×

zbMATH — the first resource for mathematics

Spectral properties of the tandem Jackson network, seen as a quasi-birth-and-death process. (English) Zbl 1078.60078
Matrix geometrical methods for quasi-birth-death processes and more general systems usually rely on having a structured state space of essentially two dimensions. One dimension is the level of the process, the other dimension represents the phase of the system usually considered as a subclassification of the levels. Standard is to have the level space infinite while the phase space is usually finite. For this situation the steady state analysis and even the speed of convergence to equilibrium are (more or less) well understood.
It is shown that there occur strange phenomena when the phase space is infinite as well. These are discussed mostly by exploiting the behavior of a two-stage (exponential) ergodic tandem system, where the queue length of the second server is the level of the system, while the queue length of the first server is considered to be the phase. Although this a queueing network which is thought to be well understood, the authors show that there are details of the behavior, that become appearent only when considering the system in the light of the quasi-birth-death formalism. E.g., truncating the first queue length (loss system) and defining in a natural way an infinite sequence of approximating systems poses problems with the interchange of limiting behavior of the decay rates for the stationary probabilities.

MSC:
60K25 Queueing theory (aspects of probability theory)
60J27 Continuous-time Markov processes on discrete state spaces
PDF BibTeX XML Cite
Full Text: DOI Euclid arXiv
References:
[1] Burke, P. J. (1956). The output of a queueing system. Oper. Res. 4 699–704.
[2] Chihara, T. S. (1978). An Introduction to Orthogonal Polynomials . Gordon and Breach, New York. · Zbl 0389.33008
[3] Fujimoto, K., Takahashi, Y. and Makimoto, N. (1998). Asymptotic properties of stationary distributions in two-stage tandem queueing systems. J. Oper. Res. Soc. Japan 41 118–141. · Zbl 1001.60100
[4] Gail, H. R., Hantler, S. L. and Taylor, B. A. (1996). Spectral analysis of \(M/G/1\) and \(G/M/1\) type Markov chains. Adv. in Appl. Probab. 28 114–165. · Zbl 0845.60092 · doi:10.2307/1427915
[5] Jackson, J. R. (1957). Networks of waiting lines. Oper. Res. 5 518–521.
[6] Kroese, D. P. and Nicola, V. F. (2002). Efficient simulation of a tandem Jackson network. ACM Transactions on Modeling and Computer Simulation (TOMACS) 12 119–141. · Zbl 1390.90231
[7] Latouche, G. and Ramaswami, V. (1999). Introduction to Matrix-Analytic Methods in Stochastic Modelling . ASA-SIAM, Philadelphia. · Zbl 0922.60001 · doi:10.1137/1.9780898719734
[8] Latouche, G. and Taylor, P. G. (2000). Level-phase independence in processes of \(GI/M/1\) type. J. Appl. Probab. 37 984–998. · Zbl 0992.60069 · doi:10.1239/jap/1014843078
[9] Latouche, G. and Taylor, P. G. (2002). Truncation and augmentation of level-independent QBD processes. Stochastic Process. Appl. 99 53–80. · Zbl 1058.60066 · doi:10.1016/S0304-4149(01)00155-7
[10] Latouche, G. and Taylor, P. G. (2003). Drift conditions for matrix-analytic models. Math. Oper. Res. 28 346–360. · Zbl 1082.60077 · doi:10.1287/moor.28.2.346.14475
[11] Neuts, M. F. (1981). Matrix-Geometric Solutions in Stochastic Models . Johns Hopkins Univ. Press. · Zbl 0469.60002
[12] Neuts, M. F. (1989). Structured Stochastic Matrices of \(M/G/1\) Type and Their Applications . Dekker, New York. · Zbl 0695.60088
[13] Ramaswami, V. (1997). Matrix analytic methods: A tutorial overview with some extensions and new results. In Matrix-Analytic Methods in Stochastic Models (S. R. Chakravarthy and A. S. Alfa, eds.) 261–296. Dekker, New York. · Zbl 0872.60067
[14] Ramaswami, V. and Taylor, P. G. (1996). Some properties of the rate operators in level dependent quasi-birth-and-death processes with a countable number of phases. Stoch. Models 12 143–164. · Zbl 0846.60086 · doi:10.1080/15326349608807377
[15] Rudin, W. (1973). Functional Analysis . McGraw-Hill, New York. · Zbl 0253.46001
[16] Seneta, E. (1981). Nonnegative Matrices and Markov Chains . Springer, New York. · Zbl 0471.60001
[17] Takahashi, Y., Fujimoto, K. and Makimoto, N. (2001). Geometric decay of the steady-state probabilities in a quasi-birth-and-death process with a countable number of phases. Stoch. Models 17 1–24. · Zbl 0985.60074 · doi:10.1081/STM-100001397
[18] Tweedie, R. L. (1982). Operator-geometric stationary distributions for Markov chains with application to queueing models. Adv. in Appl. Probab. 14 368–391. · Zbl 0484.60072 · doi:10.2307/1426527
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.