The Product Form for Sojourn Time Distributions in Cyclic Exponential Queues

Boxma, Onno; Kelly, F. P.; Konheim, Alan G.

doi:10.1145/2422.322419

Cited by 55 publications

(18 citation statements)

References 12 publications

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“…Furthermore, we are interested in the customer's cycle time distribution. These distributions are known, given in the transform domain by their respective Laplace-Stieltjes transforms (LSTs) [4], [18]. We transform these formulae in a way that allows us to prove weak convergence results for the customer's travel-time behaviour when the bottleneck dominates the travel times.…”

Section: H Daduna Et Almentioning

confidence: 99%

“…The starting instants of the TC's cycles are his successive entrance times into station Q [1]. The limiting distribution of the vector of the TC's successive sojourn times during his cycles is determined by the LST (see [4,Theorem 1]): .1)) is the steady-state probability that at the arrival instants of the TC at Q [1] there are n 1 further customers present at node Q [1] (without counting the TC himself), and n j customers at nodes…”

Section: Closed Cycle With M Stationsmentioning

confidence: 99%

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Weak Convergence Limits for Sojourn Times in Cyclic Queues Under Heavy Traffic Conditions

Daduna

Malchin

Szekli³

2008

J. Appl. Probab.

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We consider sequences of closed cycles of exponential single-server nodes with a single bottleneck. We study the cycle time and the successive sojourn times of a customer when the population sizes go to infinity. Starting from old results on the mean cycle times under heavy traffic conditions, we prove a central limit theorem for the cycle time distribution. This result is then utilised to prove a weak convergence characteristic of the vector of a customer's successive sojourn times during a cycle for a sequence of networks with population sizes going to infinity. The limiting picture is a composition of a central limit theorem for the bottleneck node and an exponential limit for the unscaled sequences of sojourn times for the nonbottleneck nodes.

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Section: H Daduna Et Almentioning

confidence: 99%

Section: Closed Cycle With M Stationsmentioning

confidence: 99%

Weak Convergence Limits for Sojourn Times in Cyclic Queues Under Heavy Traffic Conditions

Daduna

Malchin

Szekli³

2008

J. Appl. Probab.

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“…1 with state independent service rates, see Ref. 7 . As a by-product Boxma, Kelly, and Konheim showed that in steady state each pair of local sojourn times within a fixed cycle for a customer traversing the queues is negatively correlated.…”

Section: Na Of Sojourn Times In Closed Network: Cyclic Pathsmentioning

confidence: 99%

On the Correlation Structure of Closed Queueing Networks

Daduna

Szekli

2004

Stochastic Models

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Closed networks of exponential queues are prototypes of systems which show strong negative dependence properties. We use the device of a test customer traveling in the network to evaluate the qualitative behavior of the network's correlation. The results are: negative association of successive sojourn times during a cycle; negative correlation of two successive cycle times; feedbacksojourn times are positively associated; correlation between cycles vanishes over long distances geometrically fast. Generalizations to networks with general topology are given. Additionally we prove for cyclic networks that the conditional cycle time stochastically increases when the initial joint queue length vector increases with respect to the partial sum ordering. This connects the space-time correlation with space-time monotonicity.

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“…4, with one exception: all waiting rooms are infinite. In [3] exact results from [4] are used to analyse the influence of the "slowest server" (the server with largest mean service time) on cycle timeand sojourn time distributions. Of particular interest to us is the result for the mean cycle time,…”

Section: Motivation Vf the Approximation Assumptionmentioning

confidence: 99%

Throughput analysis of a flow-controlled communication network with buffer space limitations

Berg¹,

Boxma²

1987

Flow Control of Congested Networks

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This paper studies the traffic flow in a virtual circuit of a computer communication network with window flow control. Due to finite buffer capacity, overflow of data packets is possible. Lost packets have to be retransmitted. To maintain the original order of the packets, subsequently sent packets also have to be retransmitted, causing a deterioriation of throughput. An ap_proximation method, based on a queueing network model, is developed to analyse the throughput behaviour. This method leads to exact results tor the single-hop network. For the multi-hop circuit, it yields very accurate approx1mat1ons, as 1s illustrated by simulation.

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The Product Form for Sojourn Time Distributions in Cyclic Exponential Queues

Abstract: Consider a closed cyclic queuing system consisting of M exponential queues. The LaplaceStieltjes transform of the joint d~stribution of the consecutive sojourn times of a customer at the M queues is determined and shown to have a product form. The proof is based on a reverslbdity argument.

Cited by 55 publications

References 12 publications

Weak Convergence Limits for Sojourn Times in Cyclic Queues Under Heavy Traffic Conditions

Weak Convergence Limits for Sojourn Times in Cyclic Queues Under Heavy Traffic Conditions

On the Correlation Structure of Closed Queueing Networks

Throughput analysis of a flow-controlled communication network with buffer space limitations

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