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.

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[1] Bladt, M.; Neuts, M. F. (2003). "Matrix‐Exponential Distributions: Calculus and Interpretations via Flows". Stochastic Models. 19: 113. doi:10.1081/STM-120018141.

[2] 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.

[3] 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.

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v t e 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 point 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 next 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
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Software

References