The inclusion of generally distributed random variables in stochastic models is often tackled by choosing a parametric family of distributions and applying fitting algorithms to find appropriate parameters. A recent paper proposed the approximation of probability density functions (PDFs) by Bernstein exponentials, which are obtained from Bernstein polynomials by a change of variable and result in a particular case of acyclic phase-type distributions. In this paper, we show that this approximation can also be applied to cumulative distribution functions (CDFs), which enjoys advantageous properties and achieves similar accuracy; by focusing on CDFs, we propose an approach to obtain stochastically ordered approximations. The use of a scaling parameter in the approximation is also presented, evaluating its effect on approximation accuracy.

Several previous works have addressed the inherent trade-off between allocating resources in the power and time domains to pilot and data signals in multiple input multiple output systems over block-fading channels. In particular, when the channel changes rapidly in time, channel aging degrades the performance in terms of spectral efficiency without proper pilot spacing and power control. Despite recognizing non-stationary stochastic processes as more accurate models for time-varying wireless channels, the problem of pilot spacing and power control in multi-antenna systems operating over non-stationary channels is not addressed in the literature. In this paper, we address this gap by introducing a refined first-order autoregressive model that exploits the inherent temporal correlations over non-stationary Rician aging channels. We design a multi-frame structure for data transmission that better reflects the non-stationary fading environment than previously developed single-frame structures. Subsequently, to determine the optimal pilot spacing and power control within this multi-frame structure, we develop an optimization framework and an efficient algorithm based on maximizing a deterministic equivalent expression for the spectral efficiency, demonstrating its generality by encompassing previous channel aging results. Our numerical results indicate the efficacy of the proposed method in terms of spectral efficiency gains over the single frame structure.

Previous works in array processing have proposed two types of snapshot models for the angle of arrival (AoA) estimation problem in multi-antenna systems. The deterministic model assumes that the source waveforms are non-random, while the random sensor noise is white Gaussian with a known covariance matrix. The stochastic model assumes that both the waveforms and the noise are zero-mean Gaussian. Interestingly, the performance of these two models have rarely been compared in integrated sensing and communication (ISAC) systems. Therefore, in this paper, we consider the uplink of a bistatic ISAC system that uses unitary constant envelope signaling and pilot-based channel estimation while transmitting a sensing signal simultaneously with the communication signals. The base station uses both the pilot and data signals to estimate the angle of a passive source and the transmitted data symbol by an active (connected) user equipment device. For this system, we derive the classical Cramér-Rao bound for unbiased estimators of the AoA and the transmitted symbol, along with the Bayesian Cramér-Rao bound, which bounds the error of all estimators. We also derive the ISAC-aware minimum mean squared error receiver for both the deterministic and stochastic models. We study the trade-off between sensing and communication under the deterministic and stochastic waveform assumptions. Specifically, we show that the fundamental trade-off between sensing and communication power allocations is expressed differently in the deterministic and stochastic models and argue that the results serve as basic considerations when designing pilot and sensing signals for ISAC systems.

In the uplink of multiuser multiple input multiple output (MU-MIMO) systems operating over aging channels, pilot spacing is crucial for acquiring channel state information and achieving high signal-to-interference-plus-noise ratio (SINR). Somewhat surprisingly, very few works examine the impact of pilot spacing on the correlation structure of subsequent channel estimates and the resulting quality of channel state information considering channel aging. In this paper, we consider a fast-fading environment characterized by its exponentially decaying autocorrelation function, and model pilot spacing as a sampling problem to capture the inherent trade-off between the quality of channel state information and the number of symbols available for information carrying data symbols. We first establish a quasi-closed form for the achievable deterministic equivalent SINR when the channel estimation algorithm utilizes multiple pilot signals. Next, we establish upper bounds on the achievable SINR and spectral efficiency, as a function of pilot spacing, which helps to find the optimum pilot spacing within a limited search space. Our key insight is that to maximize the achievable SINR and the spectral efficiency of MU-MIMO systems, proper pilot spacing must be applied to control the impact of the aging channel and to tune the trade-off between pilot and data symbols.

