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Passenger arrival process considering batch size distribution can be modelled according to compound Poisson process (Ross 1992). The compounding batch size distribution depends on the time of day and the floor utilization (Kuusinen et al. 2012). The random passenger batch arrivals to elevator lobbies is a generalization of ordinary Poisson process {N(t), t ≥ 0} to a time‐inhomogeneous compound Poisson process with piecewise constant arrival rates

(3.7)

where X(t) is the number of passengers arriving until time t, N(t) is the number of batch arrivals and D_i is the ith batch size (Sorsa et al. 2013). Theory whether the social group size in elevator traffic follows a zero‐truncated Poisson distribution, and geometric (Johnson et al. 2005) as well as logarithmic distributions (Fisher et al. 1943) was studied. The zero‐truncated Poisson distribution for the number of occurrences, k, never gets the value zero. In generating people flow in a building, within a time interval, the number of batch arrivals would then be greater than zero. The probability that the observed group size X equals k > 0 is

(3.8)

where p is the parameter characteristic to each distribution estimated from the observed data. The mean value for zero‐truncated Poisson is . The value for p and the suggested distributions for each building type can be solved by equalizing the observed mean value to the theoretical mean.

The geometric distribution is used to model the number of independent Bernoulli trials, X, until the first success and it is defined as

(3.9)

where p is as defined as above. The mean of this distribution is .

The logarithmic distribution has been used to model, for example, the abundance of species in a catch. The common species are represented by X > 1 individuals with a decreasing probability with respect to X. The logarithmic distribution is defined as

(3.10)

The mean of logarithmic distribution is . The mean batch size was used to generate the zero‐truncated, the geometric and the logarithmic Poisson probabilities of Eqs. (3.8)–(3.10), respectively. All three given theoretical distributions are close to each other. The zero‐truncated and logarithmic distributions do not match with the mean distribution since the distributions vary considerably especially in observed hotels. The geometric distribution of Eq. (3.9) was found to fit best to the average daily batch size distribution (Sorsa et al. 2021).

If the geometric distribution is used in the compound Poisson process (Sorsa et al. 2018), the batch arrival process can be modelled as a Pólya–Aeppli process (Johnson et al. 2005):

(3.11)

In simulation, the generated passenger arrivals follow Poisson distribution, and the passengers are grouped, e.g. according to the geometric distribution. The observation of batch arrival process better explains the real passenger traffic phenomenon. It affects the basics of the design of elevators as well as the modelling of passenger traffic in buildings.