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Mathematics  2013 

Advances on the Late Arrivals Problem

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Abstract:

We study a discrete time queueing system where deterministic arrivals have i.i.d. exponential delays $\xi_{i}$. The standard deviation $\sigma$ of the delay is finite, but its value is much larger than the deterministic unit service time. We describe the model as a bivariate Markov chain and focus on the joint equilibrium distribution. We also prove that the latter decays super-exponentially fast in the quarter plane. Finally, we discuss the numerical computation of the stationary distribution, showing the effectiveness of a simple approximation scheme in a wide region of the parameters. The model, motivated by air and railway traffic, was proposed many decades ago by Kendall with the name of "late arrivals problem", but no solution has been found so far.

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