{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,1]],"date-time":"2026-02-01T04:56:49Z","timestamp":1769921809985,"version":"3.49.0"},"reference-count":38,"publisher":"Wiley","license":[{"start":{"date-parts":[[2023,12,5]],"date-time":"2023-12-05T00:00:00Z","timestamp":1701734400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Advances in Operations Research"],"published-print":{"date-parts":[[2023,12,5]]},"abstract":"<jats:p>In this review article, we consider discrete-time birth-death processes and their applications to discrete-time queues. To make the analysis simpler to follow, we focus on transform-free methods and consider instances of non-birth-death Markovian discrete-time systems. We present a number of results within one discrete-time framework that parallels the treatment of continuous time models. This approach has two advantages; first, it unifies the treatment of several discrete-time models in one framework, and second, it parallels to the extent possible the treatment of continuous time models. This allows us to draw parallels and contrasts between the discrete and continuous time queues. Specifically, we focus on birth-death applications to the single server discrete-time model with Bernoulli arrivals and geometric service times and provide the reader with a simple rigorous detailed analysis that covers all five scheduling rules considered in the literature, with attention to stationary distributions at slot edges, slot centers, and prearrival epochs. We also cover the waiting time distributions. Moreover, we cover three Markovian models that fit the global balance equations. Our approach provides interesting insights into the behavior of discrete-time queues. The article is intended for those who are familiar with queueing theory basics and would like a simple, yet rigorous introductory treatment to discrete-time queues.<\/jats:p>","DOI":"10.1155\/2023\/6620393","type":"journal-article","created":{"date-parts":[[2023,12,5]],"date-time":"2023-12-05T16:35:11Z","timestamp":1701794111000},"page":"1-22","source":"Crossref","is-referenced-by-count":1,"title":["A Review of Birth-Death and Other Markovian Discrete-Time Queues"],"prefix":"10.1155","volume":"2023","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3857-6366","authenticated-orcid":true,"given":"Muhammad","family":"El-Taha","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, University of Southern Maine, 96 Falmouth Street, Portland, ME 04104-9300, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"key":"1","doi-asserted-by":"publisher","DOI":"10.1287\/opre.6.1.96"},{"key":"2","volume-title":"Mathematical Techniques of Applied Probability, Volume II: Discrete-Time Models: Tecniques and Applications","author":"J. J. Hunter","year":"1983"},{"key":"3","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-1164-8","volume-title":"Computer Networks and Systems: Queueing Theory and Performance Evaluation","author":"T. G. Robertazzi","year":"2000","edition":"3rd"},{"key":"4","doi-asserted-by":"crossref","first-page":"1099","DOI":"10.1016\/S0362-546X(96)00124-1","article-title":"Sample-path insensitivity of symmetric queues in discrete-time","volume":"30","author":"M. El-Taha","year":"1997","journal-title":"Nonlinear Analysis: Theory, Methods and Applications"},{"key":"5","doi-asserted-by":"crossref","DOI":"10.1007\/3-540-44592-7","volume-title":"Queueing Networks with Discrete Time Scale","author":"H. Daduna","year":"2001"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1016\/s0166-5316(01)00057-8"},{"key":"7","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4939-3420-1","volume-title":"Applied Discrete-Time Queues","author":"A. S. Alfa","year":"2016","edition":"2nd"},{"key":"8","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4419-7314-6","volume-title":"Queueing Theory for Telecommunications: Discrete Time Modeling of a Single Node System","author":"A. S. Alfa","year":"2010"},{"key":"9","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4615-3130-2","volume-title":"Discrete-time Models for Communications Systems Including ATM","author":"H. Bruneel","year":"1993"},{"key":"10","volume-title":"Communications and computer networks: modeling with discrete-time