## 4 Analytical Framework for Heterogeneous Service Provisioning
### Problem Overview and Healthcare Service Requirements
#### 4.1.2 Delay-Aware Utility Adjustment  
In smart healthcare environments, many IoT-based services are highly delay-sensitive. Applications such as real-time cardiac monitoring, fall detection, and emergency alerts require not only optimal service delivery rates but also stringent latency guarantees. To account for these latency constraints, we extend the utility model to penalize delays incurred during service delivery. In this work, we model the delay function \(d_{s}(g_{s,i}^{t})\) using a hybrid approach. For analytical formulation and theoretical derivation, we assume an \(M/M/1\) queuing model, which provides a tractable and convex approximation of delay under varying arrival rates. For simulation and learning-based integration, delay is estimated using a supervised regression model trained on network telemetry data, capturing variations due to congestion, link quality, and mobility. This dual representation ensures that our optimization model remains mathematically robust while being practically deployable in real-world smart healthcare settings.  
Let \(d_{s}(g_{s,i}^{t})\) denote the average delay experienced by consumer \(i\) of service \(s\) when traversing link \(t\). The cumulative delay experienced by a consumer over the entire delivery path is represented as:  
\[D_{s,i}=\sum_{\{t\in T_{s,i}\}}d_{s}(g_{s,i}^{t}), \tag{7}\]where \(T_{s,i}\) is the set of links in the delivery path from the service provider to consumer \(i\) for service \(s\). To incorporate delay sensitivity, we define a delay-weighted utility function \(\mathcal{U}_{s}^{*}(g_{s,i})\) as follows:  
\[\mathcal{U}_{s}^{*}(g_{s,i})=\mathcal{U}_{s}(g_{s,i})-\varphi_{s}D_{s,i}, \tag{8}\]  
where \(\varphi_{s}\geq 0\) is a delay penalty coefficient that captures the importance of latency for each service \(s\). The delay penalty coefficient \(\varphi_{s}\) can be tuned according to the criticality of the service. For example, for emergency alerts or ICU monitoring, \(\varphi_{s}\) is set high to enforce strict latency constraints, whereas for routine data logging or archival services, it may be near zero. For highly delay-sensitive services, \(\varphi_{s}\) is large, ensuring that the optimization penalizes high-delay paths more heavily. For latency-tolerant services, \(\varphi_{s}\) can be set to zero or a small value.  
Substituting the delay-adjusted utility function into the optimization objective leads to:  
\[\text{P3:}\quad\underset{g\geq 0}{\text{maximize}}\quad\sum_{s\in S}\sum_{i=1} ^{n_{s}}\left(\mathcal{U}_{s}(g_{s,i})-\varphi_{s}\sum_{t\in T_{s,i}}d_{s}(g_{ s,i}^{t})\right), \tag{9}\]  
subject to the same capacity constraints on links as defined earlier. This formulation balances the trade-off between high service delivery rates and low end-to-end latency, making it well-suited for real-time healthcare applications where both factors are mission-critical.  
The delay function \(d_{s}(g_{s,i}^{t})\) can be modeled using queuing theory (e.g., M/M/1 or M/D/1 approximations), empirical measurements, or predicted via ML models, which will be discussed in the next subsection.