Another approach to asymptotic expansions for large closed queueing networks

Kogan, Yaakov

doi:10.1016/0167-6377(92)90009-r

Cited by 35 publications

(19 citation statements)

References 3 publications

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“…Using the multi-server form (12) forf i , the z-transform for the sequence {G(n), n ≥ 0} can be written as Let l = |B|.…”

Section: Assumptionmentioning

confidence: 99%

“…Each of these exact methods typically improves the computational cost over that of direct calculation but still often requires a large number of computations for networks with large populations. Most previous studies focused on either closed-form expressions [22,Chapter 1] or asymptotic approximations [11], [12], [17] for the normalizing constant under rather restrictive assumptions on network structure or parameter values. For example, Muntz and Wong [18] derived asymptotic bounds on the server utilizations and mean response times for a closed terminal network with only single-server and infinite-server stations.…”

Section: Introductionmentioning

confidence: 99%

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Exact-Order Asymptotic Analysis for Closed Queueing Networks

George¹,

Xia²,

Squillante³

2012

J. Appl. Probab.

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In this paper we study the asymptotic behavior of a general class of product-form closed queueing networks as the population size grows large. We first characterize the asymptotic behavior of the normalization constant for the stationary distribution of the network in exact order. This result then enables us to establish the asymptotic behavior of the system performance metrics, which extends a number of well-known asymptotic results to exact order. We further derive new, computationally simple approximations for performance metrics that significantly improve upon existing approximations for large-scale networks. In addition to their direct use for the analysis of large networks, these new approximations are particularly useful for reformulating large-scale queueing network optimization problems into more easily solvable forms, which we demonstrate with an optimal capacity planning example.

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“…Using the multi-server form (12) forf i , the z-transform for the sequence {G(n), n ≥ 0} can be written as Let l = |B|.…”

Section: Assumptionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Exact-Order Asymptotic Analysis for Closed Queueing Networks

George¹,

Xia²,

Squillante³

2012

J. Appl. Probab.

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“…asymptotic behavior as the network size gets large: more precisely, when both the number of nodes and customers tend to infinity with their ratio tending to a constant, say λ (see [7,13,14] and references therein). The main result is the existence of a critical value of λ such that under this value, the system is stable with queue lengths of any finite subset being asymptotically independent with geometric distributions, while above this value, some queues, those with the maximum so-called utilization, behave as bottlenecks with an infinite mean number of customers, while the others are stable with the same asymptotic property as in the subcritical regime.…”

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confidence: 99%

Two-choice regulation in heterogeneous closed networks

Fricker

Servel

2015

Queueing Syst

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“…This generating function is one-dimensional, and it has an explicit form up to the normalization constant. This allows us to derive the asymptotics of the probability mass function in a direct way similar to [13,14,4] by evaluating the Cauchy integral by the saddle-point method. The final result is derived by proving that −F (x * ) gives the logarithmic asymptotics for the normalization constant.…”

mentioning

confidence: 99%

“…Its application in our case would require consideration of an auxiliary multidimensional Markov process and complicate the analysis. Therefore we follow another approach, which is based on generating functions and integral representations [5,15,16,17,18,19,4,13,14].…”

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confidence: 99%

Distribution of Processor-Sharing Customers for a Large Closed System with Multiple Classes

Berger¹,

Kogan²

2000

SIAM J. Appl. Math.

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A closed processor-sharing (PS) system with multiple customer classes is considered. The system consists of one infinite server (IS) station and one PS station. For a system with a large number of customers, a saturated PS station, and an arbitrary number of customer classes, asymptotic approximations to the stationary distribution of the total number of customers at the PS station are derived. The asymptotics for the probability mass function is described by a quasipotential function, which defines the exponential decay for the distribution, and a state-dependent preexponential factor. Both functions have an explicit expression in terms of the solution at each point x of a polynomial equation whose order equals the number of classes and whose coefficients are explicit functions of x. The quasi-potential function at its minimum point provides the logarithmic asymptotics for the normalization constant, and the asymptotic approximation for the variance is inversely proportional to the second derivative of the quasi-potential function at its minimum point. The complementary probability distribution is computed using the normal approximation and its refinements, which do not require repeated solution of polynomial equations. Numerical results demonstrate the range of applicability of the approximations. The results can be applied to the problem of dimensioning bandwidth and of admission control for different data sources in packetswitched communication networks.1. Introduction. This paper is motivated by a new application of closed queueing networks (CQN) with a large number of customers. The application is the dimensioning of bandwidth and of admission control for different data sources subject to feedback control in packet-switched communication networks when available bandwidth at the network nodes is shared between all active sources. Data sources are modeled by an infinte server (IS) station, network nodes are modeled by processorsharing (PS) stations, and a "customer" in the PS station represents an active data source. We consider a CQN that consists of one IS station with multiple customer classes and one PS station. The distinguishing property of the new application is that this CQN model is valid only if the PS station is saturated; see [3] for further details. The saturated station is defined asymptotically as the station, where the number of customers grows proportionally to the total number of customers in the network as the latter increases with service rates at the PS station.The application includes the performance metric that the bandwidth received by an active data source at a given network node is greater than a target value with probability 1 − α, where α is in the range of 0.001 to 0.1. As the network nodes of interest have a packet-based implementation of PS, the performance metric can be restated as the number of active data sessions at a network node (the total number of customers at the PS station in the CQN model) is less than a target value with the given probability 1 − α. As the above prob...

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Another approach to asymptotic expansions for large closed queueing networks

Cited by 35 publications

References 3 publications

Exact-Order Asymptotic Analysis for Closed Queueing Networks

Exact-Order Asymptotic Analysis for Closed Queueing Networks

Two-choice regulation in heterogeneous closed networks

Distribution of Processor-Sharing Customers for a Large Closed System with Multiple Classes

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