t~4e are measures of the length of time when a node is used by s or m o r e persons. A high value to these measures means that the node is important in the sense that its 'failure' is likely to have consequences for the persons. For most
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applications, measure ~4d is more informative than measure t~4e. The expected total time spent in a node (~4f) measures the 'popularity' of the node, and may suggest something about the importance of the node. The average population size in node i (measure ~4g) is related both to different levels s of population size and to different time periods Y~, Z i and L. For example, .~i(Y/, s) is the average number of persons in node i given that there are at least s persons in the node. Another example, Xi(Li, 0) is the average number of persons in node i over the whole time period [0, T]. If it is likely that persons are hurt in overcrowded situations, it is interesting for us to know E S i max (measure (~)4h), indicating if the conditions are untolerable or not. Measure gai indicates if a node is visited by many persons, and measure ~4j give us the average time spent in the node by each visit; these are helpful measures when one tries to identify 'bottlenecks' in the system.
4.5. Measures of category 5 These measures, related to the remaining distance D(t), are interesting measures of the evacuation system performance, the complexity of the evacuation network, and the queueing problems present in the system. Define Dm(t) as the lower limit for D(t) given that the ES performs perfectly. Starting with Din(0) = D(0), and assuming that all persons walk immediately and with unimpeded speed to the nearest exit, it is easy to calculate Dm(t) for all t ~ [0, T]. If ED(t) is close to the lower limit, then the evacuation process is performing well. The expected value of No(d, t) tells something about the size of both the building and the population, and hence about the potential for fatal outcome of an accident.
4.6. Measures of category 6 For the persons in the building at time important that there exist safe paths out building. Measure ~6b gives information the average number of safe independent
t, it is of the about paths
for the persons in the building. This number will normally fall as t grows, i.e. the situation will be worse for the persons, since the accident develops and make some evacuation routes 'less attractive'. If EF(t, 0) equals zero, then the E ( K - M ( t ) ) persons in the building have a serious problem, since there are no more safe routes to follow. Then, it is interesting to know EF(t, s) to see if there are any paths which are safe at level s. Measures 4 6 . are well suited to tell whether the ES has high enough redundancy or not. This was noticed by [17] who used similar measures to describe some buildings in the USA.
5. Calculation
of measures
Many of the measures are hard to calculate, and no attempt will be made in this paper to arrive at nice analytical solutions. With a large building ( N large) and many building occupants ( K high), any calculation principle based on realistic assumptions is likely to run into computational problems. In Section 5.1 it is shown how an analytical solution may be derived - although no simple answers are given - and in Section 5.2 a simulation model is described.
5.1. An analytical approach A short sketch of an analytical solution method, based on ideas from queueing network theory (see [14, Section 4.8] or [8, Chapter 4]), is included in the following. An overview over other analytical solution methods can be found in [18]. A queueing system is a system where 'customers' arrive, spend some time (usually, but not necessarily, to get some kind of service), and then leave. Each node in G is modelled as a queueing system, inspired by Jackson [11]. Obviously it can be useful to describe a system as a queueing system even if 'queueing' in the traditional sense - waiting in lines - is not a (big) problem. Actually, if queueing at a station is certain (i.e. someone is always waiting), the departure process from the station is not related to the input process, and the apparatus of queueing theory is not really
G.G. LCv?ts/ European Journal of Operational Research 85 (1995) 352-367
needed. Therefore, the presented stochastic approach is of particular interest when N is large and K is relatively small, corresponding to a large but not overcrowded building. In the opposite case, when a building is heavily overcrowded, traditional 'hydraulic' flow models will often suffice as a method for obtaining the performance measures. As noted before, let the aggregate system state X(t) = (Xl(t) ..... XN(t)) be an element in the finite space X of all possible system states. Note that X(t) does not necessarily contain all information about the actual state of the system. The number of distinguishable states of X(t) is equal to the number of ways in which one can place K persons among N nodes, known to be the binomial coefficient N+K-1 K-1
× P r ( m o v e m e n t from / t o
=p(k,t) N
