Queueing SystemsPub Date : 2023-10-17DOI: 10.1007/s11134-023-09894-8
Herwig Bruneel, Arnaud Devos
{"title":"Asymptotic behavior of a system of two coupled queues when the content of one queue is very high","authors":"Herwig Bruneel, Arnaud Devos","doi":"10.1007/s11134-023-09894-8","DOIUrl":"https://doi.org/10.1007/s11134-023-09894-8","url":null,"abstract":"Abstract We consider a system of two parallel discrete-time single-server queues, queue 1 and queue 2. The service time of any customer in either queue is equal to 1 time slot. Arrivals during consecutive slots occur independently from slot to slot. However, the arrival streams into both queues are possibly mutually interdependent, i.e., during any slot, the numbers of arrivals in queue 1 and queue 2 need not be statistically independent. Their joint probability generating function (pgf) A ( x , y ) fully characterizes the queueing model. As a consequence of the possible intra-slot correlation in the arrival process, the numbers of customers present (“system contents”) in queues 1 and 2, at any given slot boundary, are not necessarily independent either. In a previous paper, we have already discussed the mathematical difficulty of computing their steady-state joint pgf $$U(z_1,z_2)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>z</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>z</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> ; explicit closed-form results can only be obtained for specific choices of A ( x , y ). In this paper, we therefore look at the problem from an other angle. Specifically, we study the (asymptotic) conditional steady-state behavior of the system under the condition that the content of queue 1 is (temporarily) very high (goes to infinity). For ease of terminology, we refer to the system as the “asymptotic system” in these circumstances. We prove that the asymptotic system is nearly identical to the original (unconditional) system, but with a modified joint arrival pgf $$A^*(x,y)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msup> <mml:mi>A</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> that can be computed explicitly from A ( x , y ). This fundamental result allows us to determine the stability condition of queue 2 in the asymptotic system, and explicitly compute the classical queueing performance metrics of queue 2, such as the pgf, the moments and the approximate tail distribution of its system content, when this condition is fulfilled. It also leads to accurate approximative closed-form expressions for the joint tail distribution of the system contents in both queues, in the original (unconditional) system. We extensively illustrate our methodology by means of various specific (popular) choices of A ( x , y ). In some examples, where an explicit solution for $$U(z_1,z_2)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>z</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>z</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <","PeriodicalId":20813,"journal":{"name":"Queueing Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136033606","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Queueing SystemsPub Date : 2023-10-13DOI: 10.1007/s11134-023-09895-7
Krzysztof Dȩbicki, Enkelejd Hashorva, Michel Mandjes
{"title":"Editorial introduction: special issue on Gaussian queues","authors":"Krzysztof Dȩbicki, Enkelejd Hashorva, Michel Mandjes","doi":"10.1007/s11134-023-09895-7","DOIUrl":"https://doi.org/10.1007/s11134-023-09895-7","url":null,"abstract":"","PeriodicalId":20813,"journal":{"name":"Queueing Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135855639","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Queueing SystemsPub Date : 2023-10-07DOI: 10.1007/s11134-023-09891-x
Somya Mehra, Peter G. Taylor
{"title":"Open networks of infinite server queues with non-homogeneous multivariate batch Poisson arrivals","authors":"Somya Mehra, Peter G. Taylor","doi":"10.1007/s11134-023-09891-x","DOIUrl":"https://doi.org/10.1007/s11134-023-09891-x","url":null,"abstract":"Abstract In this paper, we consider the occupancy distribution for an open network of infinite server queues with multivariate batch arrivals following a non-homogeneous Poisson process, and general service time distributions. We derive a probability generating function for the transient occupancy distribution of the network and prove that it is necessary and sufficient for ergodicity that the expected occupancy time for each batch be finite. Further, we recover recurrence relations for the transient probability mass function formulated in terms of a distribution obtained by compounding the batch size with a multinomial distribution.","PeriodicalId":20813,"journal":{"name":"Queueing Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135254983","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Queueing SystemsPub Date : 2023-09-26DOI: 10.1007/s11134-023-09892-w
Angelos Aveklouris, Amber L. Puha, Amy R. Ward
{"title":"A fluid approximation for a matching model with general reneging distributions","authors":"Angelos Aveklouris, Amber L. Puha, Amy R. Ward","doi":"10.1007/s11134-023-09892-w","DOIUrl":"https://doi.org/10.1007/s11134-023-09892-w","url":null,"abstract":"","PeriodicalId":20813,"journal":{"name":"Queueing Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134885867","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Queueing SystemsPub Date : 2023-08-17DOI: 10.1007/s11134-023-09886-8
Dongzhou Huang
{"title":"On a modified version of the Lindley recursion","authors":"Dongzhou Huang","doi":"10.1007/s11134-023-09886-8","DOIUrl":"https://doi.org/10.1007/s11134-023-09886-8","url":null,"abstract":"","PeriodicalId":20813,"journal":{"name":"Queueing Systems","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2023-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44539355","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Queueing SystemsPub Date : 2023-08-01DOI: 10.1007/s11134-023-09887-7
Bara Kim, Jeongsim Kim, Yan Su, Chia-Li Wang
{"title":"Proofs of conjectures on the competition between observable and unobservable servers","authors":"Bara Kim, Jeongsim Kim, Yan Su, Chia-Li Wang","doi":"10.1007/s11134-023-09887-7","DOIUrl":"https://doi.org/10.1007/s11134-023-09887-7","url":null,"abstract":"","PeriodicalId":20813,"journal":{"name":"Queueing Systems","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44062456","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}