On the Optimal Input Rate in Queues with Batch Service

Author:

Pagano Michele1ORCID,Tananko Igor2ORCID,Stankevich Elena2ORCID

Affiliation:

1. Department of Information Engineering, University of Pisa, Via G. Caruso 16, 56122 Pisa, Italy

2. Department of System Analysis and Automatic Control, Faculty of Computer Science and Information Technologies, Saratov State University, 83 Astrakhanskaya St., 410012 Saratov, Russia

Abstract

In recent years, queuing systems with batch service are emerging as powerful and flexible mathematical models in different frameworks. In this paper, we consider a single server queuing system with Poissonian arrivals, infinite buffers, and a constant batch size b. This paper addresses a little-studied optimization problem, namely the existence of an optimal arrival rate that minimizes the average sojourn time. Unlike the classical M/M/1 queue, for any batch size b, the problem admits a non-trivial solution that can be found by solving a polynomial equation of degree b+1. Since, in general, only numerical solutions are available, a simple first-order approximation is also derived and the corresponding deviations (in terms of input rate and sojourn time) are calculated. In more detail, it is shown that the approximation improves as the batch size increases and, in any case, the relative error for the average sojourn time is less than 0.34%. Finally, the paper provides new theoretical results about the asymptotic service rate in the equivalent birth–death process, highlighting how it depends on all queue parameters.

Funder

Italian Ministry of Education and Research (MIUR) in the framework of the FoReLab project (Departments of Excellence) and by the University of Pisa in the framework

Publisher

MDPI AG

Subject

Geometry and Topology,Logic,Mathematical Physics,Algebra and Number Theory,Analysis

Reference30 articles.

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