TRANSIENT AND STEADY STATE SOLUTION OF A SINGLE SERVER QUEUE WITH MODIFIED BERNOULLI SCHEDULE VACATIONS BASED ON EXHAUSTIVE SERVICE AND A SINGLE VACATION POLICY

Resumen

ABSTRACT
We analyze a single server queue with modified Bernoulli server vacations based on exhaustive service. Unlike other vacation policies, we assume that only at the completion of service of the last customer in the system, the server has the option to take a vacation or to remain idle in the system waiting for the next customer to arrive. The service times of the customers have been assumed to be exponential and vacations are phase type exponential. We obtain explicit steady state results for the probability generating functions of the queue length, the expected number of customers in the queue and the expected waiting time of the customer. Some earlier known results of the single server have been derived as a particular case.

Key words: Bernoulli schedule server vacations, exhaustive service, phase type vacations, steady state, stability condition

RESUMEN
Analizamos una cola servidor simple con vacaciones Bernoulli modificada basada en un servicio exhaustivo. En forma diferente a otras políticas de vacaciones, asumimos que el servidor tiene la opción de tomar vacaciones o de continuar vacante en el sistema esperando el próximo cliente,
solamente al completar el servicio del último cliente. Se asume que los tiempos de servicio de los clientes son exponenciales y las vacaciones son del tipo exponencial fásico. Obtenemos resultados explícitos para el estado estable para las funciones generatrices de probabilidad del largo de la cola, la esperanza del número de clientes en cola y la esperanza del tiempo de espera del cliente. Algunos resultados conocidos sobre servidores simples han sido derivados para este caso particular.