The BMAP/G/1 vacation queue with queue-length dependent vacation schedule

Abstract

We treat a single-server vacation queue with queue-length dependent vacation schedules. This subsumes the single-server vacation queue with exhaustive service discipline and the vacation queue with Bernoulli schedule as special cases. The lengths of vacation times depend on the number of customers in the system at the beginning of a vacation. The arrival process is a batch-Markovian arrival process (BMAP). We derive the queue-length distribution at departure epochs. By using a semi-Markov process technique, we obtain the Laplace-Stieltjes transform of the transient queue-length distribution at an arbitrary time point and its limiting distribution.

Queueing models for appointment-driven systems

“…For those relevant models reported (Yijun and Quanlin (1996) and Shin and Pearce (1998)), the length of state-dependent vacations depends on the number of customers in the system at the beginning of a vacation. The AMQ however, requires the vacation length to depend on the current time in the system (i.e.…”

Analysis of BMAP/G/1 Vacation Model of Non-M/G/1-Type

“…This subsection presents the service discipline independent solution for the mean of the stationary number of customers in the system based on its vector GF (10). To this end, we introduce the following notations.…”

“…Ferrandiz [9] used Palm-martingale calculus to analyze a flexible vacation scheme. Shin and Pearce [10] studied queue-length dependent vacation schedules by using the semi-Markov process technique. Recently Banik et al [11] studied the BMAP/G/1/N queue with vacations and E-limited service discipline.…”

On a BMAP/G/1 G-queue with setup times and multiple vacations

“…In the non-exhaustive service discipline, a vacation may start even when customers are present in the system. Shin and Pearce [25] investigated a BMAP/G/1 vacation queue with queuelength dependent vacation schedule. Wu et al [28] studied a BMAP/G/1 G-queue with second optional service and multiple vacations.…”

“…There have been extensive studies on queueing systems with BMAP input. For detailed bibliographies on BMAP, readers may refer to [3,20,21,25,26]. In particular, Lee [15] used the supplementary variable method, which is combined with the embedded Markov chain, to analyze the BMAP/G/1 queues with finite or infinite waiting room.…”