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Balancing Independent and Collaborative Service

Shuwen Lu, Mark E. Lewis, Jamol Pender

TL;DR

This work models a two-type server queue where flexible Type-I servers must decide, in real time, between independent processing at Station 1 or collaborative processing with dedicated Type-II servers at Station 2. Using a Markov decision process for a clearing system, the authors prove a threshold structure for the optimal policy and establish diagonal monotonicity across station workloads, with thresholds potentially infinite in some parameter regions. To enable real-time deployment, they devise linear-affine threshold heuristics based on an approximation $H(i,k,\ell)$, and provide theoretical bounds comparing these heuristics to the optimal policy. Numerical experiments show the heuristic policy achieving costs within 0.5% of the optimum on average and exhibiting strong robustness, outperforming benchmarks that are highly parameter-sensitive. The results offer actionable guidance for systems with limited collaboration resources (e.g., expert staff or specialized equipment) and suggest paths for extending the framework to more complex, multi-stage settings.

Abstract

We study a two-type server queueing system where flexible Type-I servers, upon their initial interaction with jobs, decide in real time whether to process them independently or in collaboration with dedicated Type-II servers. Independent processing begins immediately, as does collaborative service if a Type-II server is available. Otherwise, the job and its paired Type-I server wait in queue for collaboration. Type-I servers are non-preemptive and cannot engage with new jobs until their current job is completed. We provide a complete characterization of the structural properties of the optimal policy for the clearing system. In particular, an optimal control is shown to follow a threshold structure based on the number of jobs in the queue before a Type-I first interaction and on the number of jobs in either independent or collaborative service. We propose simple threshold heuristics, based on linear approximations, for real-time decision-making. In much of the parameter and state spaces, we establish theoretical bounds that compare the thresholds proposed by our heuristics to those of optimal policies and identify parameter configurations where these bounds are attained. Outside of these regions, the optimal thresholds are infinite. Numerical experiments further demonstrate the accuracy and robustness of our heuristics, particularly when the initial queue length is high. Our proposed heuristics achieve costs within 0.5% of the optimal policy on average and significantly outperform benchmark policies that exhibit extreme sensitivity to system parameters, sometimes incurring costs exceeding 100% of the optimal.

Balancing Independent and Collaborative Service

TL;DR

This work models a two-type server queue where flexible Type-I servers must decide, in real time, between independent processing at Station 1 or collaborative processing with dedicated Type-II servers at Station 2. Using a Markov decision process for a clearing system, the authors prove a threshold structure for the optimal policy and establish diagonal monotonicity across station workloads, with thresholds potentially infinite in some parameter regions. To enable real-time deployment, they devise linear-affine threshold heuristics based on an approximation , and provide theoretical bounds comparing these heuristics to the optimal policy. Numerical experiments show the heuristic policy achieving costs within 0.5% of the optimum on average and exhibiting strong robustness, outperforming benchmarks that are highly parameter-sensitive. The results offer actionable guidance for systems with limited collaboration resources (e.g., expert staff or specialized equipment) and suggest paths for extending the framework to more complex, multi-stage settings.

Abstract

We study a two-type server queueing system where flexible Type-I servers, upon their initial interaction with jobs, decide in real time whether to process them independently or in collaboration with dedicated Type-II servers. Independent processing begins immediately, as does collaborative service if a Type-II server is available. Otherwise, the job and its paired Type-I server wait in queue for collaboration. Type-I servers are non-preemptive and cannot engage with new jobs until their current job is completed. We provide a complete characterization of the structural properties of the optimal policy for the clearing system. In particular, an optimal control is shown to follow a threshold structure based on the number of jobs in the queue before a Type-I first interaction and on the number of jobs in either independent or collaborative service. We propose simple threshold heuristics, based on linear approximations, for real-time decision-making. In much of the parameter and state spaces, we establish theoretical bounds that compare the thresholds proposed by our heuristics to those of optimal policies and identify parameter configurations where these bounds are attained. Outside of these regions, the optimal thresholds are infinite. Numerical experiments further demonstrate the accuracy and robustness of our heuristics, particularly when the initial queue length is high. Our proposed heuristics achieve costs within 0.5% of the optimal policy on average and significantly outperform benchmark policies that exhibit extreme sensitivity to system parameters, sometimes incurring costs exceeding 100% of the optimal.
Paper Structure (24 sections, 20 theorems, 128 equations, 6 figures, 16 tables)

This paper contains 24 sections, 20 theorems, 128 equations, 6 figures, 16 tables.

Table of Contents

  1. Introduction
  2. Operational implications
  3. Organization of paper
  4. Literature Review
  5. Problem Description and Main Results
  6. Main results
  7. Supporting results
  8. Heuristics Design
  9. Motivation and definition of the linear approximation
  10. Using H to approximate decisions with D
  11. Preliminary results used to prove bounds
  12. Independent service costs higher than collaborative (h1/mu1 >= h2/mu2)
  13. No queueing for collaborative service in (i,k-1,ell+1) (ell < C2)
  14. Queueing effects for collaborative service in (i,k-1,ell+1) (ell >= C2)
  15. Collaborative service costs higher than independent (h1/mu1 <= h2/mu2)
  16. ...and 9 more sections

Key Result

Theorem 3.1

Suppose $\frac{h_1}{\mu_1} \geq \frac{h_2}{\mu_2}$ and consider any $x =(i,k,\ell ) \in X_D$, the following holds.

Figures (6)

  • Figure 3.1: System with independent and collaborative services.
  • Figure 4.1: Plot of $D(i,k,\ell)$ and $H(i,k,\ell)$ as functions of $i$ under parameter configurations in Examples \ref{['eg:best1']} and \ref{['eg:worst1']}.
  • Figure 4.2: Plot of $D(i,k,\ell)$ and $H(i,k,\ell)$ as functions of $i$ under parameter configurations in Example \ref{['eg:costly_queue']} such that $h_0>h_2$.
  • Figure 4.3: Plot of $D(i,k,\ell)$ and $H(i,k,\ell)$ as functions of $i$ under parameter configurations in Example \ref{['eg:costly_collab']} such that $h_0<h_2$.
  • Figure 4.4: Plot of $D(i,k,\ell)$ and $H(i,k,\ell)$ as functions of $i$ under parameter configurations in Example \ref{['eg:equally_costly']} such that $h_0=h_2$.
  • ...and 1 more figures

Theorems & Definitions (76)

  • Definition 3.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • Proposition 3.5
  • ...and 66 more