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Stability Results on Synchronized Queues in Discrete-Time for Arbitrary Dimension

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Document pages: 5 pages

Abstract: In a batch of synchronized queues, customers can only be serviced all at onceor not at all, implying that service remains idle if at least one queue isempty. We propose that a batch of $n$ synchronized queues in a discrete-timesetting is quasi-stable for $n in {2,3 }$ and unstable for $n geq 4$. Acorrespondence between such systems and a random-walk-like discrete-time Markovchain (DTMC), which operates on a quotient space of the original state-space,is derived. Using this relation, we prove the proposition by showing that theDTMC is transient for $n geq 4$ and null-recurrent (hence quasi-stability) for$n in {2,3 }$ via evaluating infinite power sums over skewed binomialcoefficients.Ignoring the special structure of the quotient space, the proposition can beinterpreted as a result of Pólya s theorem on random walks, since thedimension of said space is $d-1$.

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