A scheduler for a wireless system

This problem deals with the scheduling of data packets for different users who share a common wireless channel of limited capacity and who have different service time requirements. From a system point of view it is important to maintain a high throughput, that is, a high number of packets served per unit of time. However from a user point of view it is important to minimise the service time. This poses a trade-off problem, which is one of the goals we wish to analyse. The wireless transmission channel alternates between good and bad states over time. While the channel is in a bad state the transmission of a packet fails and the packet needs to be retransmitted. When the state of the channel is good, packets are transmitted successfully and do not require retransmission. The system transmission efficiency or throughput might be defined as the number of data packets that can be transmitted successfully in a given time. The scheduler we wish to study is designed to consider both the wireless channel conditions and the users' quality of service requirements. For this purpose users are assigned a credit every scheduling frame, which is a function of the wireless channel conditions and the quality of service required, which depends on the traffic class

Sensitivity of Markov chains for wireless protocols

Network communication protocols such as the IEEE 802.11 wireless protocol are currently best modelled as Markov chains. In these situations we have some protocol parameters $\alpha$, and a transition matrix $P(\alpha)$ from which we can compute the steady state (equilibrium) distribution $z(\alpha)$ and hence final desired quantities $q(\alpha)$, which might be for example the throughput of the protocol. Typically the chain will have thousands of states, and a particular example of interest is the Bianchi chain defined later. Generally we want to optimise $q$, perhaps subject to some constraints that also depend on the Markov chain. To do this efficiently we need the gradient of $q$ with respect to $\alpha$, and therefore need the gradient of $z$ and other properties of the chain with respect to $\alpha$. The matrix formulas available for this involve the so-called fundamental matrix, but are there approximate gradients available which are faster and still sufficiently accurate? In some cases BT would like to do the whole calculation in computer algebra, and get a series expansion of the equilibrium $z$ with respect to a parameter in $P$. In addition to the steady state $z$, the same questions arise for the mixing time and the mean hitting times. Two qualitative features that were brought to the Study Group’s attention were: * the transition matrix $P$ is large, but sparse. * the systems of linear equations to be solved are generally singular and need some additional normalisation condition, such as is provided by using the fundamental matrix. We also note a third highly important property regarding applications of numerical linear algebra: * the transition matrix $P$ is asymmetric. A realistic dimension for the matrix $P$ in the Bianchi model described below is 8064×8064, but on average there are only a few nonzero entries per column. Merely storing such a large matrix in dense form would require nearly 0.5GBytes using 64-bit floating point numbers, and computing its LU factorisation takes around 80 seconds on a modern microprocessor. It is thus highly desirable to employ specialised algorithms for sparse matrices. These algorithms are generally divided between those only applicable to symmetric matrices, the most prominent being the conjugate-gradient (CG) algorithm for solving linear equations, and those applicable to general matrices. A similar division is present in the literature on numerical eigenvalue problems

Acoustic scattering from a strained region

A composite material consists of a rubber filled with gas-filled microspheres. In underwater applications it is compressed hydrostatically by a pressure that may be not insignificant compared with the shear modulus of the rubber, so large strains are produced around each spherical inclusion. When these spherical inclusions scatter an incident acoustic wave, the strained region around an inclusion has had its elastic properties altered by the large static strain. Thales Underwater Systems asked the Study Group to address the question of how this strained region affects the elastic scattering, bearing in mind that the dynamic shear modulus differs from its static value

Graph colouring for office blocks

The increasing prevalence of WLAN (wireless networks) introduces the potential of electronic information leakage from one company's territory in an office block, to others due to the long-ranged nature of such communications. BAE Systems have developed a system ('stealthy wallpaper') which can block a single frequency range from being transmitted through a treated wall or ceiling to the neighbour. The problem posed to the Study Group was to investigate the maximum number of frequencies ensure the building is secure. The Study group found that this upper bound does not exist, so they were asked to find what are "good design-rules" so that an upper limit exists

Wind farm output

The problem was to devise a simulation method for the wind speeds at a set of sites, that has the correct autocorrelation, cross-correlation and distributions. The report includes one way of doing this, using a multivariate auto-regressive system, and other comments and observations that may lead to better ways of achieving the aim

Chauffeur braking

An experienced driver will `feather' the brakes so as to unwind the suspension compliance and stop the vehicle with only just enough torque in the brakes to hold the vehicle stationary on any gradient, or against the residual torque from an automatic transmission’s torque converter. An optimal stopping problem that minimises the total jerk was formulated and solved. This model was extended by including a linear relationship between the brake pressure and the acceleration of the car where the coefficients are estimated by linear regression. Finally, a Kalman filter estimates the state of the car using the tone wheel

Reaction-diffusion models of decontamination

A contaminant, which also contains a polymer is in the form of droplets on a solid surface. It is to be removed by the action of a decontaminant, which is applied in aqueous solution. The contaminant is only sparingly soluble in water, so the reaction mechanism is that it slowly dissolves in the aqueous solution and then is oxidized by the decontaminant. The polymer is insoluble in water, and so builds up near the interface, where its presence can impede the transport of contaminant. In these circumstances, Dstl wish to have mathematical models that give an understanding of the process, and can be used to choose the parameters to give adequate removal of the contaminant. Mathematical models of this have been developed and analysed, and show results in broad agreement with the effects seen in experiments

Analysis of shear forces during mash disk formation

This report concerns a forming process in which mash is forced from a spreading manifold into moulds on a rotating drum, transported in the moulds underneath a surface held flush with the drum (the shoe), and ejected from the moulds. The quality of the final product is understood to be related to the shear stresses experienced by the mash in the moulds as it is transported under the shoe. We describe and analyse mathematical models of the forming process, focusing on the fluid mechanics of mash in a mould. We treat this as a driven cavity flow and obtain flow profiles, stress profiles, and expressions for the maximum shear stress for different rheological models of the mash (Newtonian fluid, power-law fluid, and Bingham plastic)