## 6 Discussion
### Limitations  
However, several limitations were identified during this research that must be addressed in future work. First, the current mass-service (M/M/n) model--while tractable for analysis and optimization--fails to capture complex client-server dependencies and dynamic interactions observed in real-world distributed computing environments [86]. Extending the framework to a G/G/n setting would account for general arrival and service distributions, but this generalization renders closed-form analysis more challenging. To maintain rigorous performance guarantees under both M/M/n and G/G/n assumptions, exponential tail bounds on queue lengths and waiting times can be derived via Chernoff inequalities [87; 88]. Specifically, if \(Q(t)\) denotes the queue length at time \(t\), then for any \(\theta>0\),  
\[\Pr\bigl{(}Q(t)>x\bigr{)}\ \leq\ \exp\bigl{(}-\theta x+\Lambda(\theta)\,t \bigr{)},\]  
where \(\Lambda(\theta)\) is the cumulant-generating function of the net arrival process; optimizing \(\theta\) yields a tight upper bound. In the Halfin-Whitt (QED) regime [89], diffusion-limit approximations further provide asymptotic lower bounds such as \(\mathbb{E}[W]\gtrsim(1-\rho)^{-1}/\sqrt{n}\), characterizing performance under heavy traffic. For availability analysis, minimal cut-set enumeration in a Continuous-Time Markov Chain model yields complementary upper and lower bounds. If \(\mathcal{C}_{1},\cdots,\mathcal{C}_{m}\) are minimal cut-sets with component availabilities \(A_{i}\), then  
\[1-A\ \leq\ \sum_{k=1}^{m}\prod_{i\in\mathcal{C}_{k}}(1-A_{i}),\]  
while inclusion-exclusion provides corresponding lower bounds. For non-exponential or multi-state components, matrix-analytic methods in the Laplace domain invert to explicit time-dependent availability bounds. Stability and convergence under extreme loads or variability can be established using Foster-Lyapunov criteria [90] and fluid-limit arguments. By constructing a Lyapunov function \(V(x)\), for example \(V(x)=x^{2}\) where \(x\) is the queue length, and demonstrating a negative drift  
\[\mathbb{E}[V(Q(t+1))-V(Q(t))\ |\ Q(t)=x]\ \leq\ -\ e\quad\text{for }x>K,\]  
one ensures positive Harris recurrence and geometric ergodicity. Fluid-limit techniques then show that suitably scaled processes converge to deterministic trajectories, guaranteeing the existence and uniqueness of long-run performance and availability averages. Sensitivity analysis of these bounds with respect to load \(\rho\), squared coefficients of variation \(c_{a}^{2},c_{s}^{2}\), and failure or repair rates allows identification of critical thresholds beyond which performance degrades sharply or availability targets cannot be met. These closed-form bounds and convergence proofs will both validate empirical and simulation results and provide practitioners with explicit guarantees for system behavior across a wide spectrum of operational conditions. To empirically validate the proposed analytical models of performance and energy consumption, a dedicated verification section will be introduced using a controlled hardware testbed. This testbed will consist of a small cluster of heterogeneous servers--each equipped with Intel CPUs supporting RAPL power-measurement interfaces and NVIDIA GPUs exposing power metrics via NVML--so that both peak and real-time power draw can be recorded with millisecond precision. Prior to workload execution, each node's peak FLOPS capacity and base power consumption will be profiled through microbenchmarks, and network latency and bandwidth will be calibrated using tools such as iPerf3. Representative deep-learning workloads--such as training a ResNet-50 on ImageNet and a convolutional autoencoder on CIFAR-10--will be deployed across the cluster. During each run, CPU/GPU utilization, core clock frequencies, and instantaneous power consumption will be logged alongside throughput (GFLOPS) and end-to-end training time.