Asymptotics for the late arrivals problem

Carlo Lancia (), Gianluca Guadagni (), Sokol Ndreca () and Benedetto Scoppola ()
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Carlo Lancia: Mathematical Institute Leiden University
Gianluca Guadagni: University of Virginia
Sokol Ndreca: Universidade Federal de Minas Gerais
Benedetto Scoppola: Università degli Studi di Roma Tor Vergata

Mathematical Methods of Operations Research, 2018, vol. 88, issue 3, No 6, 475-493

Abstract: Abstract We study a discrete time queueing system where deterministic arrivals have i.i.d. exponential delays $$\xi _{i}$$ ξ i . We describe the model as a bivariate Markov chain, prove its ergodicity and study the joint equilibrium distribution. We write a functional equation for the bivariate generating function, finding the solution on a subset of its domain. This solution allows us to prove that the equilibrium distribution of the chain decays super-exponentially fast in the quarter plane. We exploit the latter result and discuss the numerical computation of the solution through a simple yet effective approximation scheme in a wide region of the parameters. Finally, we compare the features of this queueing model with the standard M / D / 1 system, showing that the congestion turns out to be very different when the traffic intensity is close to 1.

Keywords: Late arrivals; Exponentially delayed arrivals; Pre-scheduled random arrivals; Queues with correlated arrivals; Bivariate generating function; 60J05; 60K25 (search for similar items in EconPapers)
Date: 2018
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s00186-018-0643-3

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