Pub Date : 2023-10-17DOI: 10.1007/s11134-023-09894-8
Herwig Bruneel, Arnaud Devos
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 $$U(z_1,z_2)$$ U(z1,z2) ; 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 $$A^*(x,y)$$ A∗(x,y) 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 $$U(z_1,z_2)$$ U(z1,z2 <
考虑一个由两个并行的离散时间单服务器队列1和队列2组成的系统。两个队列中任意一个客户的服务时间等于1个时隙。在连续的时段内，每个时段的到达都是独立的。然而，进入两个队列的到达流可能是相互依赖的，即在任何时段，队列1和队列2的到达数不必在统计上独立。它们的联合概率生成函数(pgf) A (x, y)充分表征了排队模型。由于到达过程中可能存在槽内相关性，在任何给定的槽边界上，队列1和队列2中存在的客户数量(“系统内容”)也不一定是独立的。在之前的文章中，我们已经讨论了计算它们的稳态关节pgf $$U(z_1,z_2)$$ U (z1, z2)的数学难度;只有在A (x, y)的特定选择下才能得到显式的封闭结果。因此，在本文中，我们从另一个角度来看待这个问题。具体地说，我们研究了在队列1的内容(暂时)非常高(趋于无穷)的情况下系统的(渐近)条件稳态行为。为了便于术语的使用，在这种情况下，我们将该系统称为“渐近系统”。我们证明了渐近系统与原始(无条件)系统几乎相同，但具有一个修改的联合到达pgf $$A^*(x,y)$$ a∗(x, y)，该联合到达可以从a (x, y)显式计算。这一基本结果使我们能够确定渐近系统中队列2的稳定性条件，并显式地计算出该条件满足时队列2的经典排队性能指标，如pgf、矩和其系统内容的近似尾部分布。它还可以得到原始(无条件)系统中两个队列中系统内容的联合尾部分布的精确近似封闭表达式。我们通过A (x, y)的各种特定(流行)选择来广泛地说明我们的方法。在某些示例中，已知$$U(z_1,z_2)$$ U (z1, z2)或(近似)联合尾分布的显式解，我们可以轻松检索已知结果。在其他情况下，对于到达的pgfs A (x, y)发现了新的结果，直到现在还没有明确的结果。
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Pub 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}
Pub Date : 2023-10-07DOI: 10.1007/s11134-023-09891-x
Somya Mehra, Peter G. Taylor
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.
{"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}
Pub 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}
Pub 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}
Pub 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}
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