## 2 Materials and Methods
To mathematically model the system's behavior under load, we adopted a classical M/M/1 queueing framework, which assumes Poisson arrival and exponential service times--an appropriate abstraction for cloud-based telemetry pipelines. Within this model, let \(\lambda\) denote the average arrival rate of data messages (e.g., telemetry events sent by IoT devices), and let \(\mu\) represent the average service rate of the processing subsystem (comprised of Stream Analytics and SQL Database layers). We analyze four key performance indicators derived from queueing theory:  
The **average waiting time in the queue**\(Wq\)** Equation (2), which quantifies the expected time a message spends waiting to be processed after arrival, is given by:  
Figure 2:0  
The **average number of messages in the queue**\(Lq\) [Equation (3)], which reflects the instantaneous queue length in steady state, is defined as:  
Figure 2:1  
The **system utilization**\(\rho\), (Equation (4)) which captures the proportion of time the processing unit is actively busy, is calculated as:  
Figure 2:2  
The **average time in the system**\(W\) (Equation(5)), encompassing both the queueing and processing phases, is expressed as:  
Figure 2:3  
These analytical expressions were used to estimate system responsiveness under both unsecured and secured configurations. Empirical estimates for \(\lambda\) and \(\mu\) were derived from the experimental data based on interarrival intervals and measured end-to-end delays. The results provided a theoretical baseline for comparing system performance and validating the observed increase in latency after the implementation of security controls. To evaluate the impact of the implemented security measures on the overall performance of the UBT Smart City IoT Cloud system, precise timing measurements were collected at three critical stages of the data flow: (i) the moment telemetry data was sent by a simulated IoT device (via Postman), (ii) the moment it was received and enqueued by the Azure IoT Hub into Stream Analytics, and (iii) the moment the data was processed and stored in the Azure SQL Database. From these timestamps, the **Total Delay** was calculated,representing the end-to-end time required to transmit, analyze, and store a data point. This delay, expressed in both seconds and milliseconds, serves as a primary performance indicator for assessing the system's responsiveness. We applied an independent two-sample t-test to determine whether the mean delay differed significantly between the unsecured and secured configurations. The null hypothesis (H\({}_{0}\)) assumed no significant difference in mean delay, while the alternative hypothesis (H\({}_{1}\)) proposed a statistically significant increase in delay after implementing security measures. Prior to conducting the t-test, [PERSON]'s test was used to confirm homogeneity of variances. Table 1 presents the results of measurements taken **before the implementation of security measures**. As shown, the average delay across 16 consecutive test samples was **626.15 milliseconds**, reflecting a fast and efficient data pipeline, albeit one with minimal access control and exposure to potential vulnerabilities. Following this baseline, Table 2 displays the results collected **after the deployment of security mechanisms**, which include the configuration of VPN access, NSGs (Network Security Groups), and custom routing rules. In this case, the average delay increased to **745.88 milliseconds**, indicating a measurable performance overhead introduced by the added security layers. The data presented in both tables was further analyzed using mathematical modeling via the M/M/1 queueing system and statistical hypothesis testing, confirming that the observed increase in latency is statistically significant. This supports the conclusion that while security measures introduce computational overhead, the system remains within acceptable performance thresholds and benefits from significantly improved data protection, confidentiality, and controlled access.