Optimality of Finite Capacity Markovian Queues with Discouraged Arrivals and Singlhiatus with Waiting Server

We consider finite-capacity Markovian queues with a single hiatus scheme and waiting server. Customers are arriving at a Poisson arrival λ and exponential service distribution, with a mean service rate µ. In which customers join the queue according to the number of customers in the system while the hiatus is in the service-providing process. For the assumed queuing model, steady-state probabilities were derived, and some important performance measures, such as the mean number of customers in the system and mean response time in the system and queue are analysed. The expected expense function is developed and formulated as an optimization problem in order to find the minimum expense. Numerical illustrations are given to show the effect of parameters on the performance measures.