## Transient Markov arrival processes.(English)Zbl 1030.60067

A family of transient Markov arrival processes (MAPs) is considered and its basic properties are analyzed. Transient MAP is defined on the basis of a continuous-time Markov chain $$\varphi(t) \in \{0,1,\dots,m\}$$, $$t \in R^+$$, described by constant transition rates $$(D^*)_{i,j}$$, $$i,j \in \{0,1,\dots,m\}$$. The state $$\{0\}$$ is absorbing, i.e. $$(D^*)_{0,j}=0$$, $$j \neq 0$$. In any other state $$\{1,\dots,m\}$$ a new event can occur with constant rates $$(D^*_1)_{i,j}$$ depending on the current transition $$i \to j$$, events can happen even if $$\varphi(t)$$ does not change ($$i\to i$$). The total number of events up to time epoch $$t$$ is denoted by $$N(t)$$. In absorbing state $$\{0\}$$ no more events can occur, so, it is said that the catastrophe occurs when the Markov chain achieves this state. Special examples of MAPs are the Poisson process ($$m=1$$) and the Markov modulated Poisson process for which $$D_1^*$$ is a diagonal matrix. Some other examples and possible applications are discussed.
Let $$T_n$$ denote the time epoch at which the $$n$$th event occurs. For the MAP the lifetime of the process $$L$$, the time $$V$$ until the catastrophe occurs and the total number of events $$K$$ are defined as follows $L=\sup\{T_n:T_n<\infty\},\quad V=\inf\{t\geq 0:\varphi(t)=0\},\quad K=\lim_{t \to \infty} N(t).$ Distributions of $$L$$, $$V$$, $$K$$ are derived. In addition, quasistationary MAPs are especially considered and some estimations are obtained in this case.

### MSC:

 60J22 Computational methods in Markov chains 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 60J27 Continuous-time Markov processes on discrete state spaces 60K35 Interacting random processes; statistical mechanics type models; percolation theory
Full Text:

### References:

 [1] ANDERSON, W. J. (1991). Continuous-Time Markov Chains: An Applications-Oriented Approach. Springer, New York. · Zbl 0731.60067 [2] GAVER, D. P., JACOBS, P. A. and LATOUCHE, G. (1984). Finite birth-and-death models in randomly changing environments. Adv. in Appl. Probab. 16 715-731. JSTOR: · Zbl 0554.60079 [3] LATOUCHE, G. and RAMASWAMI, V. (1992). A unified stochastic model for the arrival of packets from periodic sources. Performance Evaluation 14 103-121. · Zbl 0752.94002 [4] LATOUCHE, G. and RAMASWAMI, V. (1997). Spatial point processes of phase ty pe. In Teletraffic Contributions for the Information Age. Proceedings of the 15th International Teletraffic Congress (V. Ramaswami and P. Wirth, eds.) 381-390. North-Holland, Amsterdam. [5] LATOUCHE, G. and RAMASWAMI, V. (1999). Introduction to Matrix Analy tic Methods in Stochastic Modeling. SIAM, Philadelphia. · Zbl 0922.60001 [6] LUCANTONI, D. M., MEIER-HELLSTERN, K. S. and NEUTS, M. F. (1990). A single-server queue with server vacations and a class of non-renewal arrival processes. Adv. in Appl. Probab. 22 676-705. JSTOR: · Zbl 0709.60094 [7] MAIER, R. S. and O’CINNEIDE, C. A. (1992). A closure characterisation of phase-ty pe distributions. J. Appl. Probab. 29 92-103. JSTOR: · Zbl 0758.60064 [8] NARAy ANA, S. and NEUTS, M. F. (1992). The first two moment matrices of the counts for the Markovian arrival process. Comm. Statist. Stochastic Models 8 459-477. · Zbl 0756.60095 [9] NEUTS, M. F. (1979). A versatile Markovian point process. J. Appl. Probab. 16 764-779. JSTOR: · Zbl 0422.60043 [10] NEUTS, M. F. (1981). Matrix-Geometric Solutions in Stochastic Models. An Algorithmic Approach. Johns Hopkins Univ. Press. · Zbl 0469.60002 [11] NEUTS, M. F. (1989). Structured Stochastic Matrices of M/G/1 Ty pe and Their Applications. Dekker, New York. · Zbl 0695.60088 [12] NEUTS, M. F. (1990). On the packet stream generated by a random flow of messages of random durations. Comm. Statist. Stochastic Models 6 445-470. · Zbl 0705.60042 [13] PACHECO, A. and PRABHU, N. U. (1995). Markov-additive processes of arrivals. In Advances in Queueing: Theory, Methods, and Open Problems (J. H. Dshalalow, ed.) 167-194. CRC Press, Boca Raton, FL. · Zbl 0845.60090 [14] RAMASWAMI, V. and LATOUCHE, G. (1989). Modeling packet arrivals from asy nchronous input lines. In Teletraffic Science for New Cost-Effective Sy stems, Networks and Services. Proceedings of the 12th International Teletraffic Congress (M. Bonatti, ed.) 721-727. North-Holland, Amsterdam. [15] REMICHE, M.-A. (1999). Efficiency of an IPhP3 illustrated through a model in cellular networks. In Numerical Solution of Markov Chains (NSMC’99): 3rd International Workshop (B. Plateau, W. J. Stewart and M. Silva, eds.) 296-311. Prensas Univ., Zaragoza, Spain. [16] REMICHE, M.-A. (2000a). On the exact distribution of the isotropic planar point processes of phasety pe. J. Comput. Appl. Math. 116 77-91. · Zbl 0980.60055 [17] REMICHE, M.-A. (2000b). Towards an extension of Campbell’s theorem. Technical report, AachenRWTH.
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.