On the uniqueness of solutions to the Poisson equations for average cost Markov chains with unbounded cost functions

We consider the Poisson equations for denumerable Markov chains with unbounded cost functions. Solutions to the Poisson equations exist in the Banach space of bounded real-valued functions with respect to a weighted supremum norm such that the Markov chain is geometrically ergodic. Under minor additional assumptions the solution is also unique. We give a novel probabilistic proof of this fact using relations between ergodicity and recurrence. The expressions involved in the Poisson equations have many solutions in general. However, the solution that has a finite norm with respect to the weighted supremum norm is the unique solution to the Poisson equations. We illustrate how to determine this solution by considering three queueing examples: a multi-server queue, two independent single server queues, and a priority queue with dependence between the queues.     

Optimal balanced control for call centers

In this paper we study the optimal assignment of tasks to agents in a call center. For this type of problem, typically, no single deterministic and stationary (i.e., state independent and easily implementable) policy yields the optimal control, and mixed strategies are used. Other than finding the optimal mixed strategy, we propose to optimize the performance over the set of ??balanced?? deterministic periodic non-stationary policies. We provide a stochastic approximation algorithm that allows to find the optimal balanced policy by means of Monte Carlo simulation. As illustrated by numerical examples, the optimal balanced policy outperforms the optimal mixed strategy.     

Optimal patient and personnel scheduling policies for care-at-home service facilities

In this paper we study the problem of personnel planning in care-at-home facilities. We model the system as a Markov decision process, which leads to a high-dimensional control problem. We study monotonicity properties of the system and derive structural results for the optimal policy. Based on these insights, we propose a trunk reservation heuristic to control the system. We provide numerical evidence that the heuristic yields close to optimal performance, and scales well for large problem instances.     

Optimal resource allocation for multiqueue systems with a shared server pool

We study optimal allocation of servers for a system with multiple service facilities and with a shared pool of servers. Each service facility poses a constraint on the maximum expected sojourn time of a job. A central decision maker can dynamically allocate servers to each facility, where adding more servers results in faster processing speeds but against higher utilization costs. The objective is to dynamically allocate the servers over the different facilities such that the sojourn-time constraints are met at minimal costs. This situation occurs frequently in practice, for example, in Grid systems for real-time image processing (iris scans, fingerprints). We model this problem as a Markov decision process and derive structural properties of the relative value function. These properties, which are hard to derive for multidimensional systems, give a full characterization of the optimal policy. We demonstrate the effectiveness of these policies by extensive numerical experiments.     

Structural properties of the optimal resource allocation policy for single-queue systems

This paper studies structural properties of the optimal resource allocation policy for single-queue systems. Jobs arrive at a service facility and are sent one by one to a pool of computing resources for parallel processing. The facility poses a constraint on the maximum expected sojourn time of a job. A central decision maker allocates the servers dynamically to the facility. We consider two models: a limited resource allocation model, where the allocation of resources can only be changed at the start of a new service, and a fully flexible allocation model, where the allocation of resources can also change during a service period. In these two models, the objective is to minimize the average utilization costs whilst satisfying the time constraint. To this end, we cast these optimization problems as Markov decision problems and derive structural properties of the relative value function. We show via dynamic programming that (1) the optimal allocation policy has a work-conservation property, and (2) the optimal number of servers follows a step function with as extreme policy the bang-bang control policy. Moreover, (3) we provide conditions under which the bang-bang control policy takes place. These properties give a full characterization of the optimal policy, which are illustrated by numerical experiments.     

On structural properties of the value function for an unbounded jump Markov process with an application to a processor sharing retrial queue

The derivation of structural properties for unbounded jump Markov processes cannot be done using standard mathematical tools, since the analysis is hindered due to the fact that the system is not uniformizable. We present a promising technique, a smoothed rate truncation method, to overcome the limitations of standard techniques and allow for the derivation of structural properties. We introduce this technique by application to a processor sharing queue with impatient customers that can retry if they renege. We are interested in structural properties of the value function of the system as a function of the arrival rate.     

Optimal appointment scheduling in continuous time: The lag order approximation method

We study appointment scheduling problems in continuous time. A finite number of clients are scheduled such that a function of the waiting time of clients, the idle time of the server, and the lateness of the schedule is minimized. The optimal schedule is notoriously hard to derive within reasonable computation times. Therefore, we develop the lag order approximation method, that sets the client’s optimal appointment time based on only a part of his predecessors. We show that a lag order of two, i.e., taking two predecessors into account, results in nearly optimal schedules within reasonable computation times. We illustrate our approximation method with an appointment scheduling problem in a CT-scan area.     

Optimal resource allocation in survey designs

Resource allocation is a relatively new research area in survey designs and has not been fully addressed in the literature. Recently, the declining participation rates and increasing survey costs have steered research interests towards resource planning. Survey organizations across the world are considering the development of new mathematical models in order to improve the quality of survey results while taking into account optimal resource planning. In this paper, we address the problem of resource allocation in survey designs and we discuss its impact on the quality of the survey results. We propose a novel method in which the optimal allocation of survey resources is determined such that the quality of survey results, i.e., the survey response rate, is maximized. We demonstrate the effectiveness of our method by extensive numerical experiments.