How to Maximize Efficiency in Systems with Exhausted Workers

TL;DR

This work addresses task allocation in systems where worker efficiency evolves with fatigue and recovery, modeled as a CTMC over states $\{1,2,3,1^*,2^*\}$ with recovery $\lambda_i$ and exhaustion $\mu_i$. It analyzes two scenarios under a total sampling budget $C$: (i) dispatch only when workers are in the highly efficient state $s_i(t)=3$, which yields a convex optimization with a threshold-based sampling policy; and (ii) dispatch also in moderately efficient states, leading to a non-convex sum-of-ratios problem solved by a branch-and-bound algorithm that iteratively optimizes $\alpha_i$ and the moderate-efficiency decision $p_i$. Numerical results show threshold structures that prioritize workers with favorable exhaustion/recovery dynamics and demonstrate near-optimal performance of the proposed algorithm relative to exhaustive search. The findings offer practical guidance for resource-constrained, state-aware task scheduling in fatigue-sensitive distributed systems.

Abstract

We consider the problem of assigning tasks efficiently to a set of workers that can exhaust themselves as a result of processing tasks. If a worker is exhausted, it will take a longer time to recover. To model efficiency of workers with exhaustion, we use a continuous-time Markov chain (CTMC). By taking samples from the internal states of the workers, the source assigns tasks to the workers when they are found to be in their efficient states. We consider two different settings where (i) the source can assign tasks to the workers only when they are in their most efficient state, and (ii) it can assign tasks to workers when they are also moderately efficient in spite of a potentially reduced success probability. In the former case, we find the optimal policy to be a threshold-based sampling policy where the thresholds depend on the workers' recovery and exhaustion rates. In the latter case, we solve a non-convex sum-of-ratios problem using a branch-and-bound approach which performs well compared with the globally optimal solution.

Figures (6)

• Figure 1: A centralized resource allocator (source) assigning tasks to $n$ workers with rates $\alpha_i$'s, only when each worker is found to be efficient. Each worker follows a Markov process with transitions among external states $1$, $2$, and $3$, and internal processing states $1^*$ and $2^*$, driven by recovery rate ($\lambda$) and exhaustion rate ($\mu$). Shaded states indicate current worker states.
• Figure 2: State transitions in the CTMC model of a worker. Here, we present a general worker's states as $s(t)$ and their transition rates.
• Figure 3: State transitions in the CTMC model of a worker when the source assigns tasks with probability $p$ if a worker is moderately efficient. Here, we present a general worker's states as $s(t)$ and their transition rates.
• Figure 4: Optimal sampling rate allocations for $n = 10$, $C = 10$, and $\mu_i = 1$. The uniform and heterogeneous recovery rate configurations correspond to $q = 1.0$ and $q = 0.9$, respectively. In both cases, the recovery rates are normalized such that $\sum_i \lambda_i = 20$.
• Figure 5: Optimal task assignment probabilities in the moderately efficient state as functions of $p_s$, for a system with $n = 2$ workers and total sampling budget $C = 10$. Thresholds occur at $p_s = 0.23$ and $p_s = 0.15$, respectively.