English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

The large-time behavior of a nonlinearly coupled pair of measure-valuedtransport equations with discontinuous boundary conditions, parameterized by apositive real-valued parameter $\lambda$, is considered. These equationsdescribe the hydrodynamic or fluid limit of many-server queues with reneging(with traffic intensity $\lambda$), which model phenomena in diversedisciplines, including biology and operations research. For a broad class ofreneging distributions with finite mean and service distributions with finitemean and hazard rate function that is either decreasing or bounded away fromzero and infinity, it is shown that if the fluid equations have a uniqueinvariant state, then the Dirac measure at this state is the unique randomfixed point of the fluid equations, which implies that the stationarydistributions of scaled $N$-server systems converge to the unique invariantstate of the corresponding fluid equations. Moreover, when $\lambda\ne 1$, itis shown that the solution to the fluid equation starting from any initialcondition converges to this unique invariant state in the large time limit. Theproof techniques are different under the two sets of assumptions on the servicedistribution. When the hazard rate function is decreasing, a reformulation ofthe dynamics in terms of a certain renewal equation is used, in conjunctionwith recursive asymptotic estimates. When the hazard rate function is boundedaway from zero and infinity, the proof uses an extended relative entropyfunctional as a Lyapunov function. Analogous large-time convergence results arealso established for a system of coupled measure-valued equations modeling amulticlass queue.

Large-time limit of nonlinearly coupled measure-valued equations that model many-server queues with reneging