Memory, Batch, and Resource Allocation Models

• Memory, batch, and resource allocation models are foundational frameworks that define and optimize the distribution of computational, storage, and communication resources across diverse systems.
• They integrate statistical analysis, queue scheduling, and hierarchical algorithms to tackle resource heterogeneity and trade-offs in fairness, latency, and robustness.
• Applications range from cloud service scheduling and GPU batching in deep learning to agent-based and cognitive models, improving throughput and system resilience.

Memory, batch, and resource allocation models constitute fundamental frameworks for describing and optimizing the distribution and utilization of computational, storage, and communication resources across diverse computing environments—including cloud clusters, edge servers, high-performance computing (HPC) systems, deep learning accelerators, and even human cognition in language processing. These models encompass statistical, algorithmic, and analytical approaches to characterize and predict resource heterogeneity, understand coupling between resources (such as CPU, memory, and I/O), inform optimal assignment or scheduling strategies, and evaluate trade-offs imposed by constraints such as limited capacity, fairness, latency, and robustness.

1. Statistical and Correlated Resource Modeling

Resource modeling at Internet scale requires capturing not only the distributions of individual resources (e.g., core count, memory, disk) but also their temporal evolution and inter-dependencies. Trace-based statistical models, such as those based on the SETI@home dataset (2.7 million hosts, five years), employ discrete and continuous distributions:

• Discrete Resources: Quantities like core count and memory are modeled using exponential laws to reflect technology-driven changes in distribution ratios over time (e.g., for relative 1:2 core count: $a \cdot e^{b \cdot (\text{year} - 2006)}$).
• Continuous Resources: Processor speeds (via Dhrystone and Whetstone benchmarks) are modeled as correlated normal random variables; available disk space is characterized by a log-normal distribution.
• Correlation Analysis: Empirically observed Pearson correlations (e.g., $r \approx 0.606$ for cores and total memory) motivate the use of multivariate models (utilizing Cholesky decomposition on the observed correlation matrix $R$).

Validation is performed by synthesizing host populations and matching distributional and correlation structure to real data. The correlated resource model achieves significantly lower deviation (0–10%) in application utility estimation compared to uncorrelated or simplistic models (17–31%), with broad applicability for resource allocation in Internet-distributed systems and forecasting host population trends (Heien et al., 2010).

2. Batch Allocation and Scheduling Under Resource Constraints

Effective batch allocation models focus on the relationship between batch size, queueing, and resource utilization, especially under strict memory or computational limits:

• Cloud Clusters: In stochastic queueing settings, queue dynamics are expressed as

$q_{j,m}(t+1) = q_{j,m}(t) + a_{j,m}(t) - s_j^m(t)$

where $q_{j,m}$ is the type-$j$ queue at server $m$, and $a_{j,m}$ and $s_j^m$ denote arrivals and service allocation. Multi-dimensional heavy-traffic analysis (parameterized by $\epsilon$ as the distance to capacity) exposes state space collapse and demonstrates that the combination of join-the-shortest-queue or power-of-two-choices routing with MaxWeight scheduling achieves queue length optimality:

$$\lim_{\epsilon^{(k)} \to 0} \epsilon^{(k)} \cdot \mathbb{E}[\langle \mathbf{c}^{(k)}, \mathbf{q}^{(\epsilon)} \rangle] = \frac{\zeta^{(k)}}{2}$$

This implies pooled resource efficiency and minimized delays as load approaches capacity (Maguluri et al., 2012).

• Edge Inference with Batching/Early Exiting: Joint communication and computation resource allocation is further complicated by batch-dependent memory access and execution times. NP-complete formulations treat batching (size $|S|$) and early exiting (exit point $d_k$ for task $k$) via constraints such as $f(|S|) \leq \tilde{\tau}_k$ and

$t_{cp,k} = \sum_{d=1}^{d_k} f_d(n_d)$

where $n_d$ is batch size at network block $d$. Efficient best-shelf-packing and DFS-based algorithms double throughput relative to baselines (Liu et al., 2022).

3. Memory-Constrained and Hierarchical Allocation Algorithms

Robust resource allocation under strict memory constraints in cloud and HPC requires specialized models and algorithms:

• Cloud Service Allocation: The allocation problem reflects a discrete unsplittable memory demand per VM. The objective is to minimize the number of machines $m$, subject to per-service reliability constraints under failures:

$$\sum_{j=1}^m A_{ij} - B_i \sqrt{\sum_{j=1}^m A_{ij}^2} \geq K_i$$

Algorithms employ column generation (efficient for low memory-per-machine) and rare-event estimation for failure probabilities, ensuring logarithmic complexity in $m$ and practical scaling for large platforms (Beaumont et al., 2013).

• Hierarchical Multi-Task Inference: In collaborative edge–cloud scenarios, a mixed-integer program is solved to jointly select which task models to onload (subject to per-node memory budget, e.g., $\sum_m x_m^c s_m \leq \mu^c$) and how to offload inference queries (balancing compute, memory, and communication resources). A batching-aware extension explicitly models per-batch latency as $$\tau_{t}^{e}(b_t^{e}) = \sum_m [\nu_{m}^{e} \cdot \mathbf{1}_{b_t^{e}} + \tau_{m}^{e} \cdot b_t^{e}] \cdot (x_{m}^{e} z_{m,t}^{e})$$, addressing the trade-off between throughput and delay (Cha et al., 18 Aug 2025).

