Читать книгу People Flow in Buildings - Marja-Liisa Siikonen - Страница 35

Random events, such as the number of phone calls per hour received by a call centre follow Poisson distribution. If the traffic peak period is divided into intervals of length t, the expected number of passenger arrivals within time t is N(t) = λt, where λ is the passenger arrival rate. For the Poisson distribution, the probability density function for N(t) arrivals within time t is

(3.1)

The mean and the variance σ² of Poisson distribution are λt. Poisson arrival process is illustrated in Figure 3.2. Passenger arrivals follow Poisson process, when they are independent and memoryless, i.e. when the number of arrivals within an interval does not affect the probability of arrivals during later intervals. In a homogeneous Poisson process, arrival rate, λ, is constant for intervals of different lengths.

Figure 3.2 Probability density function for Poisson arrival process.

Table 3.2 X ² ‐test results for passengers exiting through the security gates, office in Warsaw.

Time of day	Arrival interval (s)	X 2	Degrees of freedom	X 2 0.95	X 2 0.05	Suitably of H0‐hypothesis
Daily 7:00–19:00	10	4818	7	2.2	14.1	Poor
30	7265	11	4.6	19.7	Poor
60	1872	14	6.6	23.7	Poor
300	911	15	7.3	25.0	Poor
8:00–9:00	10	12.6	3	0.4	7.8	Poor
30	6.9	4	0.7	9.5	Good
60	3.7	4	0.7	9.5	Good
300	7.6	10	3.9	18.3	Good
17:00–18:00	10	579	10	3.9	18.3	Poor
30	614	15	7.3	25.0	Poor
60	6.3	14	6.6	23.7	Fair
300	107	15	7.3	25.0	Poor

Alexandris (1977) performed manual counting of passenger arrivals in three office building lobbies in morning up‐peak situations, and reported the results in his doctoral thesis. Passenger arrivals were observed in a 10‐second resolution for a one‐hour period in each building. He analysed the data in different intervals, in Figure 3.3 in a 30‐second interval, and made an assumption that passengers arrive randomly following a Poisson process. In a Poisson counting process, the time differences between passenger arrivals, i.e. inter‐arrival times, are exponentially distributed. To validate the assumption of Poisson distributed arrivals, Alexandris made a Chi‐square test for the measured data. The conclusion of the study was that passenger arrivals in elevator lobbies during morning up‐peak period can be accepted to follow a Poisson process, or at least this assumption cannot be rejected. Currently, the assumption of Poisson arrival theory is widely applied in modelling people flow in buildings. According to the Poisson process, passengers arrive randomly in elevator lobbies with a defined arrival rate.

Figure 3.3 Number of occurrences as a function of number passengers analysed in 30‐second intervals for a one‐hour period

(Source: Alexandris (1977). © John Wiley & Sons).