## 6 Discussion
### Future Work  
Directions for further work include several key areas of improvement and expansion. First, there is a need for the improvement of mathematical models. Future research should focus on enhancing the existing mathematical models to incorporate new factors, such as workload changes [102], adaptation to technological shifts [103], and the prediction of evolving needs in distributed computing systems. The current use of the M/M/n model's assumption [104] of exponential inter-arrival and service times (\(c_{a}^{2}=c_{s}^{2}=1\)) simplifies analysis via the Erlang-C formula but fails to capture observed burstiness (\(c_{a}^{2}>1\)) and heavy-tailed service times (e.g., Pareto or Weibull) in production traces [105]. Adopting a G/G/n framework allows specification of the squared coefficients of variation \(c_{a}^{2}\) and \(c_{s}^{2}\), enabling use of [PERSON]'s approximation for mean waiting time  
\[\mathbb{E}[W]\approx\frac{\rho\sqrt{2(n+1)}-1}{n(1-\rho)}\cdot\frac{c_{a}^{2}+ c_{s}^{2}}{2}\,,\]  
and diffusion-limit results in the Halfin-Whitt (QED) regime [89] for large \(n\). Such generalizations can be implemented via matrix-analytic methods or fluid approximations, yielding accurate bounds on queue lengths and server utilizations under non-Markovian traffic [106]. Transitioning to G/G/n requires estimating \(c_{a}^{2}\) and \(c_{s}^{2}\) from empirical workload logs (e.g., MapReduce job arrivals) and integrating these into discrete-event simulations or asymptotic formulas. Although this increases computational overhead, it provides tighter performance envelopes and more reliable energy-availability trade-offs. In particular, accounting for long-tail service variances can prevent underestimation of queueing delays and overcommitment of resources. Rather than revising the entire framework, future work will extend the existing M/M/n analysis by (1) fitting general arrival and service distributions to collected DCS workload data, (2) applying [PERSON]'s and Halfin-Whitt approximations to derive new performance bounds, and (3) validating these against simulation results. This incremental approach preserves analytical tractability while aligning the model with realistic, non-exponential behaviors in distributed neural-network training workloads. The existing reliability assessment assumes identical nodes and independent failure events, which understates the impact of cascading outages in multi-tier or mesh networks. In practical deployments, individual servers exhibit distinct failure rates and repair times that depend on hardware characteristics, workload, and network position, and a fault in one node can elevate the failure risk of its neighbors [107]. To address this, the model can be extended using multi-state Continuous-Time Markov Chains [108] in which each node transitions among operational, degraded, and failed states with empirically derived rates \(\lambda_{i}\) and \(\mu_{i}\), and inter-node dependencies are captured by state-dependent modifiers that increase a neighbor's failure rate by a conditional probability \(p_{ij}\). Complementing this, a fault-propagation graph represents the system as a directed network \(G=(V,E)\), where edges weighted by \(p_{ij}\) quantify the likelihood that failure of \(i\) triggers failure of \(j\), enabling system availability to be computed via minimal cut-set enumeration or Monte Carlo simulation in large meshes [109]. For hierarchical, multi-tier architectures, tier-specific redundancy factors \(r_{k}\) and inter-tier dependency coefficients \(\delta_{k,k+1}\) can be incorporated into a layered reliability block diagram [110] to derive overall uptime \(A=\prod_{k}A_{k}^{\delta_{k,k+1}}\). Finally, Dynamic Bayesian Networks can model the temporal evolution of faults and repairs, accommodating arbitrary sojourn time distributions and allowing real-time inference of unavailability and cascade likelihood [111].