Markov modulated discrete arrival processes have a wide literature, including parameter estimation methods based on expectation–maximization (EM). In this paper, we investigate the adaptation of these EM based methods to Markov modulated fluid arrival processes (MMFAP), and conclude that only the generator matrix of the modulating Markov chain of MMFAPs can be approximated by EM based method. For the rest of the parameters, the fluid rates and the fluid variances, we investigate the efficiency of numerical likelihood maximization. To reduce the computational complexity of the likelihood computation, we accelerate the numerical inverse Laplace transformation step of the procedure with function fitting.

Minimizing the symbol error in the uplink of multi-user multiple input multiple output systems is important, because the symbol error affects the achieved signal-to-interference-plus-noise ratio (SINR) and thereby the spectral efficiency of the system. Despite the vast literature available on minimum mean squared error (MMSE) receivers, previously proposed receivers for block fading channels do not minimize the symbol error in time-varying Rayleigh fading channels. Specifically, we show that the true MMSE receiver structure does not only depend on the statistics of the CSI error, but also on the autocorrelation coefficient of the time-variant channel. It turns out that calculating the average SINR when using the proposed receiver is highly non-trivial. In this paper, we employ a random matrix theoretical approach, which allows us to derive a quasi-closed form for the average SINR, which allows to obtain analytical exact results that give valuable insights into how the SINR depends on the number of antennas, employed pilot and data power and the covariance of the time-varying channel. We benchmark the performance of the proposed receiver against recently proposed receivers and find that the proposed MMSE receiver achieves higher SINR than the previously proposed ones, and this benefit increases with increasing autoregressive coefficient.

In the above article[1], the title of the article appears incorrectly. The full title should read “On the Achievable SINR in MU-MIMO Systems Operating in Time-Varying Rayleigh Fading.”

The performance of the uplink of single and multiuser multiple input multiple output (MIMO) systems depends crucially on the receiver architecture and the quality of channel state information at the receiver. Therefore, several previous works have developed minimum mean squared error (MMSE) receivers and proposed balancing the resources spent on acquiring channel state information and transmitting the payload of data packets. Somewhat surprisingly, the most popular MIMO linear MMSE receivers do not exploit the correlation structure that is present in autoregressive Rayleigh fading environments. Therefore, in this article we first develop a new linear receiver that not only takes channel state information errors into account in minimizing the MSE of the received data symbols, but it also utilizes that the subsequent noisy channel coefficients are correlated. For this new linear MMSE receiver, we derive the achieved MSE as a function of the number of receive antennas and the pilot-to-data power ratio. Interestingly, we find that the pilot power that minimizes the MSE of the data symbols does not depend on the number of antennas and that the new linear MMSE receiver outperforms previously proposed MIMO receivers when the autocorrelation coefficient of the channel is high.

Resource allocation in wireless access networks has been an intensively researched topic recently: many proposed solutions tackle radio channel access and dynamic spectrum allocation, but traditional issues of queuing, bandwidth sharing and packet processing at wireless access points have been targeted as well. In most of the related work the competition for high quality of service is usually solved by central coordination among users via optimizing a specific target aspect of the overall communication. In this paper we take a turn and provide users with the possibility of resource allocation suggestions. We propose a wireless access sharing framework, in which users have a say in optimizing their quality of service on the long term, and we tackle its analysis with the tool set of stochastic game theory. Our findings show that greedy users become polite against their counterparts when the load is relatively low with the goal of preparing for situations with high load.

We consider the uplink of a single cell multi-user multiple input multiple output (MU-MIMO) system, in which the base station acquires channel state information at the receiver by means of uplink pilot signals. Since each mobile station has a sum power budget that is used to transmit pilot and data symbols, the pilot power ratio (PPR) has a large impact on the system performance in terms of spectral and energy efficiency. We formulate the problem of PPR setting as a non-cooperative game, in which each mobile station aims at minimizing the mean squared error of the uplink received data symbols at the base station. We show that in this game a unique Nash equilibrium exists, and propose an iterative decentralized algorithm-termed best PPR algorithm (BPA)-that is guaranteed to converge to that Nash equilibrium. Since BPA dynamically responds to the measured interference, it outperforms widely used schemes that use a predetermined PPR. BPA also performs close to the global optimum, especially when mobile stations with similar path loss values are co-scheduled in the MU-MIMO system. Based on these insights, we propose a practical signaling mechanism for implementing BPA in MU-MIMO systems.

