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The problem of stability in queueing theory. (English) Zbl 0676.60090

The problem of stability is enclosed in this paper in a more general characterization problem for queueing. A queueing process is treated as a mapping F from the set $𝒰$ of input data U into the set $𝒱$ of output data V. Let $Z=\left(U,V\right)$ and $W=W\left(Z\right)$ be some mapping $𝒰×𝒱\to 𝒲$ which is called “observation”. Let us fix subsets ${𝒰}^{*}\subset 𝒰$, ${𝒲}^{*}\subset 𝒲$, ${𝒵}^{*}\subset 𝒰×𝒱$. Then a pure characterisation problem is as follows:

$\left\{Z\in {𝒵}^{*},\phantom{\rule{1.em}{0ex}}W\in {𝒲}^{*}\right\}⇔\left\{U\in {𝒰}^{*}\right\}⇒\left\{V\in F\left({𝒰}^{*}\right)\right\}·$

Delivering metrics in sets U,V,W it is easy to formulate the stability problem for this characterization. As for queues, the mentioned sets are that of probability distribitions or random variables (as a rule). So, the main instrument of stability analysis is the theory of probability metrics.

Using this approach the author shows different estimates of continuity of queues (mainly, single-server ones under different suppositions on input flow and service times), stability estimates for characterizations of input flows (using aging properties of inter-arrival times), and estimates of approximation accuracy.

Reviewer: V.Kalashnikov
##### MSC:
 60K25 Queueing theory
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