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Strategic decisions in queueing systems

Y. Dimitrakopoulos, A. Economou and S. Leonardos, Strategic customer behavior in a queueing system alternating between peak and off-peak periods, working paper.

The study of the strategic behavior of customers in service systems is a particularly active research area of Operations Research and combines elements from Queueing Theory, Game Theory and Mathematical Programming. In the proposed work, we are studying a model with varrying arrival rates and different levels of information. Specifically, we are interested in the equilibrium behavior of a queue with a single server, exponential service times, unlimited capacity and First-Come-First-Served discipline, at which the customers arrive with a Markov modulated Poisson process that alternates between two states: peak periods with high intensity of arrivals and off-peak periods with low intensity of arrivals.

The customers and the manager of the system are viewed as strategic entities. The customers decide whether they will enter the queue or balk, depending on the information they receive upon arrival to the system and which may vary: observable system, meaning an arriving customer can see how many customers are in front of him, unobservable, observable only when arrival intensity is high etc. The decision of the manager is a price that he may charge an entering customer upon arrival (sunk cost) and the level of information that he will make available depending again on the state of the Markov modulated arrival process. If a customer arrives at an observable system, he has the additional decision of reneging (or not) at the time that the system becomes observable. The situation is modeled as a 2-stage game and we are interested in determining its equilibria, studying their properties and comparing them with the socially optimal outcome.

To this end, one needs to determine firstly the stationary distribution of the system – which corresponds, under the aforementioned assumptions, to a $2$-dimensional continuous time Markov Chain – under a given strategy of the customers. This task can be accomplished with the use of probability generating functions and, if necessary, matrix-analytic methods. However, it raises several challenges and the calculations lead to problems of independent interest in the areas of probability and stochastic processes. To determine the (symmetric) equilibrium strategies, the expected utility of the customers is maximized with the use of tools from mathematical programming. Additionally, we study its monotonicity in order to characterize the situation as Avoid-The-Crowd or Follow-The-Crowd. This characterization has a direct effect on the number of equilibria of the system and is complemented – in case of multiple equilibria – with application of the concept of the evolutionary stable equilibrium in order to refine them.

After determining the equilibrium strategies – which is usually done algorithmically due to the underlying complexity of the Markov or semi-Markov process – of the customers and the system manager, one compares the equilibrium outcome with the socially optimal outcome, which is the combined profit of all entities in the system. The equilibrium and socially optimal outcomes usually differ, since the behavior of the customers when they act independently, induces negative (or positive) externalities to the utility of the other customers. The emerging inefficiencies of the system, are studied through the Price of Anarchy with the aim to derive bounds that will depend on the parameters of the system. Lastly, we are interested in optimizing the profit from the manager’s point view – in terms of the throughput of the system – which is accomplished through suitable pricing and varying information revelation.

For similar works in this field, one is referred to previous works of the scientific coordinator of the project, Professor A. Economou, which include Burnetas & Economou (2007) and Economou et al. (2014) among others.

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