Rational arrival process

In queueing theory, a discipline within the mathematical theory of probability, a rational arrival process (RAP) is a mathematical model for the time between job arrivals to a system. It extends the concept of a Markov arrival process, allowing for dependent matrix-exponential distributed inter-arrival times.^[1]

The processes were first characterised by Asmussen and Bladt^[2] and are referred to as rational arrival processes because the inter-arrival times have a rational Laplace–Stieltjes transform.

Software

Q-MAM a MATLAB toolbox which can solve queueing systems with RAP arrivals.^[3]

References

↑ Bladt, M.; Neuts, M. F. (2003). "Matrix‐Exponential Distributions: Calculus and Interpretations via Flows". Stochastic Models 19: 113. doi:10.1081/STM-120018141.
↑ Asmussen, S. R.; Bladt, M. (1999). "Point processes with finite-dimensional conditional probabilities". Stochastic Processes and their Applications 82: 127. doi:10.1016/S0304-4149(99)00006-X.
↑ Pérez, J. F.; Van Velthoven, J.; Van Houdt, B. (2008). "Q-MAM: A Tool for Solving Infinite Queues using Matrix-Analytic Methods". Proceedings of the 3rd International Conference on Performance Evaluation Methodologies and Tools (PDF). doi:10.4108/ICST.VALUETOOLS2008.4368. ISBN 978-963-9799-31-8.

Queueing theory

Single queueing nodes	D/M/1 queue M/D/1 queue M/D/c queue M/M/1 queue Burke's theorem M/M/c queue M/M/∞ queue M/G/1 queue Pollaczek–Khinchine formula Matrix analytic method M/G/k queue G/M/1 queue G/G/1 queue Kingman's formula Lindley equation Fork–join queue Bulk queue

Arrival processes	Poisson process Markovian arrival process Rational arrival process

Queueing networks	Jackson network Traffic equations Gordon–Newell theorem Mean value analysis Buzen's algorithm Kelly network G-network BCMP network

Service policies	FIFO LIFO Processor sharing Round-robin Shortest job first Shortest remaining time

Key concepts	Continuous-time Markov chain Kendall's notation Little's law Product-form solution Balance equation Quasireversibility Flow-equivalent server method Arrival theorem Decomposition method Beneš method

Limit theorems	Fluid limit Mean field theory Heavy traffic approximation Reflected Brownian motion

Extensions	Fluid queue Layered queueing network Polling system Adversarial queueing network Loss network Retrial queue

Information systems	Data buffer Erlang (unit) Erlang distribution Flow control (data) Message queue Network congestion Network scheduler Pipeline (software) Quality of service Scheduling (computing) Teletraffic engineering

Category

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