Control policies for a two-stage queueing system with parallel and single server options

TL;DR

This work analyzes a two-stage tandem queue with two flexible servers, where after the first-phase service a decision determines whether the second phase proceeds in a parallel downstream station or a preferred but potentially congested single-service facility. Through a Markov decision process with a clearing system cost structure, the authors derive the optimal policy's structural properties, showing threshold-like behavior that depends on the comparison between downstream mean service times $m_1$ and $m_2$, and on cost parameters. The study highlights significant performance gaps between the optimal policy and natural heuristics, and emphasizes the impact of blocking and queueing effects on routing decisions. These insights offer guidance for designing control policies in practical settings such as manufacturing and service systems with limited downstream resources.

Abstract

We study a two-stage tandem service queue attended by two servers. Each job-server pair must complete both service phases together, with the server unable to begin a new job until the current one is fully processed after two stages. Immediately after the first phase of service, the server decides whether to send the job/customer to a downstream station that allows parallel processing or to a single-service facility that offers faster or higher-quality service but handles only one job at a time. This choice determines whether the second phase commences immediately or (potentially) after waiting in a queue for the single-service facility to become available. The decision-making scenario is modeled via a Markov decision process formulation, of a clearing system with holding costs at each station. We fully characterize the structural properties of an optimal control policy based on the relationship between the service rates at the downstream stations. A numerical study highlights the significance of optimal control by comparing its performance against several natural heuristic policies.

Figures (1)

• Figure 1.1: Diagram of Two-stage Model.

Theorems & Definitions (23)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Lemma A.1
• proof : Proof of Statement \ref{['state1:general_rates']}
• proof : Proof of Statement \ref{['state2:general_rates']}
• proof : Proof of Statement \ref{['state3:general_rates']}
• proof : Proof of Statement \ref{['state4:general_rates']}
• Lemma A.2
• proof : Proof of Lemma \ref{['lemma:equal_rates']}