Delay analysis of a two-class batch-service queue with class-dependent variable server capacity

Abstract

In this paper, we analyse the delay of a random customer in a two-class batch-service queueing model with variable server capacity, where all customers are accommodated in a common single-server first-come-first-served queue. The server can only process customers that belong to the same class, so that the size of a batch is determined by the length of a sequence of same-class customers. This type of batch server can be found in telecommunications systems and production environments. We first determine the steady state partial probability generating function of the queue occupancy at customer arrival epochs. Using a spectral decomposition technique, we obtain the steady state probability generating function of the delay of a random customer. We also show that the distribution of the delay of a random customer corresponds to a phase-type distribution. Finally, some numerical examples are given that provide further insight in the impact of asymmetry and variance in the arrival process on the number of customers in the system and the delay of a random customer.

Delay analysis of a variable-capacity batch-server queue with general class-dependent service times

“…Similarly to the analysis in our previous paper [3], where the delay is analysed for a similar model without customerbased correlation in the arrival process, it can be proven that the branching points, where λ 1 (z) = λ 2 (z), can be removed. Defining the matrices R(z) and L(z) as the right and left eigenvectors corresponding with the matrix M(z), we can show that…”

“…Germs and van Foreest [7] studied the M(n) X(n) /G(n) Y(n) /1/K + B model where the arrival rate, service time distribution and service capacity depend on the number of customers waiting in the queue with size K + B. In our previous papers [1,2,3], we analysed a batch queueing system with a service capacity that not only depends on the number of waiting customers but also on their respective classes. Such a server is often found in production lines where a machine can process multiple product types with different characteristics like required temperature or color.…”

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.…”

“…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). Batch service type The most widely studied is the classical General Bulk Service (GBS) rule introduced in Neuts (1967): the two finite constant threshold policy states that the server starts service of a batch of size larger or equal to a ≥ 1, and can handle up to b ≥ a customers in a batch, with b finite.…”

“…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.…”