## Brazilian Journal of Probability and Statistics

### A new stochastic model and its diffusion approximation

#### Abstract

This paper considers a kind of queueing problem with a Poisson number of customers or, more generally, objects which may arrive in groups of random size. The focus is on the total quantity over time, called system size. The main result is that the process representing the system size, properly normalized, converges in finite-dimensional distributions to a centered Gaussian process (the diffusion approximation) with several attractive properties. Comparison with existing works (where the number of objects is assumed nonrandom) highlights the contribution of the present paper.

#### Article information

Source
Braz. J. Probab. Stat., Volume 31, Number 1 (2017), 62-86.

Dates
Accepted: October 2015
First available in Project Euclid: 25 January 2017

https://projecteuclid.org/euclid.bjps/1485334825

Digital Object Identifier
doi:10.1214/15-BJPS303

Mathematical Reviews number (MathSciNet)
MR3601661

Zentralblatt MATH identifier
1380.60085

#### Citation

Covo, Shai; Elalouf, Amir. A new stochastic model and its diffusion approximation. Braz. J. Probab. Stat. 31 (2017), no. 1, 62--86. doi:10.1214/15-BJPS303. https://projecteuclid.org/euclid.bjps/1485334825

#### References

• Adler, R. J. and Taylor, J. E. (2007). Random Fields and Geometry. Springer Monographs in Mathematics. New York: Springer.
• Berman, S. M. (1978). Gaussian processes with biconvex covariances. Journal of Multivariate Analysis 8, 30–44.
• Covo, S. (2011). Two-parameter Lévy processes along decreasing paths. Journal of Theoretical Probability 24, 150–169.
• Covo, S. and Elalouf, A. (2015). A simple decomposition of fractional Brownian motion with parameter $H<1/2$. Unpublished manuscript.
• Louchard, G. (1988). Large finite population queueing systems part I: The infinite server model. Communications in Statistics. Stochastic Models 4, 473–505.
• Steinsaltz, D. (1996). Deviation bounds and limit theorems for the maxima of some stochastic processes. Unpublished manuscript. Available at http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.45.4844.