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Queue or lounge: strategic design for strategic customer

Riya Sultana, Khushboo Agarwal, Veeraruna Kavitha

TL;DR

This work analyzes an M/M/1 queue augmented with a lounge facility (LF), where arriving customers choose between immediate service in the queue or pausing in the LF before joining. By modeling bounded rationality with pessimal anticipation and observable state variables, it derives a two-dimensional threshold decision rule $A,B$ that governs where customers relocate, analyzes the resulting CTMC stability and stationary distribution (including a tractable $B=1$ case and a small-$\nu$ approximation), and formulates a Stackelberg lounge-design problem. The key findings show that customers use the LF only when the queue is long and the lounge is not crowded; under high load, both system and customers favor LF with minimal occupancy, while under low load both prefer a large LF, indicating an alignment rather than a tug-of-war. The results illuminate when LF design pays off and provide a framework for optimizing LF capacity and comfort in practice, with extensions suggested for more realistic behavioral assumptions.

Abstract

Considering an M/M/1 queue with an additional lounge facility (LF), the quest of this paper is to understand the instances when LF is an attractive option, from customer perspective as well as from system perspective: will the customers choose to join the queue or prefer to detour briefly to lounge? In reality, customers do not perform complex computations for such tasks, but instead choose based on some heuristics. We further assume that the customers pessimistically anticipate the future congestion while making the choice. Our analysis reveals that the customers use the LF only when the queue is too crowded, and the lounge is relatively empty; however, strikingly, the customer choice is more inclined towards rejection for the LF in systems with higher traffic (load). We also explore an optimization problem where the system determines whether to implement an LF and what capacity it should have, while accounting for customers' behavioral responses. Under low load conditions, the system benefits from designing a high-capacity lounge, and the customers also prefer to use the LF actively. Surprisingly, neither the system prefers big LF, nor the customers prefer to use the LF profusely at high load conditions; optimal for either is to use the LF sparingly. Thus, importantly, the strategic system and the bounded-rational customers are not in a tug-of-war situation.

Queue or lounge: strategic design for strategic customer

TL;DR

This work analyzes an M/M/1 queue augmented with a lounge facility (LF), where arriving customers choose between immediate service in the queue or pausing in the LF before joining. By modeling bounded rationality with pessimal anticipation and observable state variables, it derives a two-dimensional threshold decision rule that governs where customers relocate, analyzes the resulting CTMC stability and stationary distribution (including a tractable case and a small- approximation), and formulates a Stackelberg lounge-design problem. The key findings show that customers use the LF only when the queue is long and the lounge is not crowded; under high load, both system and customers favor LF with minimal occupancy, while under low load both prefer a large LF, indicating an alignment rather than a tug-of-war. The results illuminate when LF design pays off and provide a framework for optimizing LF capacity and comfort in practice, with extensions suggested for more realistic behavioral assumptions.

Abstract

Considering an M/M/1 queue with an additional lounge facility (LF), the quest of this paper is to understand the instances when LF is an attractive option, from customer perspective as well as from system perspective: will the customers choose to join the queue or prefer to detour briefly to lounge? In reality, customers do not perform complex computations for such tasks, but instead choose based on some heuristics. We further assume that the customers pessimistically anticipate the future congestion while making the choice. Our analysis reveals that the customers use the LF only when the queue is too crowded, and the lounge is relatively empty; however, strikingly, the customer choice is more inclined towards rejection for the LF in systems with higher traffic (load). We also explore an optimization problem where the system determines whether to implement an LF and what capacity it should have, while accounting for customers' behavioral responses. Under low load conditions, the system benefits from designing a high-capacity lounge, and the customers also prefer to use the LF actively. Surprisingly, neither the system prefers big LF, nor the customers prefer to use the LF profusely at high load conditions; optimal for either is to use the LF sparingly. Thus, importantly, the strategic system and the bounded-rational customers are not in a tug-of-war situation.
Paper Structure (13 sections, 1 theorem, 38 equations, 6 figures)

This paper contains 13 sections, 1 theorem, 38 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Queueing system with lounge facility
  3. Decisions of bounded-rational customers
  4. Decision rule of customers
  5. LF for higher load factors: customer perspective
  6. Analysis
  7. M/M/1 connection and stability
  8. Stationary distribution for $B=1$
  9. Small $\nu$-approximation
  10. Lounge design problem
  11. At high load factors
  12. Numerical analysis
  13. Conclusions

Key Result

Corollary 1

Let $\Delta := 1-\beta/\alpha$. The customers always choose to join the queue if $\rho \ge \Delta.$

Figures (6)

  • Figure 1: Decision-rule for bounded-rational customers
  • Figure 2: Transition rate diagram for the system with LF
  • Figure 3: Transition rate diagram for the approximating system
  • Figure 4: Approximation of ${\bm \pi}^{(\nu)}$ improves as $\nu \to 0$; depicted in 'log scale' for $l = 0,1$
  • Figure 5: Congestion cost v/s $A$
  • ...and 1 more figures

Theorems & Definitions (4)

  • proof
  • Corollary 1
  • proof
  • Conjecture 1