[Paper Review] Strategic Arrivals into Queueing Networks: The Network Concert Queueing Game

Harsha Honnappa, Rahul Jain|arXiv (Cornell University)|Dec 13, 2011

Advanced Queuing Theory Analysis10 references4 citations

TL;DR

This paper introduces the Network Concert Queueing Game, modeling strategic user arrivals in multi-queue networks where users choose arrival times and queues to minimize a cost combining waiting and service completion times. It derives a fluid limit approximation, characterizes equilibrium arrival distributions as uniform over time intervals, and computes the price of anarchy, showing it is bounded by 2 in symmetric cases with multiple populations.

ABSTRACT

Queueing networks are typically modelled assuming that the arrival process is exogenous, and unaffected by admission control, scheduling policies, etc. In many situations, however, users choose the time of their arrival strategically, taking delay and other metrics into account. In this paper, we develop a framework to study such strategic arrivals into queueing networks. We start by deriving a functional strong law of large numbers (FSLLN) approximation to the queueing network. In the fluid limit derived, we then study the population game wherein users strategically choose when to arrive, and upon arrival which of the K queues to join. The queues start service at given times, which can potentially be different. We characterize the (strategic) arrival process at each of the queues, and the price of anarchy of the ensuing strategic arrival game. We then extend the analysis to multiple populations of users, each with a different cost metric. The equilibrium arrival profile and price of anarchy are derived. Finally, we present the methodology for exact equilibrium analysis. This, however, is tractable for only some simple cases such as two users arriving at a two node queueing network, which we then present.

Motivation & Objective

• To model strategic user arrivals in queueing networks where users choose arrival times and queues to minimize a cost function of waiting and service completion times.
• To develop a fluid limit approximation for the queueing network process, capturing non-renewal arrival dynamics due to strategic behavior.
• To characterize the equilibrium arrival profile in a population game setting, showing it is uniform over a time interval determined by system parameters.
• To quantify the efficiency loss due to strategic behavior by deriving the price of anarchy for both single and multiple populations.
• To provide an exact equilibrium analysis for simple cases, such as two users and two queues, proving uniqueness through construction.

Proposed method

• Derives a functional strong law of large numbers (FSLLN) approximation for the queue length, busy time, and virtual waiting time processes in a non-exogenous arrival setting.
• Introduces a fluid-scaled virtual waiting time process defined as $ \mathbf{W}^{n}(t) = \mathbf{V}^{n}(A^{n}(t)) - \mathbf{B}^{n}(t) - \mathbf{t_{s,-}}(t) $, and proves convergence to a fluid limit under appropriate scaling.
• Uses the Random Time Change Theorem and martingale functional central limit theorem techniques to establish convergence of the fluid-scaled processes.
• Models the strategic arrival game as a non-atomic population game where users minimize a linear cost function of waiting and service completion times.
• Derives the equilibrium arrival distribution $ f(t) $, showing it is uniform on $[-T_0, 0]$ and continuous on $[0, T]$, with time-dependent routing probabilities $ p_i(t) $.
• Solves for equilibrium parameters $ T_0 $ and $ T $ by enforcing $ \int_{-T_0}^{T} f(t) dt = 1 $ and $ f(T) = 0 $, leading to closed-form expressions for symmetric two-user, two-queue cases.

Experimental results

Research questions

• RQ1 Does a Nash equilibrium exist in the strategic arrival game where users choose both arrival time and queue in a multi-queue network?
• RQ2 What is the structure of the equilibrium arrival distribution in a fluid approximation of the queueing network?
• RQ3 How does the price of anarchy—measuring inefficiency due to strategic behavior—scale with system parameters in single and multiple population settings?
• RQ4 Can exact equilibrium analysis be performed for small, symmetric systems, and what are the conditions for uniqueness?
• RQ5 How do service start times, service rates, and cost weights affect the equilibrium arrival profile and routing decisions?

Key findings

• The equilibrium arrival profile is a uniform distribution over the interval $[-T_0, 0]$, with a continuous, time-varying routing distribution for $ t > 0 $, derived from the fluid limit.
• For two users and two parallel queues, the equilibrium is uniquely determined by solving a quadratic equation derived from normalization and boundary conditions, yielding explicit expressions for $ T_0 $ and $ T $.
• The price of anarchy is bounded above by 2 in the symmetric case with two populations and equal service rates, indicating a worst-case efficiency loss of at most 100%.
• The fluid approximation captures essential dynamics of strategic arrivals, showing that inter-arrival times are dependent and the process is not a renewal process.
• The arrival rate $ f(t) $ is piecewise continuous: constant on $[-T_0, 0]$ and a linear function of $ t $ on $[0, T]$, with $ f(T) = 0 $.
• The equilibrium routing probability $ p_i(t) $ is constant before service starts and becomes time-dependent afterward, depending on the expected idle time of each queue.