System Performance Of A Variable-Capacity Batch-Service Queue With Geometric Service Times And Customer-Based Correlation

Abstract

In many queueing systems the server processes several customers simultaneously. Although the capacity of a batch, that is the number of customers that can be processed simultaneously, is often variable in practice, nearly all batch-service queueing models in literature consider a constant capacity. In this paper, we extend previous work on a batch-service queueing model with variable server capacity, where customers of two classes are accommodated in a common first-come-first-served single-server queue. We include correlation between the classes of consecutive customers, and the service times are geometrically distributed. We establish the equations that govern the system behaviour, the stability condition, and an expression for the steady-state probability generating function of the system occupancy at random slot boundaries. In addition, some numerical results are shown to study the impact of the mean service times and of the customer-based correlation in the arrival process on the performance of the queueing system.

System occupancy in a multiclass batch-service queueing system with limited variable service capacity

“…We will end this section on the numerical results by showing the mean system occupancy at service initiation opportunities and the obtained approximations for three different combinations of same-class probabilities and maximum service capacities for a range of values for the load ρ, see Table 1. In this table, we also compare the results obtained by the analysis of this paper with the expected system occupancy E[U ] ∞ of the simplified model without maximum service capacities, which is analysed in Baetens et al (2016Baetens et al ( , 2017Baetens et al ( , 2018a. We note that in this simplified model, only 2 parameters must be solved and is therefore much less complex to compute.…”

“…This tendency for clustering, also called customer-based correlation, has been described in more detail by Bruneel et al (2012) as well. A simplified model without maximum service capacities has been studied by Baetens et al (2016Baetens et al ( , 2017Baetens et al ( , 2018a. The main contributions of this paper are the inclusion of maximum service capacities in order to model a much more realistic model and the approximations at low, high or intermediate loads (which is obtained by interpolating the first two).…”

Analysis of a Queueing Model with Batch Markovian Arrival Process and General Distribution for Group Clearance

“…Two major classes of single-server batch service models studied in recent papers are the discrete-time (Claeys et al 2010a;Claeys et al 2010b;Claeys et al 2013;Banerjee et al 2014;Yu and Alfa 2015;Baetens et al 2016;Baetens et al 2017;Baetens et al 2018;Panda and Goswami 2020) and continuous-time models (Saxena et al 2018;D'Arienzo et al 2019;Banerjee and Gupta 2012;Banerjee et al 2015;Yu and Tang 2018;Pradhan and Gupta 2017;Pradhan et al 2016;Pradhan and Gupta 2019;Gupta et al 2020;Gupta and Banerjee 2019;Maity and Gupta 2015;Banik 2015;Vadivu and Arumuganathan 2015;Chaudhry et al 2016;Jeyakumar and Senthilnathan 2017;Zeng and Xia 2017;Niranjan et al 2018;Gupta and Banerjee 2018;Panda et al 2018;Ayyappan and Karpagam 2018;Ayyappan and Nirmala 2018;Bank and Samanta 2020;Xie et al 2020). The variety of techniques used for the analysis includes Kolmogorov equations, Supplementary variable techniques, Roots method, Matrix-Analytic Method, Embedded Markov chain analysis, Spectral methods, Asymptotic Quasi-Toeplitz Markov chain technique and Game theory, to name a few.…”

“…Service time distribution In the majority of cases, the service time distribution is assumed to be general (e.g. defined by its Laplace-Stieltjes transform), but several exceptions include memoryless (Gupta and Banerjee 2019;Maity and Gupta 2015;Panda et al 2018;Panda and Goswami 2020;Baetens et al 2017) and phase-type (PH) ((D'Arienzo et al 2019)), which allow to obtain explicit results. Note that in discrete time models, single slot service is also used, see Claeys et al (2010a), Baetens et al (2016), and Baetens et al (2018).…”

“…A significant attention is also received by the systems with variable capacity or the so-called versatile Bulk Service rule, where the batch size taken for service is random (Pradhan et al 2016;Maity and Gupta 2015;Jeyakumar and Senthilnathan 2017;Bank and Samanta 2020). Another interesting setup is considered in a series of papers (Baetens et al 2016(Baetens et al 2017(Baetens et al , 2018(Baetens et al , 2019, where the so-called two-class service policy was introduced. The model though is rather different from classical single-server batch service queue, since it is essentially a multiclass model.…”