Sergey Shorgin

On truncations for weakly ergodic inhomogeneous birth and death processes

Abstract We investigate a class of exponentially weakly ergodic inhomogeneous birth and death processes. We consider special transformations of the reduced intensity matrix of the process and obtain uniform (in time) error bounds of truncations. Our approach also guarantees that we can find limiting characteristics approximately with an arbitrarily fixed error. As an example, we obtain the respective bounds of the truncation error for an Mt/Mt/S queue for any number of servers S. Arbitrary intensity functions instead of periodic ones can be considered in the same manner.

Ergodicity And Perturbation Bounds For Inhomogeneous Birth And Death Processes With Additional Transitions From And To The Origin

Abstract Service life of many real-life systems cannot be considered infinite, and thus the systems will be eventually stopped or will break down. Some of them may be re-launched after possible maintenance under likely new initial conditions. In such systems, which are often modelled by birth and death processes, the assumption of stationarity may be too strong and performance characteristics obtained under this assumption may not make much sense. In such circumstances, time-dependent analysis is more meaningful. In this paper, transient analysis of one class of Markov processes defined on non-negative integers, specifically, inhomogeneous birth and death processes allowing special transitions from and to the origin, is carried out. Whenever the process is at the origin, transition can occur to any state, not necessarily a neighbouring one. Being in any other state, besides ordinary transitions to neighbouring states, a transition to the origin can occur. All possible transition intensities are assumed to be non-random functions of time and may depend (except for transition to the origin) on the process state. To the best of our knowledge, first ergodicity and perturbation bounds for this class of processes are obtained. Extensive numerical results are also provided.

Sojourn Time Analysis for Processor Sharing Loss System with Unreliable Server

Processor sharing (PS) queuing systems and particularly their well-known class of egalitarian processor (EPS) sharing are widely investigated by research community and applied for the analysis of wire and wireless communication systems and networks. The same can be said for queuing systems in random environment, with unreliable servers, interruptions, pre-emption mechanisms. Nevertheless, only few works focus on queues with both PS discipline and unreliable servers. In the paper, compared with the previous results we analyse a finite capacity PS queuing system with unreliable server and an upper limit of the number of customers it serves simultaneously. For calculating the mean sojourn time, unlike a popular but computational complex technique of inverse Laplace transform we use an effective method based on embedded Markov chains. The paper also includes a practical numerical example of web browsing in a wireless network when the corresponding low priority traffic can be interrupted by more priority applications.

SIR Analysis In Square-Shaped Indoor Premises.

