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Open queueing systems in light traffic. (English) Zbl 0665.60105
Many quantities of interest in open queueing systems are expected values which can be viewed as functions of the arrival rate to the system. We are thus led to consider f($$\lambda)$$, where $$\lambda$$ is the arrival rate, and f represents some quantity of interest. The aim of this paper is to investigate the behavior of f($$\lambda)$$, for $$\lambda$$ near zero. This “light traffic” information is obtained in the form of f(0) and its derivatives, $$f^{(n)}(0)$$, $$n\geq 1.$$
Focusing initially on Poisson arrival processes, we provide a method to calculate $$f^{(n)}(0)$$ for any “admissible” function f. We describe a large class of queueing networks for which we show several standard quantities of interest to be admissible. The proof that the method yields the correct values for the derivatives involves an interchange of limits, whose justification requires a great deal of effort. The determination of $$f^{(n)}(0)$$ involves consideration of sample paths with $$n+1$$ or fewer arrivals over all of time. These calculations are illustrated via several simple examples.
These results can be extended to arrival processes which, although not Poisson, are “driven” by a Poisson process. We carry out the details for phase type renewal processes and nonstationary Poisson processes.

##### MSC:
 60K25 Queueing theory (aspects of probability theory) 90B22 Queues and service in operations research
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