## 4 Analytical Framework for Heterogeneous Service Provisioning
### Problem Overview and Healthcare Service Requirements
#### 4.1.1 Utility-Based Formulation for Service Delivery Rate  
To support heterogeneous healthcare services, we adopt a utility-based formulation that maps the service delivery rate to a corresponding utility value, representing user satisfaction or QoS level. In the context of healthcare IoT services, user satisfaction is quantified in terms of service-specific QoS goals. For latency-sensitive services such as ECG streaming or fall detection, it corresponds to minimizing detection and alerting delay. For bandwidth-intensive services like real-time video consultations, it translates to maintaining a continuous stream at the required bitrate. For latency-tolerant tasks (e.g., EHR access), satisfaction reflects the ability to complete data transactions within acceptable thresholds. Thus, the utility function aligns system-level delivery rates with the perceived performance at the application level.  
Let \(g_{s,i}\) denote the service delivery rate for consumer \(i\) receiving service \(s\). The utility function \(U_{s}(g_{s,i})\) is defined as a strictly increasing, continuous, and positive function over the interval \([m_{s},M_{s}]\), where \(m_{s}\) and \(M_{s}\) denote the minimum and maximum allowable rates for service \(s\), respectively. However, traditional utility functions may fail to maintain concavity for latency-sensitive services, which is essential for tractable convex optimization. Therefore, we introduce a pseudo-utility function \(\mathcal{U}_{s}(g_{s,i})\) that satisfies strict concavity, ensuring the global optimality of the rate allocation. The utility function is defined as:  
\[\mathcal{U}_{s}(g_{s,i})=\int_{m_{s}}^{g_{s,i}}\frac{1}{U_{s}(y)}dy,\quad m_{s }\leq g_{s,i}\leq M_{s}. \tag{1}\]This transformation allows us to reformulate the service provisioning task as a convex optimization problem, aiming to maximize the total utility across all services and consumers:  
\[\text{P1:}\quad\underset{\vec{g}\geq 0}{\text{maximize}}\quad\sum_{s\in S}\sum_{i=1}^ {n_{s}}\mathcal{U}_{s}(g_{s,i}), \tag{2}\]  
subject to the following constraint:  
\[\sum_{s\in S}g_{s}^{t}\leq c_{t},\quad\forall t\in T, \tag{3}\]  
where \(g_{s}^{t}=\max_{\{i|t\in T_{s,i}\}}g_{s,i}\) represents the maximum service delivery rate for service \(s\) on link \(t\), and \(c_{t}\) is the capacity of the link. This constraint ensures that the total service delivery rate on any link does not exceed its capacity. To maintain differentiability and facilitate the optimization process, we approximate the non-differentiable max operator using:  
\[g_{s}^{t}\approx\left(\sum_{\{i|t\in T_{s,i}\}}g_{s,i}^{n}\right)^{\frac{1}{n}}, \tag{4}\]  
where \(n\) is a large integer. This leads to the reformulated convex optimization problem P2:  
\[\text{P2:}\quad\underset{\vec{g}\geq 0}{\text{maximize}}\quad\sum_{s\in S} \sum_{i=1}^{n_{s}}\mathcal{U}_{s}(g_{s,i}), \tag{5}\]  
\[\text{subject to}\quad\sum_{s\in S}\left(\sum_{\{i|t\in T_{s,i}\}}g_{s,i}^{n} \right)^{\frac{1}{n}}\leq c_{t},\quad\forall t\in T. \tag{6}\]  
This utility-based formulation provides a scalable and flexible approach to optimize heterogeneous healthcare services, balancing latency sensitivity and resource efficiency. It also serves as a foundation for integrating learning-based techniques in the subsequent sections.