A finite state Markovian queue to let in impatient customers only during K-vacations

We investigate a matrix analysis study for a single-server Markovian queue with finite capacity, i.e. an M/M/1/N queue, where the single server can go for a maximum, i.e. a K number of consecutive vacation periods. During these vacation periods of the server, every customer becomes impatient and leaves the queues. If the server detects that the system is idle during service startup, the server rests. If the vacation server finds a customer after the vacation ends, the server immediately returns to serve the customer. Otherwise, the server takes consecutive vacations until the server takes a maximum number of vacation periods, e.g. K, after which the server is idle and waits to serve the next arrival. During vacation, customers often lose patience and opt for scheduled deadlines independently. If the customer's service is not terminated before the customer's timer expires, the customer is removed from the queue and will not return. Matrix analysis provides a computational form for a balanced queue length distribution and several other performance metrics. We design a 'no-loss; no-profit cost model' to determine the appropriate value for the maximum value of K consecutive vacation periods and provide a solution with a numerical illustration.