Dynamic access pricing control for fair and stable resource sharing

TL;DR

This paper tackles fairness in dynamic access pricing for a shared resource by modeling the interaction of price-sensitive and price-insensitive users as a switched nonlinear queueing system. It develops a pricing framework with three core components: a nonmonotone price function $f(q)$, a saturating service rate $\mu(q)$, and a decreasing admission rate $α(q)$, coupled through a switching signal that captures bursty unresponsive traffic. The authors establish global stability results by analyzing two- and three-dimensional modes, introduce a chattering-based admission policy and a saturating-price variant to ensure robustness, and validate the approach with simulations showing improved fairness and resilience compared to standard surge pricing. The framework yields a practical, provably stable method to balance demand while preventing inequitable exclusion of price-sensitive users, with potential extensions to multiple classes and objective-driven pricing design.

Abstract

We consider the use of pricing as a regulatory mechanism when an unknown number of autonomous agents compete for access to a shared resource (possibly limited in volume or capacity). In standard dynamic pricing control systems, an increasing price is used to balance supply and demand for a resource in a constrained environment. A major drawback of dynamic pricing is that it is socially regressive, i.e., unfair, as such systems favour price-insensitive (unresponsive) traffic and control the demand at the expense of price-sensitive (responsive) traffic. We tackle this fundamental issue by proposing a new form of pricing that strikes a balance between using price as a control mechanism to manage demand for a resource and ensuring fair access to the resource for both price-sensitive and insensitive traffic. Our system gives rise to a switched non-linear ODE model, the stability of which is equivalent to ensuring the fairness properties of the pricing control system. Simulations illustrate this stability-fairness tradeoff and with the results demonstrating the effectiveness of the overall design.

Figures (7)

• Figure 1: Price-sensitive and -insensitive traffic operating under the influence of a price signal.
• Figure 2: Sketch of the domain of attraction $\mathcal{A}$ of the stable fixed point $x_1^*=(R^*_1, q^*_1)$. Here arbitrary values are chosen for the system parameters, and an appropriate admission function of the form $\alpha(q)=\max(0, c_1q+c_2)$ is used with $c_1$ and $c_2$ calculated according to the conditions in Eqs. \ref{['eq:fp1_condition']} and \ref{['eq:fp2_condition']}. The hatched area in yellow shows the set of all points with a given $q \in \left(q^\dagger, q_2^*\right)$ and respectively $R \in (\max\{R^\dagger,\eta_2(q)\}, \eta_1(q)]$, addressed in Theorem \ref{['theo:domain']}. The area shaded in gray marks the unstable region.
• Figure 3: Phase portrait for a system based on the model in Eqs. \ref{['eq:system']} and \ref{['eq:systemq']}. The system fixed points, the inherent system constant $q_c$, the pricing design constant $q_m$, and trajectories for multiple initial states are plotted on top of the phase plane. The direction and color of arrows show the phase and magnitude at each point; with the magnitude gradually decreasing from red to yellow, green, and blue.
• Figure 4: Sketch of the chattering surface in the $(R, q)$ space.
• Figure 5: Sketch of the fixed points in the case of the saturated price function of Eq. \ref{['eq:costfunction-sat']}.

Theorems & Definitions (17)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Proposition 3.2
• proof
• Lemma 3.3
• proof
• Theorem 3.4
• proof
• Theorem 3.5