Optimal Resource Procurement and the Price of Causality

Abstract

In this article, we study the problem of procuring diverse resources in a forward market to cover a set <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\bf E}$</tex-math></inline-formula> of uncertain demand signals <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\bf e}$</tex-math></inline-formula> . We consider two scenarios: 1) <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\bf e}$</tex-math></inline-formula> is revealed all at once by an oracle and 2) <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\bf e}$</tex-math></inline-formula> reveals itself causally. Each scenario induces an optimal procurement cost. The ratio between these two costs is defined as the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">price of causality</i> . It captures the additional cost of not knowing the future values of the uncertain demand signal. We consider two application contexts: Procuring energy reserves from a forward capacity market, and purchasing virtual machine instances from a cloud service. An upper bound on the price of causality is obtained, and the exact price of causality is computed for some special cases. The algorithmic basis for all these computations is set containment linear programming. A mechanism is proposed to allocate the procurement cost to consumers who in aggregate produce the demand signal. We show that the proposed cost allocation is fair, budget-balanced, and respects the cost causation principle. The results are validated through numerical simulations.