Reliability Engineering 13 (1985) 163-174

Analysis of False Alarms Given by Automatic Fire Detection Systems

Yash Gupta~" and A v i n a s h Dharmadhikar i :~

Administrative Studies, University of Manitoba, Winnipeg, Canada

(Received: 25 April 1985)

ABSTRACT

The high frequency of Jalse alarms sounded by automatic fire detection (AFD) systems can result in substantial monetary losses and increased risk to human life. In this paper, we present an analysis of false alarm data, which were collected from six different sites, to estimate the time interval in which the next false alarm will occur. This analysis is poT[brmed on the basis of the Poisson and the non-homogeneous Poisson process.

1 I N T R O D U C T I O N

The function of the automatic fire detection (AFD) system is not just to detect f ire--i t is also to discriminate reliably between the presence and absence of fire. For A F D systems of all types, such as sprinklers and smoke detectors, the average ratio of false alarms to real ones is very h igh- -about 11 : 1.1,2 Although there are no established guidelines on the acceptable frequency of false alarms, 65 false alarms per 1000 detector heads per year is considered reasonable. 3 In an organization which experiences a high frequency of false alarms, substantial losses may occur as the area covered by the A F D systems is normally vacated and thoroughly checked before work is resumed. Moreover, repeated false alarms may cause the fire depar tment to cancel connection facilities, thus

~- Present address: 1902 Atwood Road, Toledo, Ohio 43615, USA. ~. On leave from University of Poona, Pune, India.

163 Reliability Engineering 0143-8174/85/$03.30 © Elsevier Applied Science Publishers Ltd, England, 1985. Printed in Great Britain

164 Yash Gupta, A~'inash Dharmadhikari

exposing the organization to tremendous fire risks. The most dangerous effect is that people on the premises may become desensitized to the alarms and thus may not act in accordance with fire safety procedures.

The causes of false alarms can be categorized as follows:

(1) the action of the environment on the detector head, e.g. high temperature, ambient smoke and humidity;

(2) an electrical or mechanical failure of either the detector or its control equipment;

(3) human action on the system such as unauthorized testing; (4) other or unknown causes.

It is possible to eliminate a large number of false alarms by reducing the sensitivity of the detecting device. However, if this is carried too far, the number of undetected fires would increase to an unacceptable level.

In this paper, we present the analysis of false alarms given by A F D systems in various environments with the application of Poisson and non- homogeneous Poisson processes. The data for the study were collected from the records of fire prevention officers, at six sites, the functions of which are given in Table 1.

TABLE 1 Sites from where the Data Were Collected

Site Function

1 and 6 Industrial process, laboratories 2 and 3 Experimental laboratories, orifices 4 and 5 Research reactors, laboratories

2 POISSON A N D N O N - H O M O G E N E O U S POISSON PROCESSES

Poisson processes are well known in the literature. They can be characterized in many ways. For example, refs 4 and 5 (list of symbols and definitions is given in Table 2): Let N(t) be the number of occurrences of events (such as false alarms) in (0, t]; then the process { N(t), t > 0} can be characterized as a Poisson process if, for all t and 6 $ 0,

Analysis of false alarms given by A FD systems 165

(a) (i) P[N(t, t + 3) = 1 I H(t)] = 26 + o(6) (1) (ii) P[N(t, t + 6) > 1 I H(t)] = 0(6) (2)

so that

(iii) e[N(t, t + 6) = 0 I n ( t ) ] = 1 - 26 + 0(6) (3)

Here H(t) is the history of the process at time t.

(b) Starting f rom the origin, the intervals tl, t2,..., between successive points of occurrence of events (false alarms) are independent ly exponential ly distr ibuted with parameter 2.

Further , a Poisson process has the following property: given that in (0, T,] n events have occurred, the epochs T~, T 2 . . . . . T,_ 1, of occurrence of ( n - 1 ) events are uniformly distr ibuted over (0, T,]. Hence the standardized r andom variable

n - 1

Z={n~~(T~-~)}/{T.~[I/12(n-I,] } (4,

i = 1

tends to be normal ly distr ibuted with mean 0 and variance 1 for large values ofn . It has been shown that the normal approximat ion is adequate while using the 5 ~ level of significance for n > 4 (ref. 6). This proper ty can be used to test whether a sequence of observed values of times of false alarms, T 1, T2 , . . . , T, is generated by a Poisson process. The above test has the following intuitive interpretat ion. If the sample mean of the times

