The disadvantage of the Cμ-rule when customers are strategic

The question of priorities naturally arises when optimizing service systems with heterogeneous customers. If heterogeneity is with respect to time sensitivity and social welfare is of interest, it is well known that the Cμ-rule is optimal in the sense that customers with high time sensitivity should be prioritized. This, for example, justifies prioritizing more acute patients in public health systems. Using a standard queueing model with strategic customers, we show that, as expected, if admission is centrally controlled, the Cμ-rule remains optimal. However, if admission cannot be controlled, for example, in some public healthcare systems, the Cμ-rule leads to an unusual equilibrium joining behavior. We show that this behavior may result in an inferior social welfare, compared to service disciplines that do not prioritize based on individual time sensitivity level, for example, first-come first-served (FCFS). In particular, the equilibrium strategy under the Cμ-rule is not threshold-based and all customers join with some (type-specific) positive probability. The suboptimality of the Cμ-rule is then demonstrated in a case where the potential arrival rate is high enough such that all the joining probabilities are also less than one. This implies that the resulting social welfare is zero while it is guaranteed to be positive under FCFS.