+ ~ ~ P(kij, t)P(i, t, h, k)r(i, j, k). i=lj=l
j~i
(1) Rearranging in (1), taking the limit as h ~ O, changing the order of summation and limit, we find
Op( k, t) at
N ~ (limP(i,t,h,k)/h}p(k,t) i=1 N
N
+ E E(limP(i,t,h,k)/h}
p(k, t) as below:
p(k, t)
t) = k l , . . . , X u ( t ) = kN)
= Pr(X(t) =k). The system state may change in one way only, namely when a person moves from one node to another. We assume that in a very short time-interval only one such change may take place. For the rest of this subsection, let us assume that the process is Markovian. Let P(i, t, h, k) be the probability that a person leaves node v i in the short time-interval [t, t + h ) given the present state k of the evacuation process. Assume that the moving person selects his new node vj according to the probability r(i, j, k). We require that )Zk~ = K and define k~j = k except that its i-th component is k~ + 1 and its j-th component is k j - 1. Then, the forward equation, describing the change in the population process from t to t + h, equals
p(k, t +h) =p(k, t ) P r ( n o change in [t, t+h)) N
E P(kiy, t)
i-lj=l
j¢i
~h~0
j~i
Define the state-probability
+ E
t+h))
i=1 N
i=1 j=l
N
j in It,
1- ~P(i,t,h,k)
)"
= Pr(X,(t)
359
Xr(i, j, k)p(kij , t).
(2)
The initial distribution, p(k, 0), is already known since p(k, O)= E, ~.~I(K u = k) Pu. Given p(k, 0) and (2) with P ( - ) and r ( ' ) specified, it is possible to study, at least by numerical methods, the transient development of p(k, t) in the timeinterval [0, T]. Grassmann [7] and Gross and Miller [9] solved the problem under the assumption that P(i, t, h, k) = t~ih and r(i, j, k) = 7rij. Assume now that p(k, t) is known for all vectors (k, t). Then it is easy to see how some of the performance measures can be quantified. Measure q~4b is identified as
EXi(t) = E ki p(k, t) k~x
and, measure tbaa, the distribution of be found directly as
Xi(t), can
Pr(Xi(t ) <_k)= E I(ki<-k)p(k, t). k~ X
Similarly, we can find measure q02b, the expected number of safe evacuees at time t, by
E M ( t ) = ~ ( ~, ki)P(k , t). kEX i:viEV 3
Several of the other measures can be related to
p(k, t) in the same fashion.
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5.2. Simulation methods Several simulation tools have been developed to evaluate the effectiveness of ESs. These tools differ in complexity, input requirements, underlying assumptions, and specific application areas. An overview of some existing methods has been given by Watts [27], and a discussion about different models has been presented by l_~vfis [18]. E V A C S I M is an evacuation simulation program under development in Norway and France. A short introduction to E V A C S I M can be found in [3], and a detailed description of the 'pedestrian flow' is given in [19]. It is not within the scope of this p a p e r to go into details about any simulation program, but a short description of E V A C S I M will be given. E V A C S I M simulates the evacuation process on a microscopic level, in the sense that each person is treated as a separate 'flow object'. Each person is characterized by a set of attributes, e.g. initial position and characteristic walking speed. The building is modelled as the graph G, with nodes and links, also with their own attributes, e.g. capacity of a node and width of a link. The accident scenario is represented as a dynamic scenario, specifying time and type of changes in accident effects in each node. E V A C S I M performs a process-oriented discrete-event simulation [16], using the QNAP2-tool (Queueing Network Analyzing Package). This means that each person is followed closely and that critical events in his movement process are identified. These events are then scheduled into an event calender, and the program continues by handling the most imminent event on the event calender, corresponding to an event (activity/movement) for another person. E V A C S I M performs R replications of the simulation from the same initial state (i.e. the same building topology, the same personnel localization, the same accident scenario). On each replication, values to the personal attributes are sampled from the appropriate distributions, and K individual m o v e m e n t processes are 'started'. Each movement process is stochastic, e.g. because the persons are allowed to make ' r a n d o m ' choices among alternative routes [20]. Process k
interacts with the other K - 1 processes: when 'it' evaluates the actual walking speed of person k; when the route-choice-probabilities are established; when a link is used (to control that only one person uses the same link at the same time); when a person tries to enter a new node (to see if there is more free capacity in the node); etc. Each replication is finished when all of the K movement processes have stopped or when the maximum time T is reached. Statistical results are collected during the simulation [15], and estimates to the performance measures are presented as the main results. In addition, E V A C SIM produces data which can be used to animate the evacuation process on a computer screen.
