Diffusion Approximation of Systems with Arrival Depending on Queue, and with Arbitrary Servicing

Krichagina, E. V.; Liptser, R.; Pukhalskii, A. A.

doi:10.1137/1133011

Cited by 14 publications

(7 citation statements)

References 10 publications

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“…Limiting relation (3.18) in turn follows by application of the following lemma of Krichagina et al [20]. Note that analogously to (3.17) and (3.18) we have 19) and therefore…”

Section: Asymptotic Properties Of Normalized Queue-length In Client Smentioning

confidence: 72%

“…Limiting relation (3.18) in turn follows by application of the following lemma of Krichagina et al [20].…”

Section: 14)mentioning

confidence: 91%

“…Containing two types of node these system are closer to the considered system. Such systems have been studied in Krichagina et al [20], Krichagina and Puhalskii [21], Liptser [23] and other papers. Different models, closed to the considered model, have been also studied in Whitt [34], Knessl and Tier [15], Kogan et al [17], [18], Reiman and Simon [31] and others.…”

Section: The History Of Question and The Goal Of The Papermentioning

confidence: 99%

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A Large Closed Queueing Network Containing Two Types of Node and Multiple Customer Classes: One Bottleneck Station

Abramov

2004

Queueing Systems

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Abstract. The paper studies a closed queueing network containing two types of node. The first type (server station) is an infinite server queueing system, and the second type (client station) is a single server queueing system with autonomous service, i.e. every client station serves customers (units) only at random instants generated by strictly stationary and ergodic sequence of random variables. It is assumed that there are r server stations. At the initial time moment all units are distributed in the server stations, and the ith server station contains Ni units, i = 1, 2, ..., r, where all the values Ni are large numbers of the same order. The total number of client stations is equal to k. The expected times between departures in the client stations are small values of the order O(N −1 ) (N = N1 + N2 + ... + Nr). After service completion in the ith server station a unit is transmitted to the jth client station with probability pi,j (j = 1, 2, ..., k), and being served in the jth client station the unit returns to the ith server station. Under the assumption that only one of the client stations is a bottleneck node, i.e. the expected number of arrivals per time unit to the node is greater than the expected number of departures from that node, the paper derives the representation for non-stationary queue-length distributions in non-bottleneck client stations.

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“…Limiting relation (3.18) in turn follows by application of the following lemma of Krichagina et al [20]. Note that analogously to (3.17) and (3.18) we have 19) and therefore…”

Section: Asymptotic Properties Of Normalized Queue-length In Client Smentioning

confidence: 72%

“…Limiting relation (3.18) in turn follows by application of the following lemma of Krichagina et al [20].…”

Section: 14)mentioning

confidence: 91%

Section: The History Of Question and The Goal Of The Papermentioning

confidence: 99%

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A Large Closed Queueing Network Containing Two Types of Node and Multiple Customer Classes: One Bottleneck Station

Abramov

2004

Queueing Systems

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“…Traditionally, the martingale methods are used to provide weak convergence results and diffusion approximations, and the majority of papers establish such type of results (e.g. Abramov [1], Kogan, and Liptser [28], Kogan, Liptser, and Shenfild [29], Krichagina [31], Krichagina, Liptser, and Puhalskii [32], Krylov, and Liptser [33], Mandelbaum, and Pats [39], Williams [52] and others).…”

Section: Introduction Description Of the Model Review Of The Literatu...mentioning

confidence: 99%

Analysis of multiserver retrial queueing system: A martingale approach and an algorithm of solution

Abramov

2006

Ann Oper Res

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Abstract. The paper studies a multiserver retrial queueing system with m servers. Arrival process is a point process with strictly stationary and ergodic increments. A customer arriving to the system occupies one of the free servers. If upon arrival all servers are busy, then the customer goes to the secondary queue, orbit, and after some random time retries more and more to occupy a server. A service time of each customer is exponentially distributed random variable with parameter µ1. A time between retrials is exponentially distributed with parameter µ2 for each customer. Using a martingale approach the paper provides an analysis of this system. The paper establishes the stability condition and studies a behavior of the limiting queue-length distributions as µ2 increases to infinity. As µ2 → ∞, the paper also proves the convergence of appropriate queue-length distributions to those of the associated 'usual' multiserver queueing system without retrials. An algorithm for numerical solution of the equations, associated with the limiting queue-length distribution of retrial systems, is provided.

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“…The following examples might be presented: 1) the queueing system considered by W. Whitt [26] with renewal arrival process and phase-type distribution of service times (see 3) closed networks containing one IS (infinite server) center with exponential service times and other nodes with arbitrary distributed service times ( [14], or see [12] for the one dimensional case).…”

Section: Introductionmentioning

confidence: 99%

Asymptotic analysis of queueingnetworks

Krichagina

1992

Stochastics and Stochastic Reports

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This paper presents limit theorems for the vector queue length process aszociated with a closed queueing network as the total number of customers n goes to infinity. The first theorem states that the queue process normalized by the factor n-I converges in probab~lrty as n + 7 to a deterministic function which id11 be characterized as the uniquc solution of a certain differential inclusion. Thc second theorem gives an zpprclximation via reflected diffusion process for the sequence o i deviations of normalized queue ieogih process from the deterministic limit, expanded by n * '

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Diffusion Approximation of Systems with Arrival Depending on Queue, and with Arbitrary Servicing

Cited by 14 publications

References 10 publications

A Large Closed Queueing Network Containing Two Types of Node and Multiple Customer Classes: One Bottleneck Station

A Large Closed Queueing Network Containing Two Types of Node and Multiple Customer Classes: One Bottleneck Station

Analysis of multiserver retrial queueing system: A martingale approach and an algorithm of solution

Asymptotic analysis of queueingnetworks

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