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Asymptotics for the Late Arrivals Problem
- Publication Year :
- 2013
- Publisher :
- arXiv, 2013.
-
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, we prove that it is ergodic and then we focus on the unique joint equilibrium distribution. We write a functional equation for the bivariate generating function, finding the solution of such equation on a subset of its set of definition. This solution allows us to prove that the equilibrium distribution of the Markov chain decays super-exponentially fast in the quarter plane. Finally, exploiting the latter result, 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.
- Subjects :
- Distribution (number theory)
General Mathematics
0211 other engineering and technologies
02 engineering and technology
pre-scheduled random arrivals
Management Science and Operations Research
01 natural sciences
Domain (mathematical analysis)
010104 statistics & probability
bivariate generating function
Functional equation
FOS: Mathematics
Applied mathematics
late arrivals
exponentially delayed arrivals
Queues with correlated arrivals
0101 mathematics
Mathematics
Queueing theory
021103 operations research
Markov chain
Ergodicity
Probability (math.PR)
Generating function
60J10, 60K25
Exponential function
Settore MAT/07
Software
Mathematics - Probability
Subjects
Details
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....d318a267b4f2a984a59ec06a46b1f517
- Full Text :
- https://doi.org/10.48550/arxiv.1302.1999