On the connections between some dynamic systems and systems of queuing theory. (Russian) Zbl 0577.60089

Zakusilo, O. K.

Teor. Veroyatn. Mat. Stat. 29, 41-46 (1983).

The solution \(x=x(t)\) of the stochastic differential equation \(dx(t)=-\mu x(t)dt+d\alpha (t),\mu >0\), is studied, where \(\alpha\) is a generalized Poisson process with parameter \(\lambda\) and jumps. Here x has an interpretation in queuing theory as well as in a physical context, from which the existence and some properties of x are already known.
A sufficient and necessary condition for the stationarity of the distribution \(\Pi\) of x is given, and the limiting behavior of \(\Pi\) for \(\lambda\) \(\to 0\) and \(\lambda\) \(\to \infty\) is studied. From this some limit theorems for the distribution of the virtual waiting time and the time of staying in a queuing system with changing serving rate are derived.

Reviewer: C.Baldauf

MSC:

60K25	Queueing theory (aspects of probability theory)
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
90B22	Queues and service in operations research

Keywords:

stationarity of the distribution; queuing system with changing serving rate