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Diffusion approximations for open queueing networks with service interruptions. (English) Zbl 0783.60088

The paper establishes heavy-traffic limit theorems for open single class queueing networks with service interruptions. In addition to an unlimited waiting space and the first-come first-served service discipline, each station has a single server which is alternatively up and down. When a station is down, service stops but arrivals continue, when a station comes up, service resumes where it left off. It is allowed the availability of these servers to depend on the basic arrival, service and routing variables. In particular the authors require that a joint functional central limit theorem (FCLT) holds for all the basic processes.

The authors consider two different treatments of the service interruptions. The standard treatment is based on fixed up and down times which leads to a long-run proportion of up time ${v}_{j}$ at each station $j$ with $0<{v}_{j}<1$ and FCLT for cumulative up time at each station after translation. In this case the authors obtain a limiting Brownian motion just as without disruptions. The second treatment allows the up and down times to become longer as the system enters heavy traffic. In particular the authors make the traffic intensities in the $n$-th system of order $1-{n}^{-1/2}$. Then they let the up times be of order $n$ and the down times be of order $\sqrt{n}$. Asymptotically, the long-run proportion of time each station is up is 1 but nevertheless the down times have a significant impact. In this case they establish convergence in the Skorokhod ${M}_{1}$ topology to a multidimensional reflection of multidimensional Brownian motion plus a multidimensional jump process.

##### MSC:
 60K20 Applications of Markov renewal processes 60K25 Queueing theory
##### References:
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