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Online Revenue Maximization for Server Pricing

Shant Boodaghians, Federico Fusco, Stefano Leonardi, Yishay Mansour, Ruta Mehta

TL;DR

This work addresses revenue maximization for online, non-preemptive scheduling on a single server by designing a posted-price mechanism where the menu depends on the earliest available time and offered length. By modeling the problem as a Markov Decision Process and solving with backwards induction, it provides Bayes-optimal pricing policies that are monotone in length under a log-concave distribution assumption, ensuring truthful reporting. The study also develops concentration results for revenue, and analyzes robustness to approximate or learned distributions, providing polynomial-sample procedures to learn $Q$ from observed decisions and to transfer near-optimality to the learned model. The results yield an efficient, truthful pricing framework with strong performance guarantees, supporting practical deployment in cloud-like resource markets and enabling robust revenue management under distributional uncertainty.

Abstract

Efficient and truthful mechanisms to price resources on remote servers/machines has been the subject of much work in recent years due to the importance of the cloud market. This paper considers revenue maximization in the online stochastic setting with non-preemptive jobs and a unit capacity server. One agent/job arrives at every time step, with parameters drawn from an underlying unknown distribution. We design a posted-price mechanism which can be efficiently computed, and is revenue-optimal in expectation and in retrospect, up to additive error. The prices are posted prior to learning the agent's type, and the computed pricing scheme is deterministic, depending only on the length of the allotted time interval and on the earliest time the server is available. If the distribution of agent's type is only learned from observing the jobs that are executed, we prove that a polynomial number of samples is sufficient to obtain a near-optimal truthful pricing strategy.

Online Revenue Maximization for Server Pricing

TL;DR

This work addresses revenue maximization for online, non-preemptive scheduling on a single server by designing a posted-price mechanism where the menu depends on the earliest available time and offered length. By modeling the problem as a Markov Decision Process and solving with backwards induction, it provides Bayes-optimal pricing policies that are monotone in length under a log-concave distribution assumption, ensuring truthful reporting. The study also develops concentration results for revenue, and analyzes robustness to approximate or learned distributions, providing polynomial-sample procedures to learn from observed decisions and to transfer near-optimality to the learned model. The results yield an efficient, truthful pricing framework with strong performance guarantees, supporting practical deployment in cloud-like resource markets and enabling robust revenue management under distributional uncertainty.

Abstract

Efficient and truthful mechanisms to price resources on remote servers/machines has been the subject of much work in recent years due to the importance of the cloud market. This paper considers revenue maximization in the online stochastic setting with non-preemptive jobs and a unit capacity server. One agent/job arrives at every time step, with parameters drawn from an underlying unknown distribution. We design a posted-price mechanism which can be efficiently computed, and is revenue-optimal in expectation and in retrospect, up to additive error. The prices are posted prior to learning the agent's type, and the computed pricing scheme is deterministic, depending only on the length of the allotted time interval and on the earliest time the server is available. If the distribution of agent's type is only learned from observing the jobs that are executed, we prove that a polynomial number of samples is sufficient to obtain a near-optimal truthful pricing strategy.

Paper Structure

This paper contains 24 sections, 14 theorems, 49 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Summary of Our Results
  3. Related Work
  4. Structure of the Paper
  5. Model
  6. Bayes-optimal Strategies for Sever Pricing
  7. Markov Decision Processes.
  8. Solving for the Optimal Policy with Distributional Knowledge
  9. Monotonicity of the Optimal Pricing Policies
  10. Concentration Bounds on Revenue for Online Scheduling
  11. Robustness of Pricing with Approximate Distributional Knowledge
  12. Robustness of the pricing strategy
  13. Learning the Underlying Distribution from Samples
  14. Performance of the Computed Policy
  15. Conclusions and Future Work
  16. ...and 9 more sections

Key Result

Lemma 1

For any fixed pricing policy $\pi:[T]\times\mathcal{S}\times\mathcal{L}\to \mathbb R$, where the ${U^\pi_{t}(\cdot)}$'s are as in eq:expected_gain, and the ${u^\pi_{t}(\cdot,\cdot)}$'s are from the modified MDP.

Theorems & Definitions (30)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • proof : Sketch.
  • Theorem 5
  • Theorem 6
  • Lemma 7
  • Lemma 8
  • Theorem 9: Finite Horizon
  • ...and 20 more