A Moment Approach for a Conditional Central Limit Theorem of Infinite-Server Queue: A Case of M/M X / ∞ Queue

Ayane Nakamura and Tuan Phung-Duc ()
Additional contact information
Ayane Nakamura: Graduate School of Science and Technology, University of Tsukuba, Tsukuba 305-8577, Japan
Tuan Phung-Duc: Institute of Systems and Information Engineering, University of Tsukuba, Tsukuba 305-8577, Japan

Mathematics, 2023, vol. 11, issue 9, 1-20

Abstract: Several studies have been conducted on scaling limits for Markov-modulated infinite-server queues. To the best of our knowledge, most of these studies adopt an approach to prove the convergence of the moment-generating function (or characteristic function) of the random variable that represents a scaled version of the number of busy servers and show the weak law of large numbers and the central limit theorem (CLT). In these studies, an essential assumption is the finiteness of the phase process and, in most of them, the CLT for the number of busy servers conditional on the phase (or the joint states) has not been considered. This paper proposes a new method called the moment approach to address these two limitations in an infinite-server batch service queue, which is called the M/M X / ∞ queue. We derive the conditional weak law of large numbers and a recursive formula that suggests the conditional CLT. We derive series expansion of the conditional raw moments, which are used to confirm the conditional CLT by a symbolic algorithm.

Keywords: queueing model; infinite server; asymptotic analysis; weak law of large numbers; central limit theorem; moment approach; raw moment; factorial moment; stirling number; symbolic algorithm (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
https://www.mdpi.com/2227-7390/11/9/2088/pdf (application/pdf)
https://www.mdpi.com/2227-7390/11/9/2088/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:11:y:2023:i:9:p:2088-:d:1134990

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().