Abstract
We consider the stochastic behaviour of a Markovian bivariate process {(C(t), N(t)), t≥0} whose statespace is a semi-stripS={0,1}×ℕ. The intensity matrix of the process is taken to get a limit distributionP ij =lim t→+∞ P{(C(t), N(t))=(i, j)} such that {P 0j ,j ∈ ℕ}, or alternatively {P 1j ,j ∈ ℕ}, satisfies a system of equations of ‘birth and death’ type. We show that this process has applications to queues with repeated attempts and queues with negative arrivals. We carry out an extensive analysis of the queueing process, including classification of states, stationary analysis, waiting time, busy period and number of customers served.
Zusammenfassung
Wir untersuchen einen bivariaten Markovschen Prozeß {C(t), N(t); t≥0} mit ZustandsraumS={0,1}×ℕ. Die Intensitätsmatrix des Prozesses ist so strukturiert, daß die GrenzverteilungP ij =lim t→+∞ P{(C(t), N(t))=(i,j)} bei festgehaltener erster Koordinate jeweils ein System von “Geburts- und Todesprozeßgleichungen” erfüllt. Derartige Prozesse modellieren Wartesysteme mit Wiederholversuchen bei Abweisung und Systeme mit negativen Kunden. Wir analysieren den Warteschlangenprozeß insbesondere in bezug auf Zustandsklassifikation, Gleichgewichtsverhalten, Wartezeiten, Besetztperioden und Anzahl der bedienten Kunden.
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Artalejo, J.R., Gómez-Corral, A. Generalized birth and death processes with applications to queues with repeated attempts and negative arrivals. OR Spektrum 20, 5–14 (1998). https://doi.org/10.1007/BF01545523
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DOI: https://doi.org/10.1007/BF01545523
Key words
- Birth and death processes
- convolutive equations
- negative arrivals
- queues with repeated attempts
- waiting times