Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

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Convex Analysisの二,三の進展について

The asymptotic behaviour of (Yn, n e N) is of fundamental importance in probability theory. Indeed, if the Xj have common mean fi and variance a, then by taking each an = n/u and b„ = n a, the celebrated Central Limit Theorem tells us that (Yn) converges in distribution to a standard normal. The class of random variables which appear as limits in distribution of (Yn) for more general (an) and (b„) are called the stable random variables and they have a number of elegant properties, such as the preservation of their distribution functions under convolution. A stochastic process is said to be stable if all its finite-dimensional distributions are themselves stable and it is the systematic study of such processes which is the aim of the present volume.

Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

(1987). Applied Probability and Queues. Journal of the Operational Research Society: Vol. 38, No. 11, pp. 1095-1096.

https://link.springer.com/content/pdf/10.1057%2Fjors.1987.184.pdf

Applied Probability and Queues

A method for determining the location of a mobile unit (MU) in a wireless communication system and presenting it to a remote party. The location of a remote MU is determined by comparing a snapshot of a predefined portion of the radio-frequency (RF) spectrum taken by the MU to a reference database containing multiple snapshots taken at various locations. The result of the comparison is used to determine if the MU is at a specific location. The comparison may be made in the MU, or at some other location situated remotely from the MU. In the latter case, sufficient information regarding the captured fingerprint is transmitted from the MU to the remote location. The database may be pre-compiled or generated on-line.

Location determination using RF fingerprinting

We consider a multi-class feedforward queueing network with first come first serve queueing discipline and deterministic services and routing. The large deviation asymptotics of tail probabilities of the distribution of the workload vector can be characterized by convex path space minimization problems with non-linear constraints. So far there exists no numerical algorithm which could solve such minimization problems in a general way. When the queueing network is heavily loaded it can be approximated by a reflected Brownian motion. The large deviation asymptotics of tail probabilities of the distribution of this heavy traffic limit can again be characterized by convex path space minimization problems with non-linear constraints. However, due to their less complicated structure there exist algorithms to solve such minimization problems. In this paper we show that, as the network tends to a heavy traffic limit, the solution of the large deviation minimization problems of the network approaches the solution of the corresponding minimization problems of a reflected Brownian motion. Stated otherwise, we show that large deviation and heavy traffic asymptotics can be interchanged. We prove this result in the case when the network is initially empty. Without proof we state the corresponding result in the stationary case. We present an illuminating example with two queues.

Heavy traffic approximations of large deviations of feedforward queueing networks

Many modern applications for mobile phones are based on information about the exact position of the user. There are several methods, which fulfill this task. Some require extensions of the network or the terminals; others do not, but provide little accuracy. The method presented here uses exclusively standard measurements, network parameters, and data from the network planning process. The power level measurements of the mobile reports to the network for handover and other radio resource management tasks are compared with the field strength values predicted by network planning tools. Using a single report is a well-known positioning method, which is not very accurate, because it is very sensitive to fading and other random variations of the signal levels. In the new method, a series of several reports is used. This series corresponds to a path, on which the user moves. An efficient algorithm calculates the probability of the most probable path leading to each point in the search area. The estimated position is the weighted average of all these points. Since searching the whole area of a network is not feasible in practice, the calculation effort is drastically reduced by a preprocessing step, which allows the preselection of the search area based on information about the current service cell of the user. Extensive field trials in existing GSM networks have shown that the method works well and gives substantial improvement compared to other positioning methods, such as CI/TA. An average accuracy of 130m was observed in an urban environment and about 300m in rural areas. An UTRAN provides some measurements, which are very similar to those of GSM, it is possible to apply this method in UMTS networks as well.

A new method for positioning of mobile users by comparing a time series of measured reception power levels with predictions

We consider multiclass feedforward queueing networks under first in first out and priority service disciplines driven by long-range dependent arrival and service time processes. We show that in critical loading the normalized workload, queue length and sojourn time processes can converge to a multi-dimensional reflected fractional Brownian motion. This weak heavy traffic approximation is deduced from a deterministic pathwise approximation of the network behavior close to constant critical load in terms of the solution of a Skorokhod problem. Since we model the doubly infinite time interval, our results directly cover the stationary case.

Fractional Brownian Heavy Traffic Approximations of Multiclass Feedforward Queueing Networks

By proving the continuity of multi-dimensional Skorokhod maps in a quasi-linearly discounted uniform norm on the doubly infinite time interval R, and strengthening know sample path large deviation principles for fractional Brownian motion to this topology, we obtain large deviation principles for the image of multi-dimensional fractional Brownian motions under Skorokhod maps as an immediate consequence of the contraction principle. As an application, we explicitly calculate large deviation decay rates for steady-state tail probabilities of certain queueing systems in multi-dimensional heavy traffic models driven by fractional Brownian motions.

Large deviations for multi-dimensional reflected fractional Brownian motion

We discuss a model for general single class queueing networks which allows discrete and fluid customers and lives on the time interval R. The input for the model are the cumulative service time developments, the cumulative external arrivals and the cumulative routing decisions of the queues. A path space fixed point equation characterizes the corresponding behavior of the network. Monotonicity properties imply the existence of a largest and a smallest solution. Despite the possible non-uniqueness of solutions the sets of solutions have several nice properties. The set valued solution map is partially upper semicontinuous with respect to a quasi-linearly discounted uniform metric on the input paths space. In addition to this main result, we investigate convergence of approximate solutions, measurability, monotonicity and stationarity. We give typical examples for situations where solutions are non-unique and unique.

Single Class Queueing Networks with Discrete and Fluid Customers on the Time Interval R

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