Tail asymptotics for the maximum of perturbed random walk

Consider a random walk $S=(S_n:n\geq 0)$ that is ``perturbed'' by a stationary sequence $(\xi_n:n\geq 0)$ to produce the process $(S_n+\xi_n:n\geq0)$. This paper is concerned with computing the distribution of the all-time maximum $M_{\infty}=\max \{S_k+\xi_k:k\geq0\}$ of perturbed random walk with a negative drift. Such a maximum arises in several different applications settings, including production systems, communications networks and insurance risk. Our main results describe asymptotics for $\mathbb{P}(M_{\infty}>x)$ as $x\to\infty$. The tail asymptotics depend greatly on whether the $\xi_n$'s are light-tailed or heavy-tailed. In the light-tailed setting, the tail asymptotic is closely related to the Cram\'{e}r--Lundberg asymptotic for standard random walk.Comment: Published at http://dx.doi.org/10.1214/105051606000000268 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Media Revenue Management with Audience Uncertainty: Balancing Upfront and Spot Market Sales

An important challenge faced by media broadcasting companies is how to allocate limited advertising space between upfront (forward) contracts and the spot market (referred to in advertising as the scatter market) to maximize profits and meet contractual commitments. We develop stylized optimization models of airtime capacity planning and allocation across multiple clients under audience uncertainty. In a short-term profit maximizing setting, our results provide insight for capacity planning decisions upfront and during the broadcasting season. Our results suggest that broadcasting companies should prioritize upfront clients according to marginal revenue per audience unit. We find that accepted upfront market contracts can be aggregated across clients and served in proportion to the audience demanded. Closed-form solutions are obtained in a static setting. These results remain valid in a dynamic setting, when considering the opportunity to increase allocation by airing make-goods during the broadcasting season. Our structural results characterize the impact of contracting parameters, time, and audience uncertainty on profits and capacity decisions. The results hold under general audience and spot market profit models. Overall, we find that ignoring audience uncertainty can have a significant cost for media capacity planning and allocation.media planning, TV advertising, revenue management, capacity planning