Strategic joining rules in unobservable queues with dynamic service rate.

This article investigates customers' equilibrium and socially optimal joining strategies in an unobservable single-server Markovian queue with vacations and variable service rates governed by a double-threshold \((m, N)\) policy. Two information scenarios are analyzed: the fully unobservable queue, where neither server state nor queue length is visible to arriving customers, and the almost unobservable queue, where the server state is observable but the queue length is not. The study derives the system's stationary distribution, mean queue length, and expected sojourn time using iterative and recursive methods, revealing that up to four equilibrium arrival rates can exist in the fully unobservable case and two in the almost unobservable case. Numerical examples demonstrate that the socially optimal arrival rate typically lies between the minimum and maximum stable equilibria, implying that managers may need to impose pricing or subsidies to align individual customer behavior with social welfare objectives depending on which equilibrium is realized.