6. Example To get more familiar with the different performance measures, an example will be presented. The building network G is shown in Fig. 1; the nodes are numbered i = 1 , . . . , 25 as indicated in the figure. Each link has length d(i, j ) = 10 meters. We study a population of K = 100 persons who evacuate from room 1 to room 25 (which is considered safe), i.e. V1 = {vl}, V2 = {u2,...,u24 } and V3 = {v25}. In order to simplify, no accident scenario is specified, meaning that all persons will succeed to evacuate sooner or later. We assume, in this example, that all persons start to evacuate immediately at t = 0, and we follow the evacuation process until every person has exited, i.e. T = ~. The walking speed of the persons is as-
25 20 13
15 10
1
2
3
4
5
Fig. 1. Network of nodes (dots) and links (lines).
G.G. Levis/European Journal of Operational Research 85 (1995) 352-367
361
No. of p e r s o n s 10 0
I.
~ - - - ' , .......... ,...........................................
-
Z.; /
100
200
T i m e t in seconds. Fig. 2. Measures @2b and @2c: The average value of M(t), with its observed 2% lower and upper quantiles.
sumed to be lognormally distributed with mean 1.5 m / s and standard deviation 0.3 m/s. We assume that the persons are familiar with the building, being able to select paths leading them effectively to the destination v25. The simulation program EVACSIM has been used to simulate the evacuation process. In this example, R = 50 replications have been performed. Several interesting results will be presented in the following. However, since no accident sce-
@~.
nario is specified, measures and @6. are of no interest. Many of the performance measures are functions of time, and well suited for graphical presentations. Measures @2.: Fig. 2 shows the average of M(t), with its observed upper and lower 2% quantiles. From this curve it can be seen that it is expected that approximately 90 evacuees are safe (at node 25) after 160 seconds. The distribution of M(t) varies with time, and Fig. 3 shows the
Probability O.5
....
....
/~M(80)
/I
/. y( o> y"i j "',,, j" ! M ( l O 0 )
.... o,
/
1
/
lo
20
N u m b e r of persons. Fig. 3. Measure @2a: Empirical distribution of M(t) at t = 80, 90, 100.
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362
Cum. prob. (%)
Ts0
~.50
2OO
25O
T i m e Tk in s e c o n d s . Fig. 4. Measure
t~b3a: Empirical cumulative distribution of evacuation times, from left to right, 7"5o, Tso , T90 , T95 and
distribution of M(t) at t = 80, 90, 100. These curves can be used to obtain an estimate of measure 42a. For example, less than 26 persons are safe after 100 seconds with probability 0.98. Measures 43.: The distribution of evacuation time Tk is shown in Fig. 4 for k - - 5 0 , 80, 90, 95, 100. It is interesting to observe how variable the evacuation time of the last person is (T100) compared to the median evacuation time (Ts0). Measures 44.: Fig. 5 shows the average of
X13(/) together with its observed upper and lower 2% quantiles. For the nodes on the right-hand side of Fig. 1, the average of Xi(t) is presented in Fig. 6. Similar results are collected for all nodes. These curves give information about the conditions in the node, the node usage, the average density, etc. It can, for example, be seen that no persons are present in the center node (i = 13) after 160 seconds. Some examples of the measures 44d to 44S are shown in Table 1. One could
No. o f p e r s o n s lo
yi,Jv ,, ,, g \
., ,
,"2_A_JiJ.........
0 0
Tloo.
loo
Time t in seconds. Fig. 5. Measures ~4b and ~4c: Average value of Xz3(t) with its observed lower and upper 2% quantiles.
