An estimate of the probability that the queue length exceeds the maximum for a queue that is a generalized Ornstein-Uhlenbeck stochastic process. (Ukrainian, English) Zbl 1118.60076

Yamnenko, R. E.

Teor. Jmovirn. Mat. Stat. 73, 161-172 (2005); translation in Theory Probab. Math. Stat., Vol 73, 181-194 (2006).

The author investigates the process \(A(t)=mt+\sigma\int_0^tX(u)\,du, t\geq0,\) that describes the queue length, where \(m\) and \(\sigma\) are positive constants, \(X(u)\) is a \(\varphi\)-sub-Gaussian generalized Ornstein-Uhlenbeck stochastic process. The class of \(\varphi\)-sub-Gaussian stochastic processes \(\text{Sub}_{\varphi}(\Omega)\) and the class of strictly \(\varphi\)-sub-Gaussian stochastic processes \(\text{SSub}_{\varphi}(\Omega)\) were introduced and investigated by V. V. Buldygin and Yu. V. Kozachenko [see, for example, “Metric characteristics of random variables and processes” (1998; Zbl 0933.60031)].
In this paper the author proposes an estimate of the probability that the queue length exceeds the given level
\[ p\left\{ \sup_{t\geq0}(A(T)-ct)>x\right\}\leq L(\gamma)x^{r/(r-1)}\exp \left\{-\kappa(\gamma) x^{r/2(r-1)} \right\}, \]
where \(c>m\) is the service intensity, \(x>0\) is the maximum queue length, and \(L(\gamma)\) and \(\kappa(\gamma)\) are some constants. For more results related to the problem see the articles by Y. Kozachenko, O. Vasylyk and the author [Theory Stoch. Process. 9(25), No. 3–4, 70–80 (2003; Zbl 1064.60064)], and the author [Visn., Ser. Fiz.-Mat. Nauky, Kyïv. Univ. Im. Tarasa Shevchenka 2004, No. 3, 82–86 (2004; Zbl 1064.60071)].

Reviewer: Mikhail P. Moklyachuk (Kyïv)

MSC:

60K25	Queueing theory (aspects of probability theory)
60G07	General theory of stochastic processes
60K30	Applications of queueing theory (congestion, allocation, storage, traffic, etc.)
90B22	Queues and service in operations research

Keywords:

\(\varphi\)-sub-Gaussian stochastic process; generalized Ornstein-Uhlenbeck process; distribution of supremum

Citations:

Zbl 0998.60503; Zbl 1064.60064; Zbl 1064.60071; Zbl 0933.60031