Stochastic Petri nets are widely used for the modeling and analysis of non-functional properties of critical systems. The state space explosion problem often inhibits the numerical analysis of such models. Symbolic techniques exist to explore the discrete behavior of even complex models, while block Kronecker decomposition provides memory-efficient representation of the stochastic behavior. However, the combination of these techniques into a stochastic analysis approach is not straightforward. In this paper we integrate saturation-based symbolic techniques and decomposition-based stochastic analysis methods. Saturation-based exploration is used to build the state space representation and a new algorithm is introduced to efficiently build block Kronecker matrix representation to be used by the stochastic analysis algorithms. Measurements confirm that the presented combination of the two representations can expand the limits of previous approaches.

In systems employing pilot-symbol aided channel estimation, the pilot-to-data power ratio is known to have a large impact on performance. Therefore, previous works proposed methods setting the pilot power such that either the weighted sum of the mean squared error (MSE) of the estimated data symbols is minimized or the overall spectral efficiency (SE) is maximized. However, previous works did not take into account the impact of correlated antennas and channel state information (CSI) errors on the optimal pilot power setting. In this paper, we consider the uplink of a multiuser multiple-input multiple-output (MU MIMO) system employing a receiver that minimizes the MSE of the received data symbols in the presence of CSI errors and derive closed-form expressions for the MSE and the achievable SE. These expressions take into account the impact of antenna correlation and CSI errors, and are a function of pilot power and the number of receive antennas. The analytical and numerical results can help set the pilot power, minimizing the MSE in multiple antenna systems.

We consider the uplink of a single cell single input multiple output (SIMO) system, in which the mobile stations use intra-cell orthogonal pilots to facilitate uplink channel estimation. In such systems, the problem of transmission power balancing between pilot and data is known to have a large impact on the mean square error (MSE) for the received signal and, consequently, on the achievable uplink data rate. In this paper, we derive a closed form expression of the MSE for the received signal as a function of the pilot and data power levels under a per-user sum pilot-data power constraint. As a major contribution, our model is developed for arbitrary channel covariance matrices and it enables us to study the impact of the number of antennas and antenna correlation structures, including the popular 3GPP spatial channel model. Numerical results suggest that the effect of the antenna spacing is limited, but the angle of arrival and angular spread have a strong and articulated impact on the MSE performance. Moreover, as the number of antennas at the base station grows large, we show that a higher percentage of the power budget should be allocated to pilot signals than with a lower number of antennas.

A new class of non-Markovian models is introduced that results from the combination of stochastic automata networks and a very general class of stochastic processes, namely, rational arrival processes, which are derived from matrix exponential distributions. It is shown that the modeling formalism allows a compact representation of complex models with large state spaces. The resulting stochastic process is non-Markovian, but it can be analyzed with numerical techniques like a Markov chain, and the results at the level of the automata are stochastic distributions that can be used to compute standard performance and dependability results. The model class includes stochastic automata networks with phase-type distributed and correlated event times and also includes models that have a finite state space but cannot be represented by finite Markov chains. The paper introduces the model class, shows how the descriptor matrix can be represented in compact form, presents some example models, and outlines methods to analyze the new models.

No abstract available.

Early P2P-TV systems have already attracted millions of users, and many new commercial solutions are entering this market. Little information is however available about how these systems work, due to their closed and proprietary design. In this paper, we present large scale experiments to compare three of the most successful P2P-TV systems, namely PPLive, SopCast and TVAnts. Our goal is to assess what level of "network awareness" has been embedded in the applications. We first define a general framework to quantify which network layer parameters leverage application choices, i.e., what parameters mainly drive the peer selection and data exchange. We then apply the methodology to a large dataset, collected during a number of experiments where we deployed about 40 peers in several European countries. From analysis of the dataset, we observe that TVAnts and PPLive exhibit a mild preference to exchange data among peers in the same autonomous system the peer belongs to, while this clustering effect is less intense in SopCast. However, no preference versus country, subnet or hop count is shown. Therefore, we believe that next-generation P2P live streaming applications definitively need to improve the level of network-awareness, so to better localize the traffic in the network and thus increase their network-friendliness as well.

