This paper considers a single-server queueing model in which the customers are served in batches of varying size depending on predetermined thresholds as well as available inventory. There is a finite buffer for the inventory and the service of every customer requires an inventory item. An (s, S)-type inventory system is used for the models considered in this paper. Initially, the model is studied in detail using the matrix-analytic method by assuming all the underlying random variables to be exponentially distributed. Thereafter, an outline of the model in a more general set up is also presented. Due to complexity of the model when more general assumptions are made on the underlying random variables, simulation is opted after a satisfactory validation with the analytic counterpart of the exponential model. Finally, some illustrative numerical examples are also presented to accomplish our analysis.

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This paper considers an infinite-buffer queuing system with birth-death modulated Markovian arrival process (BDMMAP) with arbitrary service time distribution. BDMMAP is an excellent representation of the arrival process, where the fractal behavior such as burstiness, correlation, and self-similarity is observed, for example, in ethernet LAN traffic systems. This model was first apprised by Nishimura (2003), and to analyze it, he proposed a twofold spectral theory approach. It seems from the investigations that Nishimura's approach is tedious and difficult to employ for practical purposes. The objective of this paper is to analyze the same model with an alternative methodology proposed by Chaudhry et al. (2013) (to be referred to as CGG method). The CGG method appears to be rather simple, mathematically tractable, and easy to implement as compared to Nishimura's approach. The Achilles tendon of the CGG method is the roots of the characteristic equation associated with the probability generating function (pgf) of the queue length distribution, which absolves any eigenvalue algebra and iterative analysis. Both the methods are presented in stepwise manner for easy accessibility, followed by some illustrative examples in accordance with the context.

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A discrete-time batch service queue with batch renewal input and random serving capacity rule under the late arrival delayed access system (LAS-DA), has recently appeared in the literature [2]. In this paper, we consider the same model under the early arrival system (EAS), since it is more applicable in telecommunication systems where an arriving batch of packets needs to be transmitted in the same slot in which it has arrived. In doing so, we derive the steady-state queue length distributions at various epochs, and show that in limiting case the result gets converted to the continuous-time queue [1]. We discuss few numerical results as well.