Ergodicity and stability of non-stationary queueing system. (Russian, English) Zbl 1050.60075

Andreev, D. B.; Elesin, M. A.; Krylov, E. A.; Kuznetsov, A. V.; Zeifman, A. I.

Teor. Jmovirn. Mat. Stat. 68, 1-10 (2003); translation in Theory Probab. Math. Stat. 68, 1-10 (2004).

The authors deal with the non-homogeneous Markov process (birth and death process) \(X(t)\), \(t\geq0\), with the intensity matrix \(A(t)=\{a_{ij}(t)\}\), \(t\geq0\),
\[ a_{ij}(t)= \begin{cases} \lambda_{i-1}(t),&\text{if }j=i-1,\\ \mu_{i+1}(t),&\text{if }j=i+1,\\ -(\lambda_{i}(t)+\mu_{i}(t)),&\text{if }j=i,\\ 0,&\text{if }\| i-j\|>1, \end{cases} \]
where \(\lambda_{n}(t)\), \(n=0,1,\ldots,N\), is the intensity of birth; \(\mu_{n}(t)\), \(t\geq0\), \(n=0,1,\ldots,N\), is the intensity of death. The authors consider the case \(\lambda_{n}(t)=\lambda_{n}a(t)\), \(\mu_{n}(t)=\mu_{n}b(t)\), \(t\geq0,\;n=0,1,\ldots,N\). The problems of ergodicity and stability for \(X(t)\) are investigated. As examples queueing systems close to \(M_{t}/M_{t}/S\) and \(M_{t}/M_{t}/S/0\) are considered.

Reviewer: A. D. Borisenko (Kyïv)

MSC:

60J27	Continuous-time Markov processes on discrete state spaces
60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
60K25	Queueing theory (aspects of probability theory)

Keywords:

ergodicity; stability; non-stationary queueing system; intensity matrix