One limit theorem for stochastic networks and its applications. (Ukrainian, English) Zbl 1050.60087

Lebedev, E. O.

Teor. Jmovirn. Mat. Stat. 68, 74-85 (2003); translation in Theory Probab. Math. Stat. 68, 81-92 (2004).

The author deals with the \([G| M|\infty]^{r}\) network model. The network consists of \(r\) nodes. Let \(\nu_{i}(t)\), \(i=1,\ldots,r\), be the number of requests in the \(i\)th node during the time interval \([0,t]\). In every node there is an infinite number of servers and service time has the exponential distribution with parameter \(\mu_{i}, i=1,\ldots,r\). Let us suppose that there exist constants \(\lambda_{i}>0, i=1,\ldots,r\), such that \(n^{-1/2}(\nu_1^{(n)}(nt)-\lambda_1nt,\ldots,\nu_{r}^{(n)}(nt)-\lambda_{r}nt)\) weakly converges in the uniform topology, as \(n\to\infty\), to \(r\)-dimensional Wiener process with mean zero and correlation matrix \(\sigma\), and let \(\lim_{n\to\infty}n\mu_{i}(n)=\mu_{i}\neq0\), \(i=1,\ldots,r\). The author considers the limit behavior of the process \(\xi^{(n)}(t)=n^{-1/2}(Q^{(n)}(nt)-nq(t))\), \(t\geq0\), \(Q^{(n)}(0)=(0,\ldots,0)\), where \(q(t)=(\theta/\mu)(I-P(t))\), \(\theta/\mu=(\theta_{1}/\mu_1,\ldots,\theta_{r}/\mu_{r})\), \(\theta=\lambda(I-P)^{-1}\), \(P(t)=\exp[\Delta(\mu)(P-I)t]\), \(\Delta(\mu)=\|\delta_{ij}\mu_{i}\|_{1}^{r}\) is the diagonal matrix; \(P\) is the matrix of transition probability; \(Q_{i}(t)\) is the number of occupied servers in the \(i\)th node at moment \(t\geq0\). Properties of the limit process are investigated. The obtained limit theorem is applied to networks with semi-Markov input flow.

Reviewer: A. D. Borisenko (Kyïv)

MSC:

60K25	Queueing theory (aspects of probability theory)
90B22	Queues and service in operations research

Keywords:

limit theorem; stochastic networks; semi-Markov input flow