TABLE 2 List of Symbols and Definitions

P [AIB] 650 n~

T~ T,o exp [A] 0 L~;

probabi l i ty of event A given event B tends to zero through positive values number of the false call at site i the epoch o f j t h false call at site i, 1 < j < n i, 1 < i < 6 0 e A

maximum likelihood est imator of 0 the epoch of j th false call, in the kth sample at the ith site, 1 _<j _< 10 (the time starts from 0 in every sample). Each sample is of size 10 Ti~ j Ti~ x o

f t h e j t h smallest observation in a collection of {T~j, 1 < k < 3, 1 < j < 10} for ~ i - - 4, 5, 6; the j th smallest observation in a collection of { T~j, 1 < k < Z, ( 1 < j < 10} for i = 1, 3

166 Yash Gupta, A~'inash Dharmadhikari

of the first (n - 1) false alarms is small (large) as compared to the mid- point of the observation interval, T,/2, the false alarms tend to occur early (late) in the observation period. A statistically significant tendency to occur early (late) indicates that the system's reliability has been improving (deteriorating) during the interval. ~ If this test rejects a Poisson hypothesis, we fit a non-homogeneous Poisson process with the intensity function

0 \ 0 / 0:/~ > 0. t >_ 0 (5)

(Weibull process). 8'9 Under the assumption that an AFD system at a given site is observed

from time 0 to the time of the nth false alarm, the maximum likelihood estimators (MLE) of fl and 0 are given as follows:

~ = ~ " (6)

~ ln(T~T,) ~=l

o=T, n~/9 (7)

To test that the epochs of false alarms under study follow a non- homogeneous Poisson process with Weibull intensity function given by eqn (5) when 0 and fl are replaced by ~ and ~, respectively, we use the Cramer-Von Mises, C~, goodness-of-fit test for each site: 8

m

l~m + Z ' m -- (8)

j = l

The method of computing Z~ in the above equation is described by C r o w . 8,9

3 ANALYSIS

As stated in Section 1, the data were gleaned from the records of fire prevention officers at six sites. These sites were different in terms of geographical location, size and availability of fire-fighting capability. It was decided not to categorize data according to the cause of false alarms. This was because of the very high frequency of the cause "unknown reason' recorded in the alarm incidence book.

Analysis of false alarms given by AFD systems 167

It is recognized that the frequency of false alarms will be biased by the effect of repeated spurious trips caused by the same fault which eludes diagnosis and hence, correction. It was decided in this analysis to record every event as a single incidence rather than setting up criteria for classing all events in a certain period as a single failure of equipment.

In addition, in some instances, increase in false alarms could be attributed to the increase in population of detector heads at any site, brought about by the current trend towards wider AFD coverage. This trend is mainly due to government regulations and the lower premium charged by insurance companies.

Other factors, such as extremely hot summers or voltage fluctuations brought about by industrial disputes in the power industry will, in general, cause some distortion in the year-to-year trend pattern.

For each site i, the following null hypothesis was tested:

Ho, the epochs of false alarms are from a homogeneous Poisson process against the alternative;

H1, they are not from a homogeneous Poisson process tested at the 5 ~ level of significance.

Using eqn (4), values of Z of 2.89, 0.078, 2-20, 2.42, 5.93 and 3.06 were obtained for sites 1 to 6, respectiv.ely. Thus, at the 5 ~o level of significance, except for site 2, the null hypothesis, H 0, is rejected. This means that on site 2, the analysis of false alarms could be performed using a homogeneous Poisson process as given in ref. 8. The data from sites 1 and 3-6 were further tested for possible fit to a non-homogeneous Poisson process with the intensity function given in eqn (5).

Following Finkelstein 1° and Crow, 8 the MLEs 0 and/~ were obtained using eqns (6) and (7). These estimates are provided in Table 3.

TABLE 3 Estimated Parameter Values of a Non-Homogeneous Poisson Process

Site ~ 0

1 1.261 6 4.9936E4 3 1"095 7 2"0021E4 4 1.375 6 3"9554E4 5 1.302 2 2-8357E4 6 1.119 7 1.9809E4

168 Yash Gupta, Avinash Dharmadhikari

To compute the Cramer-Von Mises statistics C~, as defined in eqn (8), for sites 1 and 3, samples of 20 observations from each site were selected. The selection process was based on 10 observations from the beginning of the observation period and 10 from the end of the observation period. On the other hand, for sites 4-6, the observations available were numerous enough to draw an additional sample of size 10 from each of these sites. In these cases, the middle 10 observations were treated as a third sample. The calculated and critical values of C~ are given in Table 4.

TABLE 4 Cramer-Von Mises Statistics

Site Number of samples C.2~ Critical lnJkrence (each of size 10) calculated at 1 ',~i

l 2 0.074 4 0.342 Accepted 3 2 0.079 7 0.342 Accepted 4 3 0.144 7 0.328 Accepted 5 3 0.197 0.328 Accepted 6 3 0.188 0.328 Accepted

4 PREDICTION OF FALSE ALARMS

Given the false alarms epochs, the most natural question is when the next false alarm is expected. To answer this question, the a-level lower prediction limit for Ti,,+ 1, T~.(n, 1, ~), is obtained.

