[Paper Review] Analysis of the equilibrium strategies in the Geo/Geo/1 queue with multiple working vacations

Bixuan Yang, Hou Zhenting+2 more|arXiv (Cornell University)|Apr 27, 2014

Advanced Queuing Theory Analysis11 references3 citations

TL;DR

This paper analyzes equilibrium joining and balking strategies in a discrete-time Geo/Geo/1 queue with multiple working vacations, where customers decide based on partial or full information about system state. It derives equilibrium and socially optimal strategies across observable, partially observable, and unobservable cases, showing that individual optimization leads to over-joining compared to social optimality in unobservable settings.

ABSTRACT

This paper studies the equilibrium behavior of customers in the Geo/Geo/1 queueing system with multiple working vacations. The arriving customers decide whether to join or to balk the queueing systems based on the information of the queue length and the states of the server. In observable queues, partially observable queues and unobservable queues three cases we obtain the equilibrium balking strategies based on the reward-cost structure and socially optimal strategies for all customers. Furthermore, we present some numerical experiments to illustrate the effect of the information level on the equilibrium behavior and to compare the customers' equilibrium and socially optimal strategies.

Motivation & Objective

• To investigate customer equilibrium behavior in discrete-time Geo/Geo/1 queues with multiple working vacations from an economic decision-making perspective.
• To examine how information availability—full, partial, or none—affects customer joining or balking decisions.
• To derive equilibrium threshold strategies and socially optimal strategies under different information structures.
• To compare individual equilibrium behavior with social optimization, highlighting inefficiencies in unobservable settings.
• To provide numerical validation of the impact of information levels on customer behavior and system efficiency.

Proposed method

• Models a Geo/Geo/1 queue with multiple working vacations, where the server operates at a reduced service rate during vacation periods.
• Uses a Markov chain framework to describe system state transitions, incorporating vacation states and service rates.
• Applies reward-cost structures to model customer utility, with R as reward and C as cost per unit expected sojourn time.
• Derives the expected sojourn time E[W] using decomposition techniques and steady-state analysis from Tian et al. [20].
• Solves for equilibrium threshold strategies by setting expected net benefit U(q) = 0, yielding q_e^* as the critical joining probability.
• Computes social benefit per time unit as U_s(q) = pq(R - CE[W]) and determines socially optimal strategy q* by maximizing U_s(q).

Experimental results

Research questions

• RQ1 How do equilibrium balking strategies vary across observable, partially observable, and unobservable queueing systems with multiple working vacations?
• RQ2 What is the impact of information availability on customer joining behavior and system efficiency?
• RQ3 How does the equilibrium strategy compare to the socially optimal strategy in unobservable systems?
• RQ4 What role do service rates (μ_b, μ_ν) and vacation parameters (θ, r') play in shaping customer decisions?
• RQ5 How does the arrival rate p affect equilibrium and socially optimal joining probabilities?

Key findings

• In the unobservable case, the equilibrium mixed strategy q_e is unique and given by q_e = min{q_e^*, 1}, where q_e^* solves U(q) = 0.
• The equilibrium joining probability q_e decreases with increasing arrival rate p and decreases more sharply when the regular service rate μ_b is smaller.
• The social benefit under equilibrium, U_s(q_e), first increases then decreases with respect to p, indicating a non-monotonic efficiency profile.
• The equilibrium strategy q_e consistently exceeds the socially optimal strategy q*, indicating a systemic inefficiency due to individual optimization.
• Numerical results confirm that information level significantly influences customer behavior, with higher information leading to more efficient system outcomes.
• The derived expressions for E[W] and U_s(q) are validated through numerical experiments using specific parameter sets (e.g., R=4.5, C=1, θ=0.3, μ_ν=0.5, μ_b=0.9).