Analysis of the N-policy Geo/G/1 queue with general stup times and geometric m-closedowns

We analyze a discrete-time single-server queueing system under the N-policy, meaning that the server facility remains idle until exactly N customers have accumulated in the queue. Once this happens, the server is first activated during a setup period, after which the customers in the queue are served exhaustively. At the end of a busy period, when all customers in the queue have been depleted, a closedown period is started. The idea here is that although there is no work, the server remains activated, such that a customer arriving before the closedown time expires can be served immediately without prior setup. The goal of this closedown mechanism is the same as that of the N-policy: to further reduce the number of times that the server needs to be activated and deactivated. In our model, we assume the setup and closedown times to have a general and geometric distribution respectively. Customers arrive according to a Bernoulli arrival process and the service times of the customers are independent and generally distributed. The convention for simultaneous events is the Early Arrival System (EAS). The distinctive feature of our model however is the fact that a closedown is not interrupted until m>1 arrivals have occurred during the closedown period. If this is the case, service of the customers in the queue starts immediately without setup. Otherwise, if less than m arrivals occured when the closedown expires, the server deactivates and remains idle until N customers have accumulated again in the queue. Note that the server in this system is always in either of four states: idle, setup, busy or closedown. Our aim is to derive exact expressions for the distribution of both the queue content and the customer delay in the system conditioned on the server state. More precisely, we obtain the distribution of the queue content as observed in idle slots only, in setup slots only, and so on. Likewise, the delay distribution of a customer is conditioned on the server state of the slot in which that customer enters the system. Additionally, we also discuss the length of the server state sojourn times, i.e. the number of slots that the server remains in the same state. The results are illustrated by means of some numerical examples