Analysis of Queue-Length Dependent Vacations and P-Limited Service in BMAP/G/1/N Systems: Stationary Distributions and Optimal Control

We consider a finite-buffer single server queueing system with queue-length dependent vacations where arrivals occur according to a batch Markovian arrival process (BMAP). The service discipline is P-limited service, also called E-limited with limit variation (ELV) where the server serves until either the system is emptied or a randomly chosen limit of L customers has been served. Depending on the number of customers present in the system, the server will monitor his vacation times. Queue-length distributions at various epochs such as before, arrival, arbitrary and after, departure have been obtained. Several other service disciplines like Bernoulli scheduling, nonexhaustive service, and E-limited service can be treated as special cases of the P-limited service. Finally, the total expected cost function per unit time is considered to determine locally optimal values N∗ of N or a maximum limit ?̂?∗ of ?̂? as the number of customers served during a service period at a minimum cost.