Early P2P-TV systems have already attracted millions of users, and many new commercial solutions are entering this market. Little information is however available about how these systems work. In this paper we present large scale sets of experiments to compare three of the most successful P2P-TV systems, namely PPLive, SopCast and TVAnts. Our goal is to assess what level of "network awareness" has been embedded in the applications, i.e., what parameters mainly drive the peer selection and data exchange. By using a general framework that can be extended to other systems and metrics, we show that all applications largely base their choices on the peer bandwidth, i.e., they prefer high-bandwidth users, which is rather intuitive. Moreover, TVAnts and PPLive exhibits also a preference to exchange data among peers in the same autonomous system the peer belongs to. However, no evidence about preference versus peers in the same subnet or that are closer to the considered peer emerges. We believe that next-generation P2P live streaming applications definitively need to improve the level of network-awareness, so to better localize the traffic in the network and thus increase their network-friendliness as well.

The 10 items in this special issue focus on quantitative evaluation of computer systems.

Recently the Birkhoff-von Neumann load-balanced (LB) switch has become a promising switch design due to its high scalability properties and simple control. The performance of the LB switch was studied under strong assumptions such as infinite buffers and admissible traffic conditions. However, both such assumptions may be violated in multi-hop networks since admissibility requirement cannot be maintained unless some inter-switch feedback mechanism is implemented, and infinite buffers are not feasible either. This paper considers the performance of the LB switch with finite central stage buffers under both (i) admissible and (ii) inadmissible input traffic conditions. Its contributions are two folds: firstly, by means of mathematical model we demonstrate that the load-balanced switch has a non-zero cell dropping probability due to buffer overflow even under the admissible input traffic assumptions. Secondly, cell loss probabilities are even higher and large buffers are required under inadmissible traffic conditions to cope with such behavior.

This paper presents a general closed-form analysis of the time-blocking probability in time-driven switching networks. Time-blocking occurs when transmission resources are available in both input and output, but there is no schedule, i.e., input and output resources are outside a pre-defined maximum scheduling delay that is allowed between them. This situation may happen in architectures based on pipeline forwarding of packets. The main constraints affecting the schedulability of resources are the load and the maximum scheduling delay. The analysis yields the exact blocking probabilities for all possible scheduling delays and under all load conditions for a node in isolation, as well as initial results for a network of nodes.

The representation of general distributions or measured data by phase-type distributions is an important and nontrivial task in analytical modeling. Although a large number of different methods for fitting parameters of phase-type distributions to data traces exist, many approaches lack efficiency and numerical stability. In this paper, a novel approach is presented that fits a restricted class of phase-type distributions, namely, mixtures of Erlang distributions, to trace data. For the parameter fitting, an algorithm of the expectation maximization type is developed. This paper shows that these choices result in a very efficient and numerically stable approach which yields phase-type approximations for a wide range of data traces that are as good or better than approximations computed with other less efficient and less stable fitting methods. To illustrate the effectiveness of the proposed fitting algorithm, we present comparative results for our approach and two other methods using six benchmark traces and two real traffic traces as well as quantitative results from queuing analysis

The representation of general distributions or measured data by phase-type distributions is an important and non-trivial task in analytical modeling. Although a large number of different methods for fitting parameters of phase-type distributions to data traces exist, many approaches lack efficiency and numerical stability. In this paper, a novel approach is presented that fits a restricted class of phase-type distributions, namely mixtures of Erlang distributions, to trace data. For the parameter fitting an algorithm of the expectation maximization type is developed. The paper shows that these choices result in a very efficient and numerically stable approach which yields phase-type approximations for a wide range of data traces that are as good or better than approximations computed with other less efficient and less stable fitting methods. To illustrate the effectiveness of the proposed fitting algorithm, we present comparative results for our approach and two other methods using six benchmark traces and two real traffic traces.

