Hidden Markov models, HMMs, are useful in a wide range of applications, for example telecommunications. The estimation procedures consist of two main groups; the fast but in general not very accurate moment-based methods and the more accurate but computationally much more demanding likelihood-based methods. A third group of methods are the subspace methods. Our hope is to be able to, within the subspace framework combine the simplicity of the moment-based methods with the better accuracy of the likelihood-based methods. To be able to fit the HMM into the framework of subspace methods the process needs to be formulated in state space form. The main purpose of this paper is to show that it is possible to represent an HMM as a state space system in innovations form. The reformulation is complicated by the non-minimality within the state space representation of the HMM. The reformualtion involves deriving solutions to algebraic Riccati equations which are usually treated under minimality assumptions.
The method of transmitting equally spaced pulses in time to carry discrete information is basic. The shapes of these pulses are the same and the amplitudes carry the information (e.g. antipodal in the binary case). This is referred to as Pulse Amplitude Modulation (PAM). When the pulses overlap, intersymbol interference (ISI) is present. It was not until 1972 that the optimal Maximum Likelihood Sequence Detector (MLSD) for such signals in additive white Gaussian noise (AWGN) was known (Forney 1972, Ungerboeck 1974). It is still not known how to calculate the probability of decision error. Improved lower and upper bounds for this quantity are briefly presented.
A sufficient statistic is achieved by sampling a matched filter, yielding a discrete time model. It is at this point we ask ourselves - how do we approach/solve the problem above when the pulse shape is not known.
There are two basic methods for controlling transmission errors in data communication systems, both involving coding of the message: the forward error correction (FEC) method and the automatic repeat request (ARQ) method. The differences lie in the way the codes are used. With FEC, the code is used for detecting and correcting errors. With ARC, the code is used to detect errors and if there are such, to request a retransmission. The codes used are block codes or convolutional codes. In most cases the block codes are linear. How good a code performs in error control, depends on its probability of undetected error. The talk is about how to recognise good and proper codes for error control.
We investigate the limiting behaviour of the model for the arrival process based on the renewal theory. In the model, conditions for the connection-rate similar to those known for the on-off processes determine different limiting processes. Two of the limiting processes are the same as in the on-off case: fractional Brownian motion and stable Levy motion. However, in the renewal model a third, non-Gaussian and non-stable limiting process appears, some properties of which will be considered in the talk.
We examine a simple discrete time Markov model of TCP congestion control and prove that it has a unique invariant measure. We also show that if the process is scaled by a factor $\sqrt{p}$, then the invariant measures converge to a limit as $p \to 0$, where $p$ is the probability of error in any given data packet.
If the scaled process is transformed to continuous time in a suitable way, we show that it converges in the Skorokhod space $D[0,\infty)$ to a piecewise linear limit process. The unique invariant measure of the limit process coincides with the limit of the invariant measures above and it can be easily computed.
Calculating the stationary distribution of a Jackson network with bounds on the buffer capacities can be a tedious task. Sampling such networks with ordinary simulation methods yield samples with distribution only close to the stationary distribution. The talk will show how the coupling from the past algorithm can be adapted to simulate networks with buffer constraints giving perfect samples, together with some actual simulations.
A stochastic modeling method for TCP is proposed. It is shown how throughput and the risk of experiencing timeout depends on the packet loss probability. We show how this can be done in a unified manner for various combinations of TCP algorithm (Tahoe, New Reno) and type of Active Queueing Method (DropTail, RED).
The so-called additive increase multiplicative decrease (AIMD) window based flow control schemes used in data transmission are presented in this talk. With the emergence of TCP (Transmission Control Protocol) as the ubiquitous data transfer protocol, the study of these algorithms is crucial to understand the complex behavior of modern communication networks. First, TCP is analyzed at a microscopic level, i.e. by considering the traffic of packets between two sources. We show that a nice mathematical structure gives the main performances of this algorithm. Secondly, on a macroscopic level, i.e. by considering flows, we investigate the allocation of bandwidth generated by TCP at the level of the network.
In this paper we address the problem of estimating the mean derivative when the entity containing the parameter has jumps. The methods considered are finite differences, infinitesimal pertubation analysis and the likelihood ratio score function. We calculate the difference between the differentiated mean and the mean derivative. In the case of finite differences, we compute the stepsize in the simulation that asymptotically minimize the mean square error. We also show that the two latter methods, infinitesimal pertubation analysis and likelihood ratio score function, are mathematically equivalent.
We discuss the approach to modelling of architecture of large telecommunication networks based on stochastic geometry that has been developed during the last few years. We will show how recent tools of stochastic geometry and point processes allow for an analytical treatment of rather complex models of networks, stationary or mobile. Variation analysis technique developed for Poisson processes gives rise to new steepest descent algorithms that may be useful for numerical estimation of the optimal configuration in the cases when analytical results are hardly achievable.