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

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.

### MSC:

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

### Keywords:

limit theorem; stochastic networks; semi-Markov input flow