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363
No. of p e r s o n s 8
6 2O
4
0
2O0
100
T i m e t in seconds. Fig. 6. Measure qbab: Average of Xi(t) for transportation nodes on the right hand side of Fig. 1.
for example see that the persons spend more time in node 20 than in node 2, even though the nodes have the same number of visits; this is due to queueing phenomena, restricting the flow out of node 20 (only one link, 20-25, is used) more than the flow out of node 2 (two links, 2-3 and 2-7, are used). Measures qOs.: The average total remaining distance D(t) is shown in Fig. 7 together with its lower limit Din(t). We see that the persons need a minimum of 54 seconds to evacuate (corresponding to the lower limit Din(t)). In the average simulated case, however, it is seen that the population needs about 175 seconds to walk all the total distance. Hence, the population requires
about 3 times as much time to leave the building in 'reality' as in the 'idealized uncongested world'. This is due to two factors: The links impose restrictions on the flow, leading to queues, and the slow walkers need more time to walk 80 meters than the 'average walker'.
7. Case study A real case study for a Norwegian offshore oil production platform will be briefly reported here. The actual oil platform is a very large 'building' with about four kilometers of walk-ways, with a population ranging from 150 to 300 persons.
Table 1 Node i
1 2 3 4 5 10 13 15 20
Performance measures
t~4f
t~ad
t~4e
t~4g
t~4g
t~4i
(J~aj
~4h
Eli
EY~
EL,
E.~,(Y~)
EXi(L i)
EZi(T)
E~
EXimax
maxR X/max
5534 451 219 117 64 171 342 344 603
103.7 92.4 86.3 68.9 48.0 82.2 97.3 99.8 109.6
103.7 108.6 114.5 117.5 121.4 140.8 138.5 158.9 173.2
53.4 4.9 2.5 1.7 1.3 2.1 3.5 3.4 5.5
53.4 4.2 1.9 1.0 0.5 1.2 2.5 2.2 3.5
100.0 50.5 25.1 13.5 7.4 19.4 37.8 35.4 50.7
55.3 8.9 8.7 8.6 8.7 9.0 9.1 9.7 11.9
100.0 7.0 5.2 3.8 2.8 5.0 10.3 7.3 7.8
100 8 7 6 6 7 12 10 15
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Total distance \
\
loo
200
Time t in seconds. Fig. 7. Measure ~Sb: The average total remaining distance D(t) and its lower limit Din(t).
The platform is operated throughout the hours (12 hour shifts), but the majority of the crew work at daytime. Most of the persons are located within the living quarters (LQ). The rest of the workforce is spread around the platform in many different modules. Since large areas of the platform are not continuously manned, it was decided to locate the personnel in the network model according to a specially given probability distribution. The platform was modelled as a walkway network with 219 nodes, 314 links and 6 exits, each corresponding to a lifeboat with capacity to carry 50 persons. The purpose of the study has been to analyse the effectiveness of two different evacuation strategies, explained below: Strategy 1. All personnel move from their present location to a temporary safe area within the LQ, waiting for further messages. When the last person arrives, they all move to their respective (preassigned) lifeboat positions. This strategy will not be followed in the unlikely case that a serious accident occurs within the LQ.
Strategy 2. All personell move directly from their present location to their respective (preassigned) lifeboat positions.
The second strategy is obviously quicker, and the goal of the study was to find 'how much'. However, the first strategy is more manageable, making communication between the leaders and the personnel more reliable, which is of course important in an emergency situation. 'Gas and oil blowout in the wellhead area' is the single accident scenario which contibutes most to the total risk level. A blowout may - if it is ignited - lead to a pool fire at sea. Based on knowledge of the weather conditions, the well pressure, the sea surface water flow, etc. it can be found that - in this case - the lifeboats can be safely launched within the first 4 - 8 minutes. After this time, there is a large risk that there will be a fire below the boat. Performance measures related to time are therefore very important. The differences between strategies 1 and 2 can be measured even without a model of the accident. In the case study, the strategies were considered both for the blowout scenario and under 'normal' evacuation conditions. The latter will be described here. First, it is necessary to point out that the initial response patterns of the 'building occupants' are extremely important for the outcome of the evacuation process. This is in contrast to the previous, rather academic, example. In the case study it was assumed that each individual had his re-
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365
No. of p e r s o n s
Strategy 2
//'/ o
Strategy 1
50o
1000
T i m e t in seconds. Fig. 8. Average number of persons at lifeboats at different times for the two different strategies.
