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We investigate a server's best queue disclosure strategy in a single‐server service system with an uncertain quality level (which is assumed to be binary). We consider this problem from the perspective of a Bayesian persuasion game. The server first commits to a possibly mixed strategy stating the probability that the queue length will be revealed to customers on their arrival given a realized quality level. The service quality level is then realized, and the server's corresponding queue‐disclosure action is observed by customers, who then update their beliefs regarding service quality and decide whether to join the service system. We reformulate the server's decision problem as looking for the best Bayes‐plausible distribution of posterior beliefs regarding service quality. We demonstrate that the maximal expected effective arrival rate, as a function of the prior belief, can be graphed as the upper envelope of all convex combinations of any two arbitrary points on the two effective arrival rate functions of the revealed and concealed queues. We show that when the market size is sufficiently small (large), the server always conceals (reveals) the queue, regardless of the realized service quality. Numerically, we find that in a medium‐sized market, the server's optimal commitment strategy is often hybrid or mixed, that is, randomizing queue concealment and revelation. We also extend our analysis to a situation in which the server aims to maximize social welfare. We show that under certain conditions, it is always beneficial for the welfare‐maximizing social planner to randomize queue concealment and revelation, regardless of the market size.

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Problem definition: We consider a single-server queueing system where service quality is either high or low. The server, who knows its exact quality level, can signal this quality information to customers by revealing or concealing its queue length. Based on this queue disclosure action and the observed queue length in the case of a revealed queue, customers decide whether to join the system. Academic/practical relevance: The queue disclosure action is regarded as a signal indicating the service quality. Methodology: We develop a signaling game and adopt the sequential equilibrium concept to solve it. We further apply the perfect sequential equilibrium as an equilibrium-refinement criterion. Results: In our baseline model, where all of the customers are uninformed of service quality, the pure-strategy perfect sequential equilibrium is always a pooling one, except at several discrete values of market size (measured by the potential arrival rate). When the market size is below a certain threshold, both high- and low-quality servers adopt queue concealment; otherwise, both types of servers adopt queue revelation. We also consider a general scenario in which the market is composed of both quality informed and uninformed customers. Under this setting, when the server conceals the queue, we can fully characterize customers’ equilibrium queueing strategies and the corresponding effective arrival rates. The unique sequential equilibrium outcome is still a pooling one when the market size is either below a lower threshold or above an upper threshold. A separating equilibrium can occur only when the market size falls between two thresholds; under that circumstance, the uninformed customers can infer the server’s quality from its queue disclosure behavior. Managerial implications: Under separating sequential equilibria, uninformed customers can fully infer the quality information and thus behave in an informed way. Unlike studies where queue disclosure is not regarded as a quality signal, our study reveals that the signaling effect of queue disclosure increases (decreases) the effective arrival rate of the high-quality (low-quality) server and also increases the customers’ total utility when the server is of low quality. Funding: P. Guo acknowledges the financial support from the Research Grants Council of Hong Kong [Grant 15502820]. The research of M. Haviv was funded by Israel Science Foundation [Grant 1512/19]. Z. Luo acknowledges the financial support from the Internal Start-up Fund of the Hong Kong Polytechnic University [Grant P0039035] and the National Natural Science Foundation of China[Grant 71971184]. Y. Wang’s work was supported by the Research Grants Council of Hong Kong [Grant 15505019]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/msom.2022.1170 .

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