queues","author":"M. E. Woodward","year":"1994"},{"key":"11","doi-asserted-by":"publisher","DOI":"10.1002\/j.1538-7305.1983.tb03527.x"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1016\/0166-5316(92)90014-8"},{"key":"13","article-title":"Alternating birth-death processes","author":"H. Daduna","year":"2020"},{"key":"14","article-title":"Exact asymptotics of the stationary tail probabilities in an arbitrary direction in a two-dimensional discrete-time qbd process","author":"T. Ozawa","year":"2023"},{"key":"15","doi-asserted-by":"publisher","DOI":"10.1007\/s11134-018-9586-x"},{"key":"16","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2021.125029"},{"key":"17","doi-asserted-by":"publisher","DOI":"10.1063\/5.0079228"},{"key":"18","doi-asserted-by":"publisher","DOI":"10.1016\/s0898-1221(00)00157-7"},{"key":"19","doi-asserted-by":"publisher","DOI":"10.1007\/bf00289264"},{"key":"20","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1007\/978-1-4757-4828-4_10","article-title":"On numerical computations of some discrete-time queues","volume-title":"Computational Probability","author":"M. L. Chaudhry","year":"2000"},{"key":"21","doi-asserted-by":"publisher","DOI":"10.1016\/0167-6377(95)00051-8"},{"key":"22","first-page":"1183","article-title":"Relationships among different models for discrete-time queues","volume":"4","author":"G. R. Dattatreya","year":"2005","journal-title":"WSEAS Transactions on Systems"},{"key":"23","first-page":"1","article-title":"A discrete-time single server queue with Bernoulli arrivals and constant service time","volume":"12","author":"J. R. Louvion","year":"1988","journal-title":"Proceedings of the International Teletraffic Conference (ITC)"},{"key":"24","doi-asserted-by":"publisher","DOI":"10.1007\/bf01919483"},{"key":"25","doi-asserted-by":"publisher","DOI":"10.1007\/bf01164006"},{"key":"26","volume-title":"Matrix-Geometric Solutions in Stochastic Models-An Algorithmic Approach","author":"M. F. Neuts","year":"1981"},{"key":"27","doi-asserted-by":"crossref","DOI":"10.1002\/9781118625651","volume-title":"Fundamentals of Queueing Theory","author":"D. Gross","year":"2008","edition":"4th"},{"key":"28","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4615-5721-0","volume-title":"Sample-Path Analysis of Queueing Systems","author":"M. El-Taha","year":"1999"},{"key":"29","first-page":"124","article-title":"A discrete workload conservation law with applications to discrete-time queueing systems","volume":"2","author":"M. El-Taha","year":"2019","journal-title":"Queueing Models and Service Management"},{"key":"30","first-page":"1084","article-title":"On averages seen by arrivals in discrete-time","author":"A. Makowski"},{"key":"31","doi-asserted-by":"publisher","DOI":"10.1016\/s0378-3758(01)00131-8"},{"key":"32","volume-title":"Introduction to Queueing Theory","author":"R. B. Cooper","year":"1981","edition":"2nd"},{"key":"33","article-title":"Queueing analysis, A foundation of performance evaluation","volume-title":"Discrete-Time Systems","author":"H. Takagi","year":"1993"},{"key":"34","doi-asserted-by":"publisher","DOI":"10.1016\/s0305-0548(02)00155-7"},{"key":"35","volume-title":"Renewal Theory","author":"D. R. Cox","year":"1962"},{"key":"36","doi-asserted-by":"publisher","DOI":"10.1287\/opre.23.4.830"},{"key":"37","doi-asserted-by":"publisher","DOI":"10.1287\/ijoc.13.3.172.12627"},{"key":"38","doi-asserted-by":"publisher","DOI":"10.1155\/2019\/6480139"}],"container-title":["Advances in Operations Research"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/aor\/2023\/6620393.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/aor\/2023\/6620393.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/aor\/2023\/6620393.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,12,5]],"date-time":"2023-12-05T16:35:21Z","timestamp":1701794121000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.hindawi.com\/journals\/aor\/2023\/6620393\/"}},"subtitle":[],"editor":[{"given":"Arunava","family":"Majumder","sequence":"additional","affiliation":[],"role":[{"role":"editor","vocabulary":"crossref"}]}],"short-title":[],"issued":{"date-parts":[[2023,12,5]]},"references-count":38,"alternative-id":["6620393","6620393"],"URL":"https:\/\/doi.org\/10.1155\/2023\/6620393","relation":{},"ISSN":["1687-9155","1687-9147"],"issn-type":[{"value":"1687-9155","type":"electronic"},{"value":"1687-9147","type":"print"}],"subject":[],"published":{"date-parts":[[2023,12,5]]}}}