Call centers often experience persistent congestion due to random customer arrivals and limited services capacity, making efficient queue management a critical challenge. This paper presents a steady-state analysis of a finite-capacity single-server queueing system with encouraged arrivals and server vacations. Customer arrivals follow a Poisson process with an encouragement factor, while service and vacation times are assumed to be exponentially distributed. The system operates under a first-come, first-served discipline with a maximum capacity of R customers. The underlying process is modeled as a continuous-time Markov chain, and the corresponding system of differential-difference equations is formulated. Closed-form expressions for the steady- state probabilities are derived, and standard performance measures are obtained analytically. The analytical results provide useful insights into the impact of encouraged arrivals and server vacations on system congestion and waiting behavior, which can assist in improving operational efficiency in call center environments.
Rekha et al. (Thu,) studied this question.