The increased wireless network densification has resulted in availability of wireless access points (AP) in almost each and every indoor location (room, office, etc.). To provide complete in-building coverage very often an AP is deployed per room. In this paper we analyze signal-to-interference (SIR) ratio for wireless systems operating in neighboring rooms separated by walls of different materials by explicitly taking into account the propagation and wall penetration losses. Both AP and direct device-to-device (D2D) configurations are addressed. Our numerical results indicate that the performance of such system is characterized by both the loss exponent describing the propagation environment of interest and wall materials. We provide the numerical results for typical wall widths/materials and analyze them in detail. INTRODUCTION The predicted increase in the user traffic demands places extreme requirements on the future evaluation of mobile systems, often referred to as fifth generation (5G) networks [1], [2]. In addition to physical layer improvements including advanced modulation and coding and MIMO techniques, over the last decade researchers investigated a number of network solutions providing decisive performance improvements including the use of small (micro/pico/femto) cells [3], clientrelays [4], direct in-band and out-of-band device-to-device communications [5]. All these concepts target aggressive spatial reuse of frequencies promising substantial area capacity gains. With the adoption of novel mechanism the user devices are expected to take a more active part in 5G systems and, in some cases, even take on the role of the network infrastructure in providing wireless connectivity such as offering D2D-based data relaying, proximity services, etc. This shift from the classic cellular model is dictated by the progress in communications technologies: the user devices are augmenting their capabilities, whereas the base stations (BSs) are becoming smaller as a result of the ongoing network densification [6]. The networks densification, novel networking and service mechanisms as well as the trend to use multiple access technologies to serve the users, known as heterogeneous cellular system concept, altogether lead to increased randomness of the network, where the positions of servicing stations such as BSs, relays and D2D partners are random rather than deterministic. The signal-to-interference ratio (SIR) is a universal metric specifying performance of wireless systems [7]. Once SIR is known one could describe the Shannon rate of the channel and spectral efficiency of the system. In contrast to noise-limited systems, where the bit error rate (BER) decreases exponentially with signal-to-noise ratio (SNR), the heterogeneous mobile networks are interference-limited showing linear improvement of BER with respect to SIR. Thus, the increase of the emitted power does not improve the performance of these systems. Thus, the problem of finding SIR for typical network configurations is of special importance characterizing applicability and typical scenarios of modern and future wireless technologies. The SIR performance of wireless systems is often studied using the tools of stochastic geometry [8]. The basic approach is to specify the point process on the plane modeling positions of the stations and then derive the interference at the point of interest. The last step is rather complex as we need closed-form distribution of distance to the point of interest from at least several neighboring points. For this reason typical considered models are often limited to Poisson point process on the plane for which we immediately have closed-form expressions for distributions of distances to the i-th neighbor [9]. The constantly increasing need for wireless connectivity on-the-go [10] are gradually changing the way service is provisioned in wireless networks. Nowadays, one of the trends is to deploy small wireless stations including both IEEE 802.11 or micro-LTE access points (AP) in crowded areas to benefit from increased network densification [6] and shorter propagation distances. Examples include large shopping mall, office environment, where one of few adjacent rooms is served by an AP having relatively small coverage area. In this dense environment interference between neighboring APs is inevitable and may easily lead to degraded system performance. In this paper, using the tools of stochastic geometry, we analyze performance of wireless systems operating in neighboring rooms of rectangular configuration. We consider both direct device-to-device and AP configurations assuming that Proceedings 30th European Conference on Modelling and Simulation ©ECMS Thorsten Claus, Frank Herrmann, Michael Manitz, Oliver Rose (Editors) ISBN: 978-0-9932440-2-5 / ISBN: 978-0-9932440-3-2 (CD) the systems in adjacent rooms operate at the same frequency. The analytical results are compared to simulations showing adequate agreement. Numerical results for the set of input metrics demonstrate that the system performance is dictated by the interplay between path loss exponent typical for a given environment and type of the walls used between rooms. The rest of the paper is organized as follows. First, in the next section we introduce a system model. Further, we analytically study SIR for downlink scenario. The simulation models for both downlink and D2D scenarios are introduced next. Numerical results for different sets in input variables are illustrated. Conclusions are drawn in the last section. SYSTEM MODEL In this study we focus on an indoor scenario with grid aligned rooms, see Fig. 1 that are typical for shopping malls or office buildings. In these environments rooms are often of rectangular or square shapes. Each room is assumed to be equipped with an AP deployed in the geometrical center. To take advantage of the wireless network densification trend as a solution to upgrade the degree of spatial reuse, the devices in adjacent rooms are assigned the same set of communication channels [6]. The mobile terminals (users) operating over the same channel are assumed to be uniformly distributed over the room, one per room. We concentrate on the so-called tagged user in the central room, see Fig. 1(a) and Fig. 1(b). We assume both AP and users to be equipped with omnidirectional antennas. We do not focus on a particular radio technology addressing the general case. In addition to AP scenario we also address D2D configuration, sketched in Fig. 1(c). The principal difference compared to AP case is that both transmitter and receiver are assumed to be uniformly distributed within a room. Under this assumption the configuration is symmetric, i.e., we do not have to distinguish between uplink and downlink cases. Similarly, we concentrate on D2D pair located in the central room. Focusing on SIR, as a metric of interest, for both AP and D2D configurations we calculate it for a randomly chosen receiving device, taking into account the interference from a set of neighboring rooms. Using the commonly used propagation model, we add a correction factor, accounting for the attenuation of a signal when passing through a wall

