Abstract: There are several 2-dimensional monotonicity results known in the literature (due to Hajek, Lin & Kumar and Stidham & Weber) that have the same underlying convexity structure. Many researchers have tried to extend these results to more than 2 dimensions or to multi-server queues, but to no avail. We will discuss these results in detail and show the difficulties in generalizing them.
Abstract: We focus on the class of multi-dimensional queuing system control problems. This class includes problems with controlled admission, routing or transfer between multiple queues and covers a large range of problems. It is well established that in many of these problems the optimal policy is monotone where different optimal actions are separated by switching curves (or surfaces). We show that, under certain conditions, the evolution of such switching curves is monotone when the values of several system input parameters such as transition rates and costs/rewards are changed. Our approach is based on the propagation of certain properties of the Dynamic Programming (DP) value function. In particular, we observe that the monotonicity of optimal actions as a function of the system parameters depend on the complexity of the DP formulation as well as the DP state space. We illustrate the results on two examples, a tandem queue with controlled transfers and an admission control problem for a multiple customer class queueing system.
Abstract: Length of Stay Control (LOSC) is one of the hospitality Revenue Management (RM) levers to optimize the marginal room yield, by controlling the quantity sold. It consists in finding a policy to limit the number of rooms sold for short stays within a busy period, in order to increase occupancy on shoulders nights (i. e. nights immediately before or after the overcrowded nights). Price-based revenue management levers may be inadequate, due to law restrictions or fierce concurrency. Known as "network optimization" in other fields, this kind of control has been abundantly discussed in the literature, but implementations of such a policy under uncertainty conditions - reinforced in the case of hotels of relatively small size - are not straight-forward. We present here two approaches of LOSC policy optimization under uncertainty conditions. In the first one the stochastic dimension is captured into scenarios, and the problem is solved through mathematical programming. Though non linear per se, the problem can be reformulated as a mixed integer program and efficiently solved. An optimal threshold for short stays is then computed that maximizes the marginal revenue expectancy considering the likelihood of occurrence of the scenarios. In the second approach the demand and occupancy are modeled with Discrete-time Markov Decision Processes, with time steps being days, and the problem can be solved through dynamic programming. Also, we use simulation to evaluate threshold policies with smaller time steps. We discuss about the strengths and weaknesses of such approaches, and illustrate their use on real industry data of French budget hotels.
Abstract: We address the problem of dynamically scheduling jobs with abandonment. Execution times and abandonment times are exponentially distributed. Release dates can be abitrarily distributed. We consider either holding costs or abandonment costs. We provide simple conditions under which strict priority rules are optimal.
Abstract: Markov decision processes (MDP) are powerful tools but often suffer from the curse of dimensionality: the complexity of their resolution explodes as the number of objects to be controlled grows. In this talk, I will show that under some conditions, a MDP reduces to a problem of optimal control on a ODE when the number of objects goes to infinity. I will show how to apply this to an infection problem. This allows one to obtain structural and quantitative properties on the optimal policy.