sponse time drawn from a lognormal distribution, where the mean and standard deviation varied between the different areas of the platform. Response from offices was assumed quicker than response from bedrooms or noisy workplaces. The persons were also assumed to communicate with each other, e.g. meaning that persons in the same node influence each others response times. For each strategy, 100 replicated simulations were performed with EVACSIM, assuming a con-
stant population size of 200 persons. A short summary of the main results are shown in Table 2. These numbers clearly indicate that the strategies are significantly different with respect to performance measures ~3.. The curves in Fig. 8 even better visualize the differences between the strategies, showing measure ~2b for both strategies. At an offshore platform, the time until the last person has reached the lifeboats is very important, because it determines how long time
Probability density rategy 2
60O
BOO
I000
T i m e t in seconds. Fig. 9. Probability distribution for the time needed to evacuate all persons under the two strategies.
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Table 2 Main results (All times in seconds) Person Average Confidenceinterval (95%) number evac.-time of average evac.-time Strategy 1 First 521 (489, 553) Median 632 (599, 665) Last 741 (708, 774) Strategy 2 First 44 (42, 46) Median 174 (173, 175) Last 447 (417, 477)
many persons have to wait before a lifeboat can be launched. Fig. 9 shows measure ~3a, the probability distribution of this time for the two strategies. The bimodal shape of the density curves is due to the fact that there is a r a n d o m n u m b e r of persons, normally one or none, in the most remote region of the platform. The final conclusion from the case study t e a m was that the effectiveness of the standard evacuation strategy (number 1) is significantly lower than that of strategy 2. Based on this, the oil company started an internal discussion about the validity of the results and about the possibility to draw conclusions of relevance to the offshore installation management. At the time of the writing of this article, the m a n a g e m e n t conclusions and decisions are not known.
8. Concluding remarks W h e n we study evacuation systems, we do so because we want to see if the building can be evacuated in an acceptable way. Hence, the performance measures form a basis for all quantitative studies of evacuation systems. If the performance measures indicate that the system will (or may) perform unacceptably, then some corrective actions are needed to 'speed up' the evacuation process (or 'slow down' the accident development). The performance of the evacuation system may be improved by one of the following actions: • Reduce reaction time of personnel (e.g. better detector- and alarm systems). • Reduce interpretation time (e.g. better messages to building occupants).
• Reduce walking time by improving the escapeway system: - shorter routes (e.g. new connections, new doors); - higher flow capacities (e.g. wider doors); - easy wayfinding (e.g. better marking and orientation help). • Train personnel (e.g. react correctly, walk faster, follow planned routes, etc.). • Improve organization of the evacuation process (e.g. 'optimal' routing of evacuees). • Delay accident development (e.g. sprinkler systems, fire-doors). • Minimize accident effects (e.g. use materials producing non-toxic combustion products).
Acknowledgements Many thanks to Professors Terje Aven and Bent Natvig for helpful discussions over the presented topic; their experience with reliability thepry, especially that of multistate systems and network flows, has been very inspiring. Thanks also to Jo Wiklund, Quasar Consultants, for useful ideas, critical comments, and for his kind permission to let me present the case-study, in which he was responsible for the modelling and simulations. Many thanks to a referee for valuable comments on a previous version of the manuscript. The author gratefully acknowledges the financial support of this work by the Royal Norwegian Council for Scientific and Industrial Research.