On Capturing Spatial Diversity of Joint M2M/H2H Dynamic Uplink Transmissions in 3GPP LTE Cellular System

While queuing theory has indeed been instrumental to various communication problems for over half a century, the unprecedented proliferation of wireless technology in the last decades brought along novel research challenges, where user location has become a crucial factor in determining the respective system performance. This recent shift turned important to characterize large cellular macrocells, as well as the emerging effects of network densification. However, the latter trend also called for increased attention to the actual user loading and uplink (UL) traffic dynamics, accentuating again the necessity of queuing analysis. Hence, by combining queuing theory and stochastic geometry in a feasible manner, we may quantify the dependence of system-level performance on the traffic loading.

Polling System with Threshold Control for Modeling of SIP Server under Overload

The main purpose of this research is the development of the methods for realizing an overload control mechanism on the Session Initiation Protocol servers by the application of polling systems with different service disciplines. The mathematical model is studied by means of numerical methods of the queuing theory and allows analyzing the behavior of different control parameters depending on the load in network of SIP servers. The polling system consists of two queues of finite capacity and implements the threshold control of loading by low-priority customers. The exhaustive and gated service disciplines are studied under Markov assumptions; formulas for calculation of the main probability measures of the polling system are derived. By performing simulations we demonstrate that the polling system with a threshold in the priority queue is a possible solution for loss-based overload control scheme at the SIP server. In some cases from the viewpoint of server utilization we found that the gated discipline is more preferable.

On M n (t)/M n (t)/S queues with catastrophes

We consider the Mn(t)/Mn(t)/S queue with catastrophes. The bounds on the rate of convergence to the limit regime and the estimates of the limit probabilities are obtained.

Modeling and analyzing licensed shared access operation for 5G network as an inhomogeneous queue with catastrophes

The framework of licensed shared access (LSA) to spectrum seems to become one of the trends of 5G wireless networks. The framework assumes the simultaneous access to spectrum by at least two parties - the primarily owner (incumbent), which has the highest priority, and several secondary users (licensees), which have lower priorities. The critical up-to-date problem is the development of the corresponding radio admission control and load balancing algorithms that form an essential part of the LSA agreement between the parties. The algorithm of binary use of spectrum gives an absolute priority to the incumbent, e.g. the airport using spectrum for aeronautical telemetry purposes. In the paper, capturing the inhomogeneous in time nature of rates of requests for access to spectrum and average times of spectrum use, we propose a queuing model of binary access to spectrum as seen from the licensees point of view. The queue is described by an inhomogeneous birth and death process with catastrophes and repairs. The main aim of the paper is to find the bounds on the rates of convergence to the limiting characteristics of the queue - average number of users, blocking probability, and probability of service interruption due to the incumbents need for spectrum. Not only the acceptable upper thresholds on the limiting characteristics are important for consideration but also the corresponding bounds showing the moment in time when the system becomes stable and the LSA licensee could really access to spectrum.

A two-priority queueing system with trunk reservation, infinite capacity buffers for customers of both priorities, and different service intensities for high-priority and non-priority customers

A two-priority queueing system with trunk reservation, Poisson input flows of both priorities customers, infinite buffers and different exponential distributions of both priorities customers service times is considered. Trunk reservation means that there are some channels which can be used only by customers of high priority. Analytic relations to calculate the main stationary distributions of both priorities customers quantities are obtained.

On the Bayesian approach to the analysis of queueing systems and reliability characteristics

The paper discusses the statements or problems and the results obtained by the authors since 2005 within the bounds of application of Bayesian approach to some problems of queueing theory and reliability theory, as well as some directions of further research in this area. The method provides the randomization of system characteristics with regard of a priori distributions of input parameters. This approach could be used, for instance, for calculating moment and quantile characterics for performance and reliability characteristics of large groups of systems or devices.