Required service rate for mixed traffic

В данной работе нами получены границы для скорости обслуживания при некоторых ограничениях на характеристики обслуживания в неоднородной модели входящего трафика, основанной на сумме независимых фрактального броуновского движения и симметричного $\alpha$-устойчивого движения Леви с разными коэффициентами Херста $H_1$ и $H_2=1/\alpha$. Хорошо известно, что для процессов, приращения которых имеют тяжёлые хвосты, методы расчета эффективной пропускной способности, основанные на производящей функции моментов входящего потока, не применимы. Однако существуют простые соотношения между характеристиками потока, скоростью обслуживания $C$ и вероятностями $\varepsilon(b)$ переполнения для конечного и бесконечного буфера, из которых при фиксированном значении $\varepsilon(b)$ можно выразить $C$. In this paper we analyse the nonhomogenous traffic model based on sum of independent Fractional Brownian motion and symmetric $\alpha$-stable Levy process with different Hurst exponents $H_1$ and $H_2=1/\alpha$ and present bounds for the required service rate under QoS constraints. It is well known that for the processes with long-tailed increments effective bandwidths are not expressed by means of the moment generating function of the input flow. However we can derive simple relations between the flow parameters, service rate $C$ and overflow probabilities $\varepsilon (b)$ for finite and infinite buffer. In this way it is possible to find required service rate $C$ under a constraint on maximum overflow probability.

Large deviations for acyclic networks of queues with correlated Gaussian inputs

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.

Alternative estimate of curve exceeding probability of sub-Gaussian random process

Investigation of sub-gaussian random processes are of special interest since obtained results can be applied to Gaussian processes. In this article the properties of trajectories of a sub-Gaussian process drifted by a curve a studied. The following functionals of extremal type from stochastic process are studied: $\sup_{t\in B}(X(t)-f(t))$, $\inf{t\in B}(X(t)-f(t))$ and $\sup_{t\in B}|X(t)-f(t)|$. An alternative estimate of exceeding by sub-Gaussian process a level, given by a continuous linear curve is obtained. The research is based on the results obtained in the work \cite{yamnenko_vasylyk_TSP_2007}. The results can be applied to such problems of queuing theory and financial mathematics as an estimation of buffer overflow probability and bankruptcy