Strategic Queues with Priority Classes

TL;DR

This paper analyzes a two-class M/M/1 queue under FCFS with class A obtaining priority over class B. Customers can balk and, for B, reneging is allowed, and the authors derive equilibrium strategies where A uses a class-specific threshold $M_A$ and B uses a threshold on the total queue length, with the precise threshold depending on system parameters; the fully strategic case yields a split decision depending on whether $R_B/C_B$ is below or above $E(M_A,0)$. The social optimum is examined both without and with priority: without priority, the planner favors the class with higher reward-to-cost ratio and uses an Naor-type threshold; with priority, two planners (one per class) interact and the analysis relies on random-walk-based computations, including a reduction to a 2D ruin problem and, in some regimes, a LCFS-inspired solution. Overall, the work provides exact equilibrium thresholds via ruin-theory tools and clarifies how priority constraints alter social efficiency, offering insights for the design and regulation of prioritized service systems.

Abstract

We consider a strategic M/M/1 queueing model under a first-come-first-served regime, where customers are split into two classes and class $A$ has priority over class $B$. Customers can decide whether to join the queue or balk, and, in case they have joined the queue, whether and when to renege. We study the equilibrium strategies and compare the equilibrium outcome and the social optimum in the two cases where the social optimum is or is not constrained by priority.

Figures (4)

• Figure 1: Random-walk representation of a $B$-customer strategy
• Figure 2: Random-walk representation of a $B$-planner strategy
• Figure 3: Random-walk in a rectangle
• Figure 4: Bounded Random-walk

Theorems & Definitions (21)

• Definition 2.1
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Theorem 4.1
• Theorem 4.2
• proof
• Claim 4.3
• proof