# Identification of Bottlenecks in Passenger Handling Processes Using Data-Driven Tools
## 3 Development and Excel-Based Implementation of the Simulation Model
### Simulation of the Check-In Process
Random service times are generated using an exponential probability distribution, with the cumulative distribution function given by:  
\[\text{F(x)}=1-\text{e}^{-\lambda\times\text{x}}\text{, for x}\geq 0. \tag{2}\]  
where:  
* F(x) is the probability that the service is completed within time x;
* \(\lambda\) is the service rate (i.e., intensity of service);
* x is the service time in hours. The service rate \(\lambda\) represents the number of passengers that can be served per hour and is defined as the reciprocal of the mean service time t_obs, as follows:  
\[\lambda=1/(\text{ t_obs}) \tag{3}\]To generate random values that follow the exponential distribution, the inverse transform method is used. Assuming a uniformly distributed random number u (0,1) we set the following:  
\[\text{u}=\text{F(x)}=1-\text{e}^{-\lambda\times\text{x}} \tag{4}\]  
Solving for x:  
\[\text{e}^{-\lambda\times\text{x}}=1-\text{u} \tag{5}\]  
\[-\lambda\times\text{x}=\text{ln}(1-\text{u}) \tag{6}\]  
\[\text{x}=-\frac{1}{\lambda}\times\text{ln}(1-\text{u}) \tag{7}\]  
Substituting the service rate from Equation (3), we obtain the final form, as follows:  
\[\text{x}=-\text{t}\_\text{obs}\times\text{ln}(1-\text{u}) \tag{8}\]  
Since the RAND() function in Microsoft Excel returns a uniformly distributed value in the interval (0, 1) and substituting the variable x as service time t_sn, we can directly write the following:  
\[\text{t}\_\text{sn}=-\text{t}\_\text{obs}\times\text{LN}(\text{RAND}0) \tag{9}\]  
where:  
* t_sn is the generated service time of the n-th passenger;
* t_obs represents the average service time of a facility as an input parameter;
* LN0 is an Excel-based function of the natural logarithm;
* RAND() is an Excel-based function for generating random numbers within the interval (0,1). This Excel formula enables the simulation to generate random service durations for each passenger in accordance with the exponential distribution. Most service times will cluster around shorter durations, but occasional longer service times may occur, reflecting the realistic variability of passenger handling at check-in desks. Therefore, in order to generate values t_sn in the simulation, it is necessary to define t_obs for the check-in desk. In the simulation model, the same t_obs value is applied to both check-in desks to simplify calculations. This parameter can be derived from empirical observation or obtained through data collection and statistical analysis, and it can be adjusted to test different operational scenarios. In this phase of the handling process, the simulation model evaluates several key time-related characteristics for each passenger. These parameters are calculated using a combination of generated input data and logical sequencing rules based on queueing theory. Each of them is computed in Microsoft Excel and collectively form the basis for performance evaluation at the check-in desks. The specific parameters are as follows:  
* \"Transfer duration in hours\"--time needed to reach the check-in area after entering the terminal, generated using a normal distribution. * \"Arrival of PAX\"--actual time of arrival at the check-in area, calculated as the sum of airport arrival time and transfer duration. * \"Arrival in order\"--arrival times sorted in ascending order to determine the sequence of processing. * \"Service time\"--randomly generated processing time at the check-in desk based on exponential distribution. * \"Start time\"--time when service begins, determined by passenger arrival time and availability of service channels. * \"End time\"--time when check-in is completed, calculated as the sum of start time and service time. * \"Time in system (min)\"--total time spent from arrival at the check-in area to completion of service. * \"Queue time (min)\"--time spent waiting in queue before service starts. To simplify the model, it is assumed that both check-in desks have the same average service time for a single passenger.