Resource Allocation Under Stochastic Demands

Military theater settings may involve many competing missions whose whose requirements tax the coalition’s resources in multiple dimensions—i.e., requiring different amounts of many different kinds of resources, e.g., fuel, bandwidth, personnel, etc., each of which is available in limited supply— and in uncertain or stochastic amounts. To serve the overall mission, we need to satisfy resource requirements in multiple dimensions, subject to capacity constraint of each dimension. Therefore, a given theater, as the unit with which some limited resources are associated, can only admit a subset of the users/tasks at the same time. If weights or profits can be associated with tasks, representing their importance, a natural goal is to maximize the total profit admitted to the system, consistent with the resource limitations. If tasks have deterministic demands we can think of the capacity of each resource as the capacity of an abstract (onedimensional) knapsack; together, these tasks and the capacity bounds form an instance of the multidimensional knapsack problem. In practice, however—think of fast-changing battlefield conditions—demands for resources may change over time and only their statistics or distributions may be known a priori. In this case, the demand of each task is a random vector, the elements of which indicate its varying needs for multiple resources. Certain important distributions on the demand vectors are of particular interest, such as a Bernoulli distribution of (s, q), under which, at each step, a mission either requires s of a resource with the probability of q, or none. Other examples may be an exponential distribution (implying a certain service time), a Poisson distribution (indicating an arrival rate of event) or combinations of these distributions. A simple, naive approach to dealing with the stochastic demands would be to assign tasks based on their maximum possible demands. This not only leads to low utilization of resources but also applies to few distributions, however; tasks with unbounded distributions such as the exponential distribution may never be admitted to the system. Statistical multiplexing [1] is more suitable in that uncorrelated demands are packed together allowing occasional violations of the capacity constraints. Particularly, the overflow probability, which indicates the chance that admitted tasks violate the capacity constraints, should be bounded by a predetermined value p. In general, solving such a problem relates to solving a chance-constrained program, which involves nonlinear optimization techniques. Recently, Sample Average Approximation (SAA) has been applied to such problems [2], [3]. Those works provide promising results for a more general problem but the solution is based on Monte Carlo simulation. We use SAA here to derive an upper bound to evaluate solutions to an abstract formulation of the problem. The main idea in solving this problem lies in the development of an effective bandwidth or effective size representing the stochastic demands. The concept of effective bandwidth has been widely used in the field of admission control and bandwidth management. This is a special case of our problem, because in these prior works bandwidth is the only resource to allocate. Kleinberg et al. [4] provided another definition for some restricted cases and proposed approximation algorithms. Our task is to extend this to the multidimensional case. For a d-dimensional random vector v, its effective size βp(v) takes as input an allowable overflow probability p and can be either a deterministic d-dimensional vector or simply a scalar. Once such a metric is developed, one can turn the problem into a deterministic (multidimensional) knapsack problem. Via comprehensive numerical experiments including settings in which demands follow a mix of different distributions, we show that the general effective bandwidth paired with multidimensional knapsack heuristics can generate satisfactory results. However, in many cases, the allocations are conservative. Therefore, we propose an effective way to relax the constraints in the resulting deterministic problem, in order to achieve near-optimal solutions (details omitted). Our main contributions include: