Evaluating Error when Estimating the Loss Probability in a Packet Buffer
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In this thesis we explore precision in measurement of buffer overﬂow and loss probability. We see how buffer overﬂow probability compares with queuing delay measurements covered in the literature . More speciﬁcally, we measure the overﬂow probability of a packet buffer for various sampling rates to see the effect of sampling rate on the estimation. There are various reasons for measurement in networks; one key context assumed here is Measurement Based Admission Control. We conduct simulation experiments with analytically derived VoIP and bursty trafﬁc parameters,in Matlab, while treating the buffer under consideration as a two-state Markov Chain. We note that estimation error decreases with increase in sampling gap (or in other words precision improves/variance decreases with decrease in sampling rate). We then perform experiments for VoIP and bursty data using NS-2 simulator and record the buffer states generated therein. We see a similar trend of increase in precision with increase in sampling gap. In our simulations, we have mainly considered static trafﬁc passing through the buffer, and we use elastic trafﬁc (TCP) for comparison. We see from our results that that the sampling error becomes constant beyond certain asymptotic level. We thus look into asymptotic error in estimation,for the lowest sampling gap,to establish a lower bound on estimation error for buffer loss probability measurement. We use formulae given in recent literature  for computing the experimental and theoretic asymptotic variance of the buffer state traces in our scenarios. We ﬁnd that the theoretical and experimental asymptotic variance of overﬂow probability match when sampling a trace of buffer states modelled as a two-state Markov Chain in Matlab. We claim that this is a new approach to computing the lower bound on the measurement of buffer overﬂow probability, when the buffer states are modelled as a Markov process. Using Markov Chain modelling for buffer overﬂow we further explore the relationship between sampling rate and accuracy. We ﬁnd that there is no relationship between sampling gap and bias of estimation. Crucially we go on to show that a more realistic simulation of a packet buffer reveals that the distribution of buffer overﬂow periods is not always such as to allow simple Markov modelling of the buffer states; while the sojourn periods are exponential for the smaller burst periods, the tail of the distribution does not ﬁt to the same exponential ﬁtting. While our work validates the use of a two-state Markov model for a useful approximation modelling the overﬂow of a buffer, we have established that earlier work which relies on simple Markovian assumptions will thereby underestimate the error in the measured overﬂow probabilities.
AuthorsWahid, Amna Abdul
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