On Normalizing Constants in Multiclass Queueing Networks

Gerasimov, A. I.

doi:10.1287/opre.43.4.704

“…For details on how to derive the third case, see [9]. For details on how to derive the third case, see [9].…”

Section: Exact-order Asymptotic Results For Closed Queueing Networkmentioning

confidence: 99%

Exact-Order Asymptotic Analysis for Closed Queueing Networks

George¹,

Xia²,

Squillante³

2012

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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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“…The derivation of the first two cases is obvious. For details on how to derive the third case, see [9]. Note that in (10) the expression for the single-server and infinite-server cases off i (z) can be viewed as special cases of the multi-server case, obtained by setting s i = 1 or letting s i → ∞, respectively.…”

Section: Exact-order Asymptotic Results For Closed Queueing Networkmentioning

confidence: 99%

Exact-Order Asymptotic Analysis for Closed Queueing Networks

George

¹

,

Xia

²

,

Squillante

³

2012

Journal of Applied Probability

13

0

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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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“…It is also worth noting that since the integrand of J(θ, N ) is a polynomial, the cubature rules proposed in [18] provide alternative explicit expressions for G(θ, N ), which become exact for a sufficiently high interpolation degree, corresponding to a O((N/2) d ) time complexity. We point to [3,15,16] for other works on explicit expressions for the normalizing constant.…”

Section: Case D >mentioning

confidence: 99%

A note on integrating products of linear forms over the unit simplex

Casale¹

2017

Preprint

0

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Integrating a product of linear forms over the unit simplex can be done in polynomial time if the number of variables n is fixed [1]. In this note, we highlight that this problem is equivalent to obtaining the normalizing constant of state probabilities for a popular class of Markov processes used in queueing network theory. In light of this equivalence, we survey existing computational algorithms developed in queueing theory that can be used for exact integration. For example, under some regularity conditions, queueing theory algorithms can exactly integrate a product of linear forms of total degree N by solving N systems of linear equations.

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On Normalizing Constants in Multiclass Queueing Networks

Abstract: Analytical expressions (formulas) for normalizing constants in closed queueing networks with different classes of customers are presented and discussed in this article.

Cited by 11 publications

References 5 publications

Exact-Order Asymptotic Analysis for Closed Queueing Networks

Exact-Order Asymptotic Analysis for Closed Queueing Networks

Exact-Order Asymptotic Analysis for Closed Queueing Networks

A note on integrating products of linear forms over the unit simplex

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