Abstract
We consider an acyclic network of single-server queues with heterogeneous processing rates. It is assumed that each queue is fed by the superposition of a large number of i.i.d. Gaussian processes with stationary increments and positive drifts, which can be correlated across different queues. The flow of work departing from each server is split deterministically and routed to its neighbors according to a fixed routing matrix, with a fraction of it leaving the network altogether. We study the exponential decay rate of the probability that the steady-state queue length at any given node in the network is above any fixed threshold, also referred to as the ‘overflow probability’. In particular, we first leverage Schilder’s sample-path large deviations theorem to obtain a general lower bound for the limit of this exponential decay rate, as the number of Gaussian processes goes to infinity. Then, we show that this lower bound is tight under additional technical conditions. Finally, we show that if the input processes to the different queues are nonnegatively correlated, non-short-range dependent fractional Brownian motions, and if the processing rates are large enough, then the asymptotic exponential decay rates of the queues coincide with the ones of isolated queues with appropriate Gaussian inputs.
Highlights
Modern communication networks are complex, and handle huge amounts of data
(iii) We show that if the input processes to the different queues are nonnegatively correlated, non-short-range-dependent fractional Brownian motions, and if the processing rates are large enough, the asymptotic exponential decay rates of the queues coincide with those of isolated queues with appropriate Gaussian inputs (Theorem 7)
Remark 6 It is worth highlighting that, even if the bound of Theorem 3 is not tight, it provides an upper bound for the asymptotic exponential decay rate of overflow probability that can be used as a performance guarantee in applications
Summary
Modern communication networks are complex, and handle huge amounts of data. This is especially true closer to the backbone of the networks, where large numbers. For the special case of two queues in tandem, with work arriving only to the first queue and all the departing work of the first queue going into the second one, a useful trick involving subtracting the first queue (which has Gaussian input) from the sum of both queues (which behaves exactly as a single-server queue with a Gaussian input) yields a tractable analysis of the second queue in the tandem [7], even if it does not have a Gaussian input; see the more refined approach in [8] based on the delicate busy-period analysis developed in [9] This trick does not work for more complex networks (not even for two queues in tandem with inputs to both queues, or when not all departures from the first queue join the second one [10]). In that paper, there were certain limitations regarding the correlation structure of the input processes (in that they have to be independent across different queues), and regarding the structure of the network (in that any two directed paths cannot meet in more than one node)
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