Strategic behavior and social optimization in Markovian working vacation queues with delayed observations

This article investigates the strategic behavior of customers in a single-server Markovian queue with working vacations and infinite waiting space. Customers, upon arrival, face an unobservable system but receive periodic announcements (delayed announcements) about system’s current state and their position, enabling them to strategically decide whether to stay or renege based on potential benefits. We define customers’ strategic choices using three parameters: q, representing the probability of joining upon arrival, and the reneging thresholds $$n_0$$ n 0 and $$n_1$$ n 1 , applicable during the working vacation and normal periods, respectively. We characterize and compute equilibrium strategies $$(n_0^e, n_1^e, q^e)$$ ( n 0 e , n 1 e , q e ) of customers as well as socially optimal strategies $$(n_0^{soc}, n_1^{soc}, q^{soc})$$ ( n 0 soc , n 1 soc , q soc ) . Additionally, coordination between equilibrium and socially optimal strategies has been studied through the implementation of entrance and service fees, along with the optimal selection of the announcement rate in both the periods. Furthermore, comprehensive numerical experiments yield insightful qualitative observations about the model.