Kogan, Y., and Birman, A. Asymptotic analysis of closed queueing networks with bottlenecks. In Proceedings of the International Conference on Performance of Distributed Systems and Integrated Communication Networks (Kyoto, 1991), T. Hasegawa, H. Takagi, and Y. Takahashi, Eds., pp. 237--252. 33  Home/Search   Document Not in Database   Summary   Related Articles   Check

....many chains, many queues or large populations. Thus, special approaches for analyzing large closed networks also have been developed, such as the algorithm PANACEA based on asymptotic expansions of integral representations [30 33] Other asymptotic methods for large models have also been developed [4,21,24,25]. In this paper we propose a radically different algorithm for calculating the performance measures of closed queueing networks and related product form models, which we believe usefully complements existing algorithms, because it applies to both large and small models. In contrast to the ....

....of generating functions is by Williams and Bhandiwad [39] Kelly [19] also briefly discusses generating functions. More recently, in the tradition of the statistical mechanics, generating functions have been used to do asymptotics by Birman, Ferguson and Kogan [4] Kogan [24] and Kogan and Birman [25]. Gordon [16] and Bertozzi and McKenna [3] have also recently used the generating functions to derive closed form expressions for the normalization constant by applying Cauchy s theorem and the theory of residues. This extends the classical closed form expression for the normalization constant in ....

KOGAN, Y. and BIRMAN, A. Asymptotic analysis of closed queueing networks with bottlenecks. preprint.

....especially in those involving the famous normalizing constant. To circumvent these drawbacks, the idea is to compute asymptotic expansions of the characteristic values of the network, when m and n both tend to infinity. This approach has been considered by Knessl and Tier [5] Kogan and Birman [6, 7, 1] and Malyshev and Yakovlev [10] However, it relies on purely analytical tools, which are difficult to use in a more general setting and, in our opinion, do not really give a structural explanation of the phenomena involved. The method proposed hereafter has direct connections with the Central ....

....P(X k;n = q k ) P(X 1;n = q 1 ; X ;n = q jS n = m n ) P(S n = m n ) RR n 2754 8 Guy FAYOLLE, Jean Marc LASGOUTTES 2.2 Informal description of the method Most of the derivations obtained in the paper are based on the various representations given in Lemma 2.1. Whereas the studies [6, 7, 1, 10] use mainly saddle point methods, our approach relies on direct limit theorems for the distribution of S n . For example, assume that S n m n satisfies a local limit theorem such as: Under suitable conditions, there exists a distribution with density f and a sequence a n such that, for any ....

Y. KOGAN AND A. BIRMAN, Asymptotic analysis of closed queueing networks with bottlenecks, in Proceedings of the International Conference on Performance of Distributed Systems and Integrated Communication Networks, T. Hasegawa, H. Takagi, and Y. Takahashi, eds., Kyoto, 1991, pp. 237--252.

....of queue length at each single server for large closed queueing networks with bottlenecks are the main interest in this paper. The asymptotes of the product form stationary distribution for large closed networks as N 1 have been studied by different techniques in a number of papers [3] 5] [6], 8] 10] 13] The limit of nonstationary behavior of such networks has been investigated in the case of one bottleneck and non IS server [13] and in the case of one single server forming bottleneck [4] The case of all servers being uniformly loaded bottlenecks is discussed in [7] We ....

Y. Kogan and A. Birman, Asymptotics analysis of closed queueing networks with bottlenecks, Proc. Int. Conf. on Performance of Distributed Systems and Integrated Communication Networks, eds. T. Hasegawa and H. Takagi and Y. Takahashi (Kyoto, 1991) pp.237-252.

....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 A. Berger, L. Bregman, Y. Kogan Bottleneck analysis 2 simultaneously with service rates at PS stations. It is known [7] that for a single class CQN (K = 1) consisting of IS and PS stations, in general, only one PS station may be a bottleneck, and the bottleneck node can be easily identified from the network parameters. Moreover, the asymptotics for the mean queue length at the bottleneck station is found from a ....

....a polynomial of order K, where K is the number of classes. The relative simplicity of the results for K = 1; M 1 or K 1; M = 1 is explained by their derivation from asymptotic expansions of one dimensional integral representations for the partition function (normalization constant) in complex [2, 6, 7] or real [9] space. In general, the integral representations in complex and real space are K and M dimensional respectively, and their asymptotics can be relatively easy derived only in the case of normal traffic [9, 7] when neither a PS station nor or a group of PS stations forms a bottleneck. ....

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Y. Kogan and A. Birman, "Asymptotic analysis of closed queueing networks with bottlenecks", in T.Hasegawa, H.Takagi and Y. Takahashi (eds.), IFIP Transactions C-5, Performance of Distributed Systems and Integrated Communication Networks, 265-280, North-Holland, Amsterdam, 1992.

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Kogan, Y., and Birman, A. Asymptotic analysis of closed queueing networks with bottlenecks. In Proceedings of the International Conference on Performance of Distributed Systems and Integrated Communication Networks (Kyoto, 1991), T. Hasegawa, H. Takagi, and Y. Takahashi, Eds., pp. 237--252. 33

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KOGAN, Y. and BIRMAN, A. Asymptotic analysis of closed queueing networks with bottlenecks. preprint.

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