Hou, Zhenting; Yuan, Chenggui; Zou, Jiezhong; Liu, Zaiming; Luo, Jiaowan; Liu, Guoxin; Shi, Peng Transient distribution of the length of \(GI/G/N\) queueing systems. (English) Zbl 1054.60095 Stochastic Anal. Appl. 21, No. 3, 567-592 (2003). The continuous time Markov skeleton process \(X\) is introduced as a process where an embedded Markov chain may be constructed at a sequence of stopping times with respect to a natural filtration of the process \(X\). Some properties of the Markov skeleton process, first of all measurability, are studied. Connections between transition kernels of the underlying Markov skeleton process and the embedded Markov chain are established by means of the backward equation which is reduced to Chapman-Kolmogorov backward equation in a special case. The concept of Markov skeleton process is illustrated by an \(M/M/N\) queue. Reviewer: Evsei Morozov (Petrozavodsk) Cited in 3 Documents MSC: 60K25 Queueing theory (aspects of probability theory) 90B22 Queues and service in operations research 60K15 Markov renewal processes, semi-Markov processes Keywords:\(GI/G/N\) queueing system; Markov skeleton process; backward equation; embedded Markov chain; \((H; Q)\)-pair PDFBibTeX XMLCite \textit{Z. Hou} et al., Stochastic Anal. Appl. 21, No. 3, 567--592 (2003; Zbl 1054.60095) Full Text: DOI References: [1] DOI: 10.1017/S0269964800142068 · Zbl 0979.60088 · doi:10.1017/S0269964800142068 [2] DOI: 10.1007/BF02882537 · Zbl 0883.60101 · doi:10.1007/BF02882537 [3] DOI: 10.1007/BF02884605 · Zbl 0983.60066 · doi:10.1007/BF02884605 [4] Hou Z.T., Markov Skeleton Processes–Mixed Systems Models (2000) [5] Kendall G., J. R. Stat. Soc. Ser. B 13 pp 151– (1951) [6] DOI: 10.1214/aoms/1177728975 · Zbl 0051.10505 · doi:10.1214/aoms/1177728975 [7] DOI: 10.1137/1109001 · doi:10.1137/1109001 [8] Gross D., Fundamentals of Queueing Theory (1998) · Zbl 0949.60002 [9] Liu G.X., Piecewise Deterministic Markov Skeleton Processes (2000) [10] Taka’cs L., Adv. Math. 2 pp 45– (1959) [11] DOI: 10.2307/1426144 · Zbl 0357.60021 · doi:10.2307/1426144 [12] Yan J., Introduction to Martingale and Stochastic Integral [13] Yue M., Acta Math. Sinica 9 pp 494– (1959) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.