This paper considers the analysis of second-order Markov reward models. In these systems the reward accumulation during state sojourns is not deterministic, but follows a Brownian motion with a state dependent drift and variance parameter. We give the differential equations that describe the density function and the moments of the accumulated reward, and show the similarities compared to the first-order (ordinary) case. A randomization based numerical method is also presented which is numerically stable, has an error bound to control the precision, and allows the efficient analysis of large models. The computational cost of the proposed procedure is practically the same as the one of the analysis of first-order reward models, while the modeling power of second-order models is clearly larger.

PhFit, a new phase-type fitting tool is presented in this paper. PhFit allows for approximating distributions or set of samples not only by continuous but by discrete phase-type distributions as well. The implemented algorithms separate the fitting of the body and the tail part of the distribution which results in satisfactory fitting also for heavy-tail distributions. Moreover, PhFit allows the user to choose the distance measure according to which the fitting is performed. The tool is equipped with a graphical interface that visualizes the goodness of the provided approximation from several aspects.

This paper introduces a unified approach to phase-type approximation in which the discrete and continuous phase-type models form a common model set. The models of this common set are assigned with a non-negative real parameter, the scale factor. The case when the scale factor is strictly positive results in discrete phase-type distributions and the scale factor represents the time elapsed in one step. If the scale factor is 0, the resulting class is the class of continuous phase-type distributions. Applying the above view, it is shown that there is no qualitative difference between the discrete and the continuous phase-type models. Based on this unified view of phase-type models one can choose the best phase-type approximation of a stochastic model by optimizing the scale factor.

The paper presents an analytical model for a GSM-based cellular mobile network that applies an intelligent under-overlay (IUO) scheme to increase capacity by increasing frequency reuse while maintaining service quality. The IUO is a multi-layer cell structure that is based on dividing the frequency band into super layer and regular layer frequency groups. The super frequencies (channels) can be used by mobile stations with good C/I (carrier/interferer) ratio, while the regular frequencies can be used over the whole cell. The use of IUO is expected to provide up to 40% gain of capacity (see Nokia Telecommunications, www.nokia.com). We study the effect of various parameters on the performance of networks using IUO and provide practical planning support based on the analytical results. The considered parameters include network parameters, like super area coverage, and mobile user mobility parameters, like moving mobile ratio and average mobile speed.

The time domain analysis of non-Markovian stochastic Petri nets with pre-emptive repeat identical (PRI) type transitions is considered in this paper. The set of "time domain" equations describing the evolution of the marking process is provided. The relation of the time domain and formerly available transform domain description is discussed. Based on the time domain description of the process, a simple numerical procedure is provided to analyze the transient behavior. Two examples are calculated to illustrate the proposed numerical method.

We develop a call level model of UMTS core networks where calls belonging to one of the four UMTS service classes arrive randomly. Arriving calls are granted service depending on the call's service class, the required maximum and minimum-bandwidth, and the available network resources at the arrival instance. We use a Markov model of transmission links to derive GoS (blocking probability) and QoS (throughput) measures under two reasonable and technologically feasible bandwidth sharing policies. We conclude that one of these policies is able to provide GoS/QoS guarantees for a wide range of traffic mixes. We argue that the results are applicable to the all IP/MPLS based new UMTS architecture.

Petri nets represent a useful tool for performance, dependability, and performability analysis of complex systems. Their modeling power can be increased even more if nonexponentially distributed events are considered. However, the inclusion of nonexponential distributions destroys the memoryless property and requires to specify how the marking process is conditioned upon its past history. We consider, in particular, the class of stochastic Petri nets whose marking process can be mapped into a Markov regenerative process. An adequate mathematical framework is developed to deal with the considered class of Markov Regenerative Stochastic Petri Nets (MRSPN). A unified approach for the solution of MRSPNs where different preemption policies can be defined in the same model is presented. The solution is provided both in steady-state and in transient condition. An example concludes the paper.

In this paper the integration of available results on semi-Markov decision processes and on Markov regenerative processes is attempted, in order to define the mathematical framework for solving decision problems where the underlying structure state process is a Markov regenerative process, referred to a Markov regenerative decision process. The essential question investigated here is which description of Markov regenerative processes is needed to analyze the investigation decision model.

