The Price of Anarchy in the Markovian Single Server Queue

The Price of Anarchy (PoA) is a measure for the loss of optimality due to decentralized behavior. It has been studied in many settings but, surprisingly, not in the most fundamental queueing system involving customers' decisions, namely, the single server Markovian queue. We find that the loss of efficiency in such systems is bounded by 50% in most practical cases, in which the arrival rate of the customers is significantly lower than the service rate. We also find that the loss of efficiency has an interesting behavior in two aspects: first, it sharply increases as the arrival rate comes close to the service rate; second, it becomes unbounded exactly when the arrival rate is greater than the service rate, a surprising behavior because the system is always stable. Knowing these bounds is important for the queue controller, for example when considering an investment in added service capacity.