# SCS

## Seminar

### Timescaling results for Markov modulated infinite-server systems and OU processes

Seminar Room 1, Newton Institute

#### Abstract

Co-authors: J. Blom (CWI), K. de Turck (Ghent), O. Kella (Jerusalem), D. Anderson (Wisconsin), H. Thorsdottir (CWI), G. Huang (Amsterdam), P. Spreij (Amsterdam)In this talk I'll consider an infinite-server queue whose input is modulated by a Markovian background process, with a focus on the analysis of the number of customers in the system, both in the central-limit and the large deviations regime. In the scaling that we study, the background process is sped up by a factor N^f, while the arrival rate are scaled by N. Interestingly, in the CLT regime crucially different results come out for f > 1 and for f < 1; in the former case the input process essentially behaves as a (normal) Poisson process, while in the latter case the CLT involves deviation matrices. A similar distinction applies in the LD context. In the second part of the talk I'll address similar issues for the Markov modulated Ornstein-Uhlenbeck process. The last part of the presentation is about multiple infinite-servers systems (or multiple OU systems), modulated by a single background process, thus creating correlation between the individual components. A PDE that characterizes the joint distribution of all coordinates is derived, leading to recursive expressions for all moments. Also a multidimensional CLT is established.

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