References [1] Chalmet, L.G., Francis, R.L., and Saunders, P.B. "Network models for building Evacuation", Management Science 28/1 (1982) 86-105. [2] Choi, W., Hamacher, H.W., and Tufekci, S., "Modeling of building evacuation problems with side constraints", European Journal of Operational Research 35 (1988) 98110. [3] Drager, K.H., LOves, G.G., Wiklund, J., et al. "EVACSIM. A comprehensive evacuation simulation tool", in: J. Sullivan (ed.), Proc. of the 1992 Int. Emergency Management and Engineering Conference, Florida, April 1992,
The Society for Computer Simulation (SCS), 1992, 101108. [4] Francis, R.L., "A uniformity principle' for evacuation
G.G. LOt,as/European Journal of Operational Research 85 (1995) 352-367 route allocation", Journal of Research of the National Bureau of Standards 86/5 (1981) 509-513. [5] Fruin, J.J., Pedestrian Planning and Design, Metropolitan Association of Urban Designers and Environmental Planners Inc., New York, 1971. [6] Glowacki, A., "Using simulation to prepare for emergency mine fire evacuations", in: J. Sullivan (ed.), Proc. of the 1992 Int. Emergency Management and Engineering Conference, April 1992, The Society for Computer Simulation, 1992, 79-83. [7] Grassmann, W.K., "Transient solutions in Markovian queueing systems", Computers and Operations Research 4 (1977) 47-53. [8] Gross, D., and Harris, C.M., Fundamentals of Queueing Theory, 2nd ed., Wiley, New York, 1985. [9] Gross, D., and Miller, D.R., "The randomization technique as a modeling tool and simulation procedure for transient Markov processes", Operations Research 32/2 (1984) 341-361. [10] Hamacher, H.W., and Tufekci, S., "On the use of lexicographic min cost flows in evacuation modeling", Naval Research Logistics 34 (1987) 487-503. [11] Jackson, J.R., "Jobshop-like queueing systems", Management Science 10/1 (1963) 131-142. [12] Karbowicz, C.J., and Smith, J.M., "A K-shortest paths routing heuristic for stochastic network evacuation models", Engineering Optimization 7 (1984) 253-280. [13] Kisko, T.M., and Francis, R.L., "EVACNET+: A network model of building evacuation", in: Computer Simulation in Emergency Planning, The Society for Computer Simulation (SCS), 1981. [14] Kleinrock, L., Queueing Systems: Volume I: Theory, Wiley, New York, 1975. [15] Law, A.M., "Statistical analysis of the output data from terminating simulations", Naval Research Logistics Quarterly 27 (1980) 131-143. [16] Law, A.M., and Kelton, W.D., Simulation Modeling and Analysis, McGraw-Hill, New York, 1991.
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[17] Levin, B.M., "A modeling procedure for analyzing the effect of design on emergency escape potential", Part of the Proceedings from a Seminar on Human Behavior in Fire Emergencies, National Bureau of Standards, NBSIR 80-2070, 1980, 13-41. [18] LCv~s, G.G., Wiklund, J., and Drager, K.H., "Evacuation models and objectives", in: J. Sullivan (ed.), Proc. of the International Emergency Management and Engineering Conference, March 1993, Arlington, VA, The Society for Computer Simulation, 1993. [19] Levis, G.G., "Modeling and simulation of pedestrian traffic flow", Transportation Research B 28B/6 (1994) 429-443. [20] Levis, G.G., "Wayfinding in a complex labyrinth", to appear in European Journal of Operational Research. [21] Mori, M., and Tsukaguchi, H., "A new method for evaluation of level of service in pedestrian facilities", Transportation Research. A 21A/3 (1987) 223-234. [22] Natvig, B., "Multistate coherent systems", in: N.L. Johnson and S. Kotz (eds.), Encyclopedia of Statistical Sciences, Vol. 5, Wiley, New York, 1985, 732-735. [23] Ozel, F., "Simulation modeling of human behavior in buildings", Simulation 58/6 (1992) 377-384. [24] Smith, J.M., "State-dependent queueing models in emergency evacuation networks", Transportation Research. B 25B/6 (1991) 373-389. [25] Smith, J.M., "Multi-objective routing in stochastic evacuation networks", in: J. Sullivan (ed.), Proc. of the 1992 lnt. Emergency Management and Engineering Conference, Florida, April 1992, The Society for Computer Simulation, 1992, 113-117. [26] Talebi, K., and Smith, J.M., "Stochastic network evacuation models", Computers & Operations Research 12/6 (1985) 559-577. [27] Watts, J.M., "Computer models for evacuation analysis", Fire Safety Journal 12 (1987) 237-245.