Preventive maintenance of operational software systems, a novel technique for software fault tolerance, is used specifically to counteract the phenomenon of software "aging". However, it incurs some overhead. The necessity to do preventive maintenance, not only in general purpose software systems of mass use, but also in safety-critical and highly available systems, clearly indicates the need to follow an analysis based approach to determine the optimal times to perform preventive maintenance. In this paper, we present an analytical model of a software system which serves transactions. Due to aging, not only the service rate of the software decreases with time, but also the software itself experiences crash/hang failures which result in its unavailability. Two policies for preventive maintenance are modeled and expressions for resulting steady state availability, probability that an arriving transaction is lost and an upper bound on the expected response time of a transition are derived. Numerical examples are presented to illustrate the applicability of the models.

In a client-server type system, the server software is required to run continuously for very long periods. Due to repeated and potentially faulty usage by many clients, such software "ages" with time and eventually fails. (Huang et al., 1995) proposed a technique called "software rejuvenation" in which the software is periodically stopped and then restarted in a "robust" state after proper maintenance. This "renewal" of software prevents (or at least postpones) the crash failure. As the time lost (or the cost incurred) due to the software failure is typically more than the time lost (or the cost incurred) due to rejuvenation, the technique reduces the expected unavailability of the software. We present a quantitative analysis of software rejuvenation. The behavior of the system is represented through a Markov Regenerative Stochastic Petri Net (MRSPN) model which is solved both for steady state as well as transient conditions. We provide a closed-form analytical solution for the steady state expected down time (and the expected cost incurred) due to system unavailability. We also evaluate the optimal rejuvenation interval which minimizes the expected unavailability of the software.

Non-Markovian models allow us to capture a very wide range of circumstances in which it is necessary to model phenomena whose times to occurrence is not exponentially distributed. Events such as timeouts in a protocol, service times at a machine performing the same task on each part, and memory access or instruction execution in a low-level h/w or s/w model, have durations which are constant or with a very low variance. Phase-type distributions can be used to approximate a non-exponential, but they increase the size of the state space.The analysis of stochastic systems with non-exponential timing is of increasing interest in the literature and requires the development of suitable modeling tools. Recently, some effort has been devoted to generalize the concept of Stochastic Petri Nets (SPN), by allowing the firing times to be generally distributed.A particular case of non-Markovian SPN, is the class of Deterministic and SPN (DSPN) [1]. A DSPN is a non-Markovian SPN where, in each marking, at most one transition is allowed to have a deterministic firing time with enabling memory policy.A new class of stochastic Petri nets has recently been defined [2, 3] by generalizing the deterministic firing times of the DSPN to generally distributed firing times. The underlying stochastic process for these classes of Petri nets is a Markov Regenerative Process (MRGP). This observation has opened a very fertile line of research aimed at the definition of solvable classes of models whose underlying marking process is an MRGP, and therefore referred to as Markov Regenerative Stochastic Petri Nets (MRSPN).Some of the results in this filed will be described in the session. In particular, Ciardo investigates stochastic confusion by defining the selection probability for transitions attempting to fire at the same time. German introduces the "method of supplementary variables" for the derivation of state equations describing the transient behavior of the marking process. Puliafito describes how, under some constraints, concurrent enabling of several generally distributed timed transitions is allowed. Bobbio and Telek discuss how age memory policy can be included to capture preemptive mechanisms of the resume (prs) type.

PLANET is an integrated software package aiding several steps of the transmission network planning, as well as the traffic simulation. The database of PLANET consists of the structural and the cost models of the transmission systems, the topological alternatives, the circuit routing and grouping rules, etc. The topological and transmission plans of the networks are obtained based on the traffic planning results checked by simulation and they are stored systematically with the input data in the database of plans. The results describe the network topology, the circuit routes meeting different kinds of routing conditions, the grouping solutions and the equipment assignments to the circuit demands of different services. The users of PLANET are provided with lists and graphical documentation that can be directly used in the development of the network. The global structure of the software package as well as the algorithms of specific phases are discussed and the results obtained by the application of PLANET are demonstrated.>