4. Memory, Batch, and Scheduling in Deep Learning Inference and Training

Modern large-scale deep learning setups are characterized by intricate coupling between batch size, memory, and allocation policies:

• Memory-Aware Dynamic Batching for LLMs: Batch size $b_t$ is dynamically adapted based on real-time GPU memory (from per-request input/output tokens):

$$\mu_S = b_t \cdot (E[l_{in,i}] + E[l_{out,i}]), \quad \sigma_S^2 = b_t \cdot (\text{Var}(l_{in,i}) + \text{Var}(l_{out,i}))$$

Subject to probabilistic memory constraints $P(S > \eta) \leq \epsilon_M$, and feedback mechanisms that ensure SLA-constrained decoding latency $D(b_t) \leq D_{\text{SLA}}$ (Pang et al., 7 Mar 2025). Similar approaches (e.g., Past-Future scheduling in LightLLM) leverage historical distributions of output lengths for per-step peak memory estimation:

$$M^* = \max_i\; M_i = \left(\sum_{j=1}^i [l_p^j + l_t^j]\right) + (\hat{l}_t^i - l_t^i) \cdot i$$

yielding 2–3$\times$ improvements in throughput under strict SLAs (Gong et al., 14 Jul 2025).

• Memory-Limited Training Strategies: For training under GPU memory constraints, micro-batch processing splits large batches into sequential sub-batches, applying loss normalization:

$$\mathcal{L}_{\text{norm}} = \mathcal{L}_{\mu} / N_{S_{\mu}}$$

with gradient accumulation ensuring equivalence to single large-batch updates (Piao et al., 2021). Gradient caching allows storage of only representation-layer activations to attain nearly constant memory use during contrastive learning, permitting arbitrarily large effective batch sizes (Gao et al., 2021).

5. Theoretical Frameworks and Agent-Based Allocation

Resource allocation is analyzed via both physical analogies and game-theoretic models:

• Statistical Mechanics of Allocation: Agent-based frameworks (Minority Game, KPR, stable marriage) model memory (via history windows), batch (via update schemes), and competition. Statistical physics tools—replica trick, generating functional analysis—reveal regimes (e.g., phase transitions in predictability), the impact of “quenched disorder” (agent strategy diversity), and macroscopic observables such as utilization and volatility (Chakraborti et al., 2013).
• Imperfect Information and Memory: The SPII framework formalizes how memory at the decision-maker affects resource stabilization capacity in stochastic networks. The “capacity factor” $\rho_{k,v}^*(C)$, defined as the maximal scaling of the full-information capacity region achievable under noisy observation and bounded-memory encoder (size $k$) and receiver (size $v$), establishes that performance improves with receiver memory but that encoder memory is only useful up to a defined threshold. Constructive policies (episodic Max-Weight, greedy learning) achieve the specified bounds (Xu et al., 2019).

6. Memory Allocation in Cognitive and Reinforcement Learning Models

Resource allocation models also extend to memory management in RL agents and human cognition:

• Memory-Constrained RL Agents: An RL agent with fixed memory $N$ allocates $N = N_{\hat{p}} + N_{\pi}$ between world-model estimation and planning. Empirical and theoretical results show an inverse-U performance trade-off, with optimal settings at a balanced split, but with finer nuances for episodic MCTS and continual DQN agents (Tamborski et al., 9 Jun 2025).
• Strategic Allocation in Language Processing: Resource-rational models of memory encoding propose that working memory capacity is efficiently allocated in proportion to surprisal, computed as $S_t = -\log p(w_t | w_{-t})$. This strategic process bolsters representations of unexpected words, mitigating locality/decay effects in syntactic dependencies. Empirical data confirms reduced locality costs for high-surprisal antecedents; cross-linguistic patterns indicate language-specific modulations of this universal efficiency principle (Xu et al., 18 Mar 2025).

7. Practical Implications and Systems Integration

Resource allocation models have direct consequences for system design and deployment:

• Batch Parallelism and GPU Sharing: Mechanisms such as NVIDIA Multi-Process Service (MPS) enable finer granularity in batch GPU assignment by fusing process contexts, dramatically improving throughput (up to nearly 5–9$\times$ over default) and reducing per-job energy consumption. Integration with schedulers like HTCondor involves partitioning the GPU into “slots” with device memory limits, balancing flexibility and monitoring (Voigtländer et al., 13 May 2025).
• Hybrid Streaming/Batch Execution: Distributed data processing benefits from streaming batch models (as in Ray Data), where dynamic partitioning, pipelined operator execution, and centralized memory budgeting (e.g., budget $+= \text{outputPartitionSize}/P$, with $P = \sum_i (T_i/E_i \times \alpha_{i-1})$) yield 3–8$\times$ throughput gains for heterogeneous clusters while maintaining lineage-based fault tolerance (Luan et al., 16 Jan 2025).

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Collectively, these models and algorithms provide a multi-layered foundation for understanding memory, batch, and resource allocation across computational and cognitive systems. By incorporating statistical structure, algorithmic optimality, and empirical validation, they enable robust design, efficient utilization, and adaptive policy development for future large-scale, heterogeneous, and mission-critical environments.