Fluid Approximations for Revenue Management under High-Variance Demand

Abstract

One of the most prevalent demand models in the revenue management literature is based on dividing the selling horizon into a number of time periods such that there is at most one customer arrival at each time period. This demand model is equivalent to using a discrete-time approximation to a Poisson process, but it has an important shortcoming. If the mean number of customer arrivals is large, then the coefficient of variation of the number of customer arrivals has to be small. In other words, large demand volume and large demand variability cannot co-exist in this demand model. In this paper, we start with a revenue management model that incorporates general mean and variance for the number of customer arrivals. This revenue management model has a random selling horizon length, capturing the distribution of the number of customer arrivals. The question we seek to answer is the form of the fluid approximation that corresponds to this revenue management model. It is tempting to construct the fluid approximation by computing the expected consumption of the resource capacities in the constraints and the total expected revenue in the objective function through the distribution of the number of customer arrivals. We demonstrate that this answer is wrong in the sense that it yields a fluid approximation that is not asymptotically tight as the resource capacities get large. We give an alternative fluid approximation, where, perhaps surprisingly, the distribution of the number of customer arrivals does not play any role in the constraints. We show that this fluid approximation is asymptotically tight as the resource capacities get large. A numerical study also demonstrates that the policies driven by the latter fluid approximation perform substantially better, so there is practical value in getting the fluid approximation right under high-variance demand.

Keywords: Revenue Management, Fluid Approximations, High-Variance, Dynamic Programming