Busy period analysis of the level dependent \(PH/PH/1/K\) queue
The paper studies the transient behaviour of a state-dependent \(PH/PH/1/K\) queueing system. The author derives a closed form for the joint transform of the length of a busy period, the number of customers served during a busy period and the number of losses during a busy period. Earlier results on the analysis of the transient behaviour of traditional queueing systems (such as \(M/M/1/N\), \(M/GI/1/N\), \(GI/M/1/N\), etc.) and results on the joint transform of characteristics during a busy period have been obtained by \textit{L. Takács} [Introduction to the theory of queues. University Texts in the Mathematical Sciences. New York: Oxford University Press (1962; Zbl 0106.33502)], \textit{J. W. Cohen} [The single server queue. North-Holland Series in Applied Mathematics and Mechanics, Vol. 8. Amsterdam etc.: North-Holland Publishing Company (1982; Zbl 0481.60003)] and \textit{S. I. Rosenlund} [Adv. Appl. Probab. 8, 195--208 (1976; Zbl 0332.60070); Nav. Res. Logist. Q. 25, 107--119 (1978; Zbl 0378.60091)]. In this paper, the author distinguishes between two different types of losses: overflow losses that are due to an overflow of the buffer and losses due to an admission controller. Relatively simpler expressions for the joint transforms of the particular queueing systems \(M/PH/1/K\), \(M/PH/1/K\) (under a threshold policy) and \(PH/M/1/K\) are obtained.
scientific article; zbMATH DE number 5909322
scientific article