T~(n, 1, ~) satisfies the relation

Furthermore,

P[Ti,+ 1 >- TiL(n, 1, ~)] = ~ (9)

TiL(n, 1, ~) = Ti, exp {(~ i,.~,- ~) _ 1)//~i} (10)

where/~i is MLE of/~ of site i. Assuming that some initial observations of false alarms are available on

a site, 90 ~ lower prediction limits of the time of the next false alarm were computed for a given site and are shown in Figs 1-5. Similarly, the upper limits can be computed using the method suggested by Crow.5

Analysis of false alarms given by AFD systems 169

Fig. 1.

2 0 0 0 0 0 0 -

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50 55 60 @5 70 75 80 85 ~0

8BSERV~T[ON NUNBER

Predicted times of next &lse alarm and 90 % lower confidence limit for site 1. Solid line, predicted times; broken line, 90% lower confidence limit.

170 Yash Gupta, Arinash Dharmadhikari

lqO0000-

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55 60 65 70 75 80 85 90

OBSERVRTI@N NUNBER Predicted times of next &lse alarm and 90 % lower confidence limit for site 3. Solid line, predicted times; broken line, 90% lower confidence limit.

Analysis of false alarms given by AFD systems 171

Fig. 3.

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~5 50 55 ~0 65 70 75 80 85 90

OBSEBVRTION NUNBER

Predicted times of next f~Ise alarm a~d ~0 ~ Io~ co~d~c~ limR fo~ si~ 4.

~olid line, predicted times; broken line, 90~ lower confidence limit.

172 Yash Gupta, Aeinash Dharmadhikari

I I00000-

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O8SERVFIT I @rt HUMBER

~ 0 ~ ~ 33

Predicted times of next false alarm and 90 ~0 lower confidence limit for site 5. Solid line, predicted times; broken line, 90 % lower confidence limit.

Analysis of false alarms given by AFD systems 173

Fig. 5.

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55 60 @5 70 75 80 85 90

OBSERV~TI@N NUMBER

Predicted times of next ~ l s e alarm and 90 % lower confidence limit for site 6. Solid line, predicted times; broken line, 90 % lower confidence limit.

174 Yash Gupta. Arinash Dharmadhikari

5 C O N C L U S I O N

An unacceptable level of false alarms given by an A F D system could not only cause losses to an organization through loss in production, but also exposes the organization to high levels of risk. Statistical analysis can be used to predict the false alarms of a given A F D system. Should a homogeneous Poisson process not fit statistically well, it is appropriate to consider a non-homogeneous Poisson process with Weibull intensity.

A C K N O W L E D G E M E N T

The authors wish to acknowledge the financial support provided by the Natural Sciences and Engineering Research Council of Canada and the University of Manitoba Research Administration. The second author wishes to thank the authorities of the University of Poona for granting him leave from the University of Poona, Pune, India.

R E F E R E N C E S

1. Burry, P. E. Current research on fire detection at the fire research station, Building Research Establishment Current Paper, CP 38/75, Department of the Environment, London, UK.

2. Fry, J. F. and Eveleigh, C. The behaviour of automatic fire detection systems, Building Research Establishment Current Paper, CP32/75, Department of the Environment, London, UK.

3. Peacock, S. T. and Wagstaff, T. A statistical analysis of false fire alarms from hospitals, Proc. 7th Advances in Reliability Technology Symposium, University of Bradford, UK, 14 16 April 1982.

4. Cox, D. R. and Isham, V. Point Processes, Chapman Hall, London, 1980. 5. Bain, L. J. Statistical Analysis of Reliability and Life Testing Model, Marcel

Dekker Inc., New York, 1978. 6. Bates, G. E. Joint distribution of time intervals for the occurrences of

successive accidents in a generalized polya schema, Ann. Math. Stats, 26 (1955), pp. 705-20.

7. Cox, D. R. and Lewis, P. A. W. The Statistical Analysis ojSeries of Events, Methuen & Co. Ltd, London, 1966.

8. Crow, L. H. Reliability analysis for complex, repairable systems. In Reliability and Biometry (Proschan, F. and Serfling, R. J. (Eds)), SIAM, Philadelphia, 1974, pp. 379-410.

9. Crow, L. H. Confidence interval procedures for the Weibull process with applications to reliability growth, Technometrics, 24 (1982), pp. 67-72.

10. Finkelstein, J. M. Confidence bounds on the parameters of the Weibull process, Technometrics, 18 (1976), pp. 115-17.