Analysis of a retrial queue with group service of impatient customers
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ORIGINAL RESEARCH
Analysis of a retrial queue with group service of impatient customers M. P. D’Arienzo1 · A. N. Dudin2,3 · S. A. Dudin2,3 · R. Manzo1 Received: 8 November 2018 / Accepted: 6 May 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract A single-server retrial queue with a MAP flow, PH service times and a pool of finite capacity for accumulation of the customers and their group service is considered. Service to the next group is not provided until the number of customers in the pool will reach a certain predefined threshold value. The service time of a group depends on its size and it is less than the sum of the individual service times. The dependencies of the basic performance measures of the system on the capacity of the pool and the threshold are obtained. Numerical results are presented. Keywords Markovian arrival process · Phase-type distribution of the service time · Group service · Impatient customers · Retrial
1 Introduction 1.1 Related literature survey Queueing theory is effectively used to solve optimization problems of various real world systems where arriving at random epochs customers should obtain service at some of devices or servers. Essential features of the model considered in our paper are: (1) group (or batch) service of customers; (2) absence of the input buffer; (3) impatience of customers.
* R. Manzo [email protected] M. P. D’Arienzo [email protected] A. N. Dudin dudin‑[email protected] S. A. Dudin [email protected] 1
Department of Information Engineering, Electrical Engineering and Applied Mathematics, University of Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano, Italy
2
Department of Applied Mathematics and Computer Science, Belarusian State University, 4, Nezavisimosti Ave., 220030 Minsk, Belarus
3
Department of Applied Probability and Informatics, RUDN University, 6, Miklukho‑Maklaya st., 117198 Moscow, Russia
Queueing models with batch services have many real life applications notably in transportation, manufacturing, telecommunications, group testing, and other areas, see e.g. Abolnikov and Dukhovny (2003), Bar-Lev et al. (2007), Bruneel et al. (2010) and Harten et al. (2001). Among numerous papers dealing with batch service discipline, we mention Alfa and Chakravarthy (1994), Bailey (1954), Bin and Chakravarthy (1997), Chakravarthy (1992), Chakravarthy (1996), Chakravarthy (1998), Chaudhry and Templeton (1983), Deb and Serfozo (1973), Downton (1955) and Xia and Zeng (2017) as well as the surveys in Banerjee et al. (2015), Gupta et al. (2017) and Indhira and Sasikala (2016) and a series of other important papers written by the research group of S.R. Chakravarthy. In the analysis of the majority of the systems with batch service, it is assumed that the system has a finite or infinite buffer, the minimal and the maximal size of a batch are predefined and the service time of the batch does not depend on the size of the batch. However, in many important real-world systems, the buffer does not exist and an arriving customer that me
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