Точки, прямі, площини,....аксіоми і теореми: введення в евклідову геометрію

Посилаючись на поняття математичної структури, доповнено істинно важливими аспектами, роз’яснено схему розбудови теорії евклідової геометрії на основі аксіоматики 0.В. Погорєлова, розкрито науковий стиль подання системи аксіом, продемонстровано методичні особливості строго логічного наповнення підручника фактами

User-level performance of channel-aware scheduling algorithms in wireless data networks

Channel-aware scheduling strategies, such as the Proportional Fair algorithm for the CDMA 1xEV-DO system, provide an effective mechanism for improving throughput performance in wireless data networks by exploiting channel fluctuations. The performance of channel-aware scheduling algorithms has mostly been explored at the packet level for a static user population, often assuming infinite backlogs. In the present paper, we focus on the performance at the flow level in a dynamic setting with random finite-size service demands. We show that in certain cases the user-level performance may be evaluated by means of a multiclass Processor-Sharing model where the total service rate varies with the total number of users. The latter model provides explicit formulas for the distribution of the number of active users of the various classes, the mean response times, the blocking probabilities, and the mean throughput. In addition we show that, in the presence of channel variations, greedy, myopic strategies which maximize throughput in a static scenario, may result in sub-optimal throughput performance for a dynamic user configuration and cause potential instability effects

MIMO communication system with user scheduling based on reduced channel state information (publication date 09-10-2008)

A multiple-input, multiple-output (MIMO) communication system is configured to perform user scheduling with reduced channel station information. The system includes multiple terminals and at least one base station configured to communicate with the terminals. The base station is operative to obtain channel vector magnitudes for respective ones of the terminals, to identify a subset of the terminals based on the channel vector magnitudes, to obtain channel vector phase information for the identified subset of terminals, and to utilize the channel vector phase information to control transmission to the identified subset of terminals. The system may be, for example, a multi-user MIMO system in which the multiple terminals comprise autonomous sin le-antenna terminals

Performance bounds for dynamic channel assignment schemes operating under varying re-use constraints

We derive bounds for the performance of dynamic channel assignment (DCA) schemes which strengthen the existing Erlang bound. The construction of the bounds is based on a reward paradigm as an intuitively appealing way of characterizing the achievable carried traffic region. In one-dimensional networks, our bounds closely approach the performance of maximum packing (MP), which is an idealized DCA scheme. This suggests not only that the bounds are extremely tight, but also that no DCA scheme, however sophisticated, can be expected to outperform MP in any significant manner, if at all. Our bounds extend to scenarios with varying re-use which may arise in the case of dynamic re-use partitioning or measurement-based DCA schemes. In these cases, the bounds slightly diverge from the performance of MP, which inflicts higher blocking on outer calls than inner calls, but not to the extent required to maximize carried traffic. This reflects the trade-off that arises in the case of varying re-use between efficiency and fairness. Asymptotic analysis confirms that schemes which minimize blocking intrinsically favor inner calls over outer calls, whereas schemes which do not discriminate among calls inevitably produce higher network-average blocking

MIMO communication system with user scheduling based on reduced channel state information (publication date 09-10-2008)

A multiple-input, multiple-output (MIMO) communication system is configured to perform user scheduling with reduced channel station information. The system includes multiple terminals and at least one base station configured to communicate with the terminals. The base station is operative to obtain channel vector magnitudes for respective ones of the terminals, to identify a subset of the terminals based on the channel vector magnitudes, to obtain channel vector phase information for the identified subset of terminals, and to utilize the channel vector phase information to control transmission to the identified subset of terminals. The system may be, for example, a multi-user MIMO system in which the multiple terminals comprise autonomous single-antenna terminals

Dynamic rate control algorithms for HDR throughput optimization

The relative delay tolerance of data applications, together with the bursty traffic characteristics, opens up the possibility for scheduling transmissions so as to optimize throughput. A particularly attractive approach, in fading environments, is to exploit the variations in the channel conditions, and transmit to the user with the currently 'best' channel. We show that the 'best' user may be identified as the maximum-rate user when the feasible rates are weighed with some appropriately determined coefficients. Interpreting the coefficients as shadow prices, or reward values, the optimal strategy may thus be viewed as a revenue-based policy. Calculating the optimal revenue vector directly is a formidable task, requiring detailed information on the channel statistics. Instead, we present adaptive algorithms for determining the optimal revenue vector on-line in an iterative fashion, without the need for explicit knowledge of the channel behavior. Starting from an arbitrary initial vector, the algorithms iteratively adjust the reward values to compensate for observed deviations from the target throughput ratios. The algorithms are validated through extensive numerical experiments. Besides verifying long-run convergence, we also examine the transient performance, in particular the rate of convergence to the optimal revenue vector. The results show that the target throughput ratios are tightly maintained, and that the algorithms are well able to track changes in the channel conditions or throughput targets

Polling:past, present, and perspective

\u3cp\u3eThis is a survey on polling systems, focussing on the basic single-server multi-queue polling system in which the server visits the queues in cyclic order. The main goals of the paper are: (i) to discuss a number of the key methodologies in analyzing polling models; (ii) to give an overview of recent polling developments; and (iii) to present a number of challenging open problems.\u3c/p\u3

Polling models with and without switchover times

We consider two different single-server cyclic polling models: (i) a model with zero switchover times, and (ii) a model with nonzero switchover times, in which the server keeps cycling when the system is empty. For both models we relate the steady-state queue length distribution at a queue to the queue length distributions at server visit beginning and visit completion instants at that queue; as a by-product we obtain a short proof of the Fuhrmann-Cooper decomposition. For the large class of polling systems that allow a multitype branching process interpretation, we expose a strong relation between the queue length, as well as waiting-time, distributions in the two models. The results enable a very efficient numerical computation of the waiting-time moments under different switchover time scenarios

A stochastic network with mobile users in heavy traffic

We consider a stochastic network with mobile users in a heavy traffic regime. We derive the scaling limit of the multidimensional queue length process and prove a form of spatial state space collapse. The proof exploits a recent result by Lambert and Simatos (preprint, 2012), which provides a general principle to establish scaling limits of regenerative processes based on the convergence of their excursions. We also prove weak convergence of the sequences of stationary joint queue length distributions and stationary sojourn times