# Multi-formalism Models for Performance Engineering
## 4 Multi-Formalism QN/PN Techniques
### Batching in IoT-Based Healthcare
#### 4.2.3 Model Results  
We considered several configurations of the system by modifying the global arrival rate \(\lambda\), the threshold \(\beta\) of Urgent requests, and the empty buffer cycle time. In this section, we limit the description of the results to those that emphasize the impact of the dynamic management of the buffer of requests on the performance of the system and related costs. The arrival rates of the requests considered in the study are \(\lambda=\ 0.1\div 1.9\,req/s\). The inter-arrival times are exponentially distributed. These values were assumed to be representative of the potential number of patients in an emergency ward of small- or medium-sized hospitals. Figure 21 shows the behavior of the System Response Time R, i.e., the end-to-end time required by a signal from its detection to the completion of its processing by the hospital server, for the complete range of arrival rates. The family of curves refers to different values of \(\beta\), from 2 to 20, i.e., the threshold for the identification of Urgent batches of requests will be transmitted immediately. With a \(\lambda\) increase, we model an increase in the number of sensors or in the detection rate and the workload managed by Edge nodes is also greater. For small values of \(\beta\) (2 and 5), there is an initial decrease of R. This is because, with such small sizes, nearly all batches are considered Urgent and therefore most requests are transmitted almost immediately when they join the Buffer. With larger values of \(\beta\) (10, 15, and 20) this initial decrease of R as \(\lambda\) increases is not present. When \(\lambda\) becomes greater than 0.1 the value of R starts to increase from the beginning. With these batches sizes, R begins to decrease for larger \(\lambda\) values than those obtained with smaller \(\beta\). The motivation for this behavior is that, as \(\lambda\) increases, the number of requests in the buffer increases as well, and if \(\beta\) has higher values the threshold is reached with less probability and the number of batches transmitted when the period expire is greater. However, with further increases of \(\lambda\), the threshold is reached more easily and therefore there is a shorter waiting time for the requests in the buffer, i.e., the Urgent requests increase. When \(\lambda\) is greater than 1.5 req/s, R increases for all \(\beta\) since the response time of the highly utilized hospital server becomes the dominant part of its value. According to the objectives of the study, the R values should be analyzed together with the cost (energy consumption) for transmissions. It is assumed that the cost is directly proportional to the number of times a batch is transmitted, i.e., a buffer is emptied. Indeed, the greater are the number of times and the sizes of the transmitted requests at the same time, the better is the energy efficiency of the system. In Figure 22, the transmissions per second are shown for all the values of the considered \(\lambda\)  
Figure 21: System eesponse time vs. arrival rate of requests for various values \(\beta\) of the batch sizes for Urgent requests. As expected, with batch size \(\beta=2\), the number of transmissions per second are the highest, while the minimum for \(\beta=20\). However, to have a meaningful result, these values should be considered together with the system response times. To this end, the metric system power, introduced in [33] and combining the throughput X of a system with its response time R, is considered. This metric is the ratio X/R of throughput and response time, and captures the level of efficiency in executing a workload. The maximum power corresponds to the optimal operating point for the system, i.e., the point at which the throughput is maximized with the minimum response time. In our system, we have considered the ratio of transmissions/s and aystem response time. Figure 23 shows the ratio of the two metrics of Figures 22 and 21.