Zero-automatic queues and product form

Mairesse, Jean

doi:10.1239/aap/1183667618

Cited by 3 publications

(3 citation statements)

References 25 publications

(35 reference statements)

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“…The Markovian structure here is somewhat similar to random walks on free products of cyclic groups and to zero-automatic queues and product form recently discussed in [7], [8], [18], and [19].…”

Section: Theorem 2 If the Markov Chains Of The Unmatched Words Are Ementioning

confidence: 69%

“…Analogously, if we match successive customers, the leftover server words will consist of type-2 servers and the Markov chain which describes this, Y n , will count the number of unmatched type-2 servers. Analogous to (8), the steady-state probabilities for Y n are…”

Section: Example 1: the 'N'-modelmentioning

confidence: 99%

“…(with an obvious modification if k = ), and if there are no more type-i customers in the sequence, the new state will be X n+1 = ((i 1 , x 1 It is not currently clear how useful these Markov chains may be in answering our main questions: how to calculate the r i,j and how to prove convergence. As we mentioned before (Section 4), the work of Mairesse et al [7], [8], [18], [19] may be relevant here.…”

Section: Further Specification Of the Markov Chains For Systems With mentioning

confidence: 99%

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Fcfs infinite bipartite matching of servers and customers

2009

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We consider an infinite sequence of customers of types C = {1, 2, . . . , I } and an infinite sequence of servers of types S = {1, 2, . . . , J }, where a server of type j can serve a subset of customer types C(j ) and where a customer of type i can be served by a subset of server types S(i). We assume that the types of customers and servers in the infinite sequences are random, independent, and identically distributed, and that customers and servers are matched according to their order in the sequence, on a first-come-first-served (FCFS) basis. We investigate this process of infinite bipartite matching. In particular, we are interested in the rate r i,j that customers of type i are assigned to servers of type j . We present a countable state Markov chain to describe this process, and for some previously unsolved instances, we prove ergodicity and existence of limiting rates, and calculate r i,j .

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Section: Theorem 2 If the Markov Chains Of The Unmatched Words Are Ementioning

confidence: 69%

Section: Example 1: the 'N'-modelmentioning

confidence: 99%

Section: Further Specification Of the Markov Chains For Systems With mentioning

confidence: 99%

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Fcfs infinite bipartite matching of servers and customers

2009

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Zero-Automatic Networks

Mairesse

2008

Discrete Event Dyn Syst

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We continue the study of zero-automatic queues first introduced in DaoThi and Mairesse (Adv Appl Probab 39 (2): 2007). These queues are characterized by a special buffering mechanism evolving like a random walk on some infinite group or monoid. The simple M/M/1 queue and Gelenbe's G-queue with positive and negative customers are the two simplest 0-automatic queues. All stable 0-automatic queues have an explicit "multiplicative" stationary distribution and a Poisson departure process (Dao-Thi and Mairesse, Adv Appl Probab 39 (2): 2007). In this paper, we introduce and study networks of 0-automatic queues. We consider two types of networks, with either a Jackson-like or a Kelly-like routing mechanism. In both cases, and under the stability condition, we prove that the stationary distribution of the buffer contents has a "product-form" and can be explicitly determined. Furthermore, the departure process out of the network is Poisson.

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Zero-automatic networks

Mairesse¹

2006

Proceedings of the 1st International Conference on Performance Evaluation Methodolgies and Tools - Valuetools '06

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"Best Student Paper award" of the conference.International audienceWe continue the study of zero-automatic queues first introduced in "Zero-automatic queues and product form", T.-H. Dao-Thi and J. Mairesse, Adv. Appl. Probab., vol. 39, n. 7, 2007. These queues are characterized by a special buffering mechanism evolving like a random walk on some infinite group or monoid. The simple M/M/1 queue and Gelenbe's G-queue with positive and negative customers are the two simplest 0-automatic queues. All 0-automatic queues are quasi-reversible. In this paper, we introduce and study networks of 0-automatic queues. We consider two types of networks, with either a Jackson-like or a Kelly-like routing mechanism. In both cases, and under the stability condition, we prove that the stationary distribution of the buffer content has a "product-form" and can be explicitly determined

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Zero-automatic queues and product form

Cited by 3 publications

References 25 publications

Fcfs infinite bipartite matching of servers and customers

Fcfs infinite bipartite matching of servers and customers

Zero-Automatic Networks

Zero-automatic networks

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