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 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 of or a maximum limit of as the number of customers served during a service period at a minimum cost. 1. Introduction Queueing systems with vacations have found wide applications in the modelling and analysis of computer and communication networks and several other engineering systems in which single server is performing more than one type of jobs. Modelling such systems as single server queues with vacations allows one to analyze each queue in relative isolation since the time the server is attending to other jobs in the system may be modeled as vacation. For more details and versatile implementation of vacation models, one can refer to the comprehensive survey by Doshi [1]. Vacation models are distinguished by their scheduling disciplines, that is, the rules governing when a service stops and a vacation begins. Several service disciplines in combination with vacations are possible, for example, exhaustive, limited, gated, exhaustive limited (E-limited), gated limited (G-limited), and so forth. In fact, there is an extensive amount of literature available on infinite- and finite-buffer M/G/1 and GI/M/1 type vacation models that can be found in Takagi [2, 3], Tian and Zhang [4], and so forth. Traditional teletraffic analysis using Poisson process is not powerful enough to capture the correlated and bursty nature of traffic arising in the present high-speed networks where packets or cells of voice, video, images and data are sent over a common transmission channel on statistical multiplexing basis. The performance analysis of statistical multiplexers whose input consists of superposition of several packetized sources has been done through some analytically tractable arrival process, for
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