An invariance property of sojourn times in cyclic networks

Malchin, Christian; Daduna, Hans

doi:10.1016/j.orl.2004.03.006

“…This can be expressed as a conditional invariance property of the sojourn times; see [14] for a discussion. We state the result here without further comments because we shall need it for the derivations later on.…”

Section: The Influence Of the Slowest Servermentioning

confidence: 99%

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

Daduna

¹

,

Malchin

²

,

Szekli³

2008

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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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“…Although the quantitative results in (3.1) are completely in line with intuition, there is still something behind the first-and second-order picture expressed there. This can be expressed as a conditional invariance property of the sojourn times; see [14] for a discussion. We state the result here without further comments because we shall need it for the derivations later on.…”

Section: The Influence Of the Slowest Servermentioning

confidence: 99%

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

Daduna

¹

,

Malchin

²

,

Szekli³

2008

Journal of Applied Probability

Self Cite

View full text Add to dashboard Cite

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.

show abstract