A Large Deviations Problem for Compound Poisson Processes in Queueing Theory

#### S. Aspandiiarov, E.A. Pechersky

1997, v.3, №3, 333-366

ABSTRACT

In this paper we find the logarithmic asymptotics of the probability $P \{\omega > at, \eta > bt\}$ with non-negative $a, b$ as $t \rightarrow \infty$, where $\omega$ is the virtual waiting time and $\eta$ the number of requests queued by this time in the stationary regime of an underloaded queueing system with a single device, an incoming Poisson flow of requests and the simplest service discipline FIFO. The proof is based on the large deviations principle for multidimensional compound Poisson processes on the positive half-line $[0, \infty)$ derived recently in [R.L. Dobrushin and E.A. Pechersky, Large deviations for random processes with independent increments on infinite intervals. Preprint, 1994]. We apply this result to an optimal division of the buffer into two parts corresponding to two types of information on messages entering in a single device queueing system.

Keywords: FIFO system,virtual waiting time,large deviations principle