# Analyzing the Economic Performance of a TLS-Based Structural Safety Diagnosis Process
## 2 Materials and Methods
### Research Methodology
#### 2.4.2 Queue Method
The equations for \(L\), \(W\)and \(W_{q}\) in the M/D/1 model are the same as Equations (2)-(4) in the M/M/1 model, but \(L_{q}\) is represented by the following Equation (5):  
\[L_{q}=\frac{\rho^{2}}{2(1-\rho)}=\frac{\left(\frac{\lambda}{\mu}\right)^{2}}{2 \left(1-\frac{\lambda}{\mu}\right)} \tag{5}\]
* Probabilistic Analysis of Customer Counts and Wait Times  
Queueing models allow for probabilistic analysis of the state of queueing systems, including the number of customers and waiting times. When server utilization (\(\rho\)) is high, the state of the system may deteriorate, leading to waiting costs that reduce economic efficiency. The performance of the server can be evaluated by probabilistically predicting how many customers are present in the queueing system. \(P_{n}\) represents the steady-state probability that there are \(n\) customers in the system. The equations describing \(P_{n}\) are represented in the following Equations (6) and (7):  
\[P_{n}=C_{n}\times P_{0} \tag{6}\]  
\[C_{n}=\frac{\lambda_{n-1}\,\lambda_{n-1}\,\lambda_{n-1}\,\cdots\lambda_{0}}{ \mu_{n}\,\mu_{n-1}\,\mu_{n-2}\cdots\mu_{1}} \tag{7}\]  
Additionally, the probability associated with customer waiting time (\(t\)) can be represented by Equations (8) and (9). Equation (8) expresses the probability that a customer's waiting time in the queueing system exceeds \(t\), whereas Equation (9) indicates the probability that the customer's waiting time, excluding service time, exceeds \(t\), as follows:  
\[P(W>t)=e^{-\mu t}\left[1+\frac{P_{0}\left(\frac{\lambda}{\mu}\right)^{s}}{s!(1 -\rho)}\left(\frac{1-e^{-\mu(s-\frac{\lambda}{\mu}-1)}}{s-\frac{\lambda}{\mu} -1}\right)\right] \tag{8}\]  
\[P\big{(}W_{q}>t\big{)}=\left(1-\sum_{n=0}^{s-1}P_{n}\right)e^{-s\mu(1-\rho)t} \tag{9}\]
* Analyze Economic Performance Considering Cost of Service and On-call Costs Economic performance analysis using queueing models makes it possible to derive economic outcomes by considering both the server's service cost and the customers' waiting costs incurred due to delays. For economic analysis based on queueing theory, daily and total costs (\(TC\)) are estimated using basic performance indicators (e.g., average number of customers, average waiting time, etc.) derived from the analytical models. By simultaneously considering service costs (\(SC\)) and waiting costs (\(WC\)), comparisons of costs between workforce-based approaches and TLS-based 3D scanning reverse modeling methods, as well as effects on reducing process duration, can be calculated. \(C_{s}\) represents the input cost per server unit per unit time, \(s\) represents the number of servers, and \(C_{w}\) represents the waiting cost per customer per unit time. The equations used for the analysis are represented in the following Equations (10)-(12):  
\[L_{q}=\frac{\lambda^{2}}{\mu(\mu-\lambda)} \tag{1}\]0  
\[L_{q}=\frac{\lambda^{2}}{\mu(\mu-\lambda)} \tag{1}\]1  
\[L_{q}=\frac{\lambda^{2}}{\mu(\mu-\lambda)} \tag{1}\]2  
* Analyze the ROI of adopting digital technologiesFinally, the return on investment (ROI) evaluation is conducted. Investment costs, such as purchasing scanning equipment and software licenses, are defined, and the project revenue generated by adopting the TLS-based 3D scanning reverse modeling approach is calculated. Cumulative ROI at both the individual project level and company level (multiple projects) is analyzed to provide a basis for long-term investment decision-making. The outcomes from three case projects (A, B, and C) are summarized to compare performance before and after the introduction of digital technologies. This allows for a long-term economic evaluation of the company's investment in equipment and its continuous application across multiple projects.