# Analyzing the Economic Performance of a TLS-Based Structural Safety Diagnosis Process
## 2 Materials and Methods
### Research Methodology
#### 2.4.2 Queue Method  
The waiting-line model is a model that probabilistically represents the state of a waiting system and mathematically analyzes the process by which customers or tasks arrive at and are served in a queue [80; 81; 82]. Each diagnostic method (manpower-based visual inspection, TLS-based 3D scanning reverse modeling) is defined as a production  
Figure 1: Overall research methodology flowchart. system, from which key parameters, such as the customer (inspection area, length, etc.) and the server (diagnostic team), are derived. We apply two models to quantify the waiting time, service time, service cost, and waiting cost that occur in the production system using the queuing theory. In the existing human-based diagnostic method, the M/M/1 model, which follows an exponential distribution, is applied in consideration of service deviations due to server uncertainties. In the TLS-based 3D scanning reverse modeling diagnostic method, the M/D/1 model, which follows a deterministic distribution, is applied in consideration of the constant provision of services by the server through the introduction of technology. In a queueing system, the time between the arrival of a customer is called the interarrival time. In a typical queueing system, it is almost impossible to predict when the next customer will arrive. However, if sufficient data are provided to the customers arriving at the queueing system, the average number of customers arriving per unit time can be estimated. This is called the average arrival rate (\(\lambda\)). Assuming that the server operates without interruption during the service time, the average number of customers that the server can process per unit time is called the average service rate (\(\mu\)). Once the customer arrival rate and server service rate values are determined, the server utilization rate (\(\rho\)), which is the ratio of the time spent processing customers by the server during the entire business hours, can be obtained. The server utilization rate (\(\rho\)) can be calculated by dividing the average arrival rate (\(\lambda\)) by the average service rate (\(\mu\)). * M/M/1 Model  
The first symbol (M) in the M/M/1 model, which is a model that takes into account the uncertainty of the server, is the probability distribution of the time between customer arrivals, and the second symbol (M) is the probability distribution of service time. The third symbol (1) refers to the number of servers. In the M/M/1 model, there are four indicators for analyzing the basic performance of a production system, namely \(L\), \(L_{q}\), \(W\), and \(W_{q}\). \(L\) is the average length of customers waiting in the entire queueing system, \(L_{q}\) is the average length of customers in the queueing system, excluding customers being served, \(W\) is the average waiting time from the arrival of a customer unit to the end of service in the queueing system, and \(W_{q}\) is the average waiting time from the arrival of a customer unit to the end of service, excluding the service time just before service begins. The equations describing these four indicators are shown in the following Equations (1)-(4):  
\[L_{q}=\frac{\lambda^{2}}{\mu(\mu-\lambda)} \tag{1}\]  
\[W_{q}=\frac{L_{q}}{\lambda} \tag{2}\]  
\[W=\frac{L}{\lambda}=W_{q}+\frac{1}{\mu} \tag{3}\]  
\[L=L_{q}+\frac{\lambda}{\mu} \tag{4}\]
* M/D/1 Model  
In the M/D/1 model, which considers the deterministic service provided by the server, the first symbol (M) denotes the probability distribution of interarrival times between customers, while the second symbol (D) denotes the deterministic probability distribution of service times. The third symbol (1) indicates the number of servers. In the M/D/1 model, there are four performance indicators for analyzing the basic performance of a production system, namely \(L\), \(L_{q}\), \(W\), and \(W_{q}\).