# Miklós Telek's research while affiliated with Budapest University of Technology and Economics and other places

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## Publications (236)

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 M...

Among the numerical inverse Laplace transformation (NILT) methods, those that belong to the Abate-Whitt framework (AWF) are considered to be the most efficient ones currently. It is a characteristic feature of the AWF NILT procedures that they are independent of the transform function and the time point of interest.
In this work we propose an NILT...

Among the numerical inverse Laplace transformation (NILT) methods, those that belong to the Abate–Whitt framework (AWF) are considered to be the most efficient ones currently. It is a characteristic feature of the AWF NILT procedures that they are independent of the transform function and the time point of interest.
In this work we propose an NILT...

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 subse...

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...

Piecewise homogeneous Markov fluid models are composed by homogeneous intervals where the model is governed by an interval dependent pair of generators and the model behaviour changes at the boundaries. The main difficulty of the transient analysis of piecewise homogeneous Markov fluid models is the appropriate description of the various boundary c...

In this paper, we consider a retrial queuing system with unreliable servers and analyze the distribution of the stationary generalized service time which includes also the unavailable periods (setup times) occurring during service of the customer. We consider three service interruption disciplines: preemptive resume (PRS), preemptive repeat differe...

Among the numerical inverse Laplace transformation (NILT) methods, those that belong to the Abate-Whitt framework (AWF) are considered to be the most efficient ones currently. It is a characteristic feature of the AWF NILT procedures that they are independent of the transform function and the time point of interest. In this work we propose an NILT...

We propose a numerical method to obtain the transient and first passage time distributions of first- and second-order Multi-Regime Markov Fluid Queues (MRMFQ). The method relies on the observation that these transient measures can be computed via the stationary analysis of an auxiliary MRMFQ. This auxiliary MRMFQ is constructed from the original on...

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 some parameters of MMFAPs can be approximated this way.

We propose a throughput value function (TVF) based solution for providing multi time-scale (MTS) fairness for broadband traffic in access-aggregation networks. The primary goal of MTS fairness is a dynamic control of resource sharing that considers the usage history of the broadband connection. We present a flow level description of the multi time-...

Numerical inverse Laplace transformation (NILT) is an important tool in the field of system modelling and performance analysis. The recently introduced CME method has many important advantages over the alternative numerical inverse Laplace transformation (NILT) methods. It avoids Gibbs oscillation (i.e., does not generate overshoot and undershoot),...

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...

Concentrated random variables are frequently used in representing deterministic delays in stochastic models. The squared coefficient of variation ( $\mathrm {SCV}$ ) of the most concentrated phase-type distribution of order $N$ is $1/N$ . To further reduce the $\mathrm {SCV}$ , concentrated matrix exponential (CME) distributions with complex eigenv...

This paper presents numerical methods for finding high order concentrated matrix-exponential (ME) distributions, whose squared coefficient of variation (SCV) is very low. Due to the absence of symbolic construction to obtain the most concentrated ME distributions, non-linear optimization problems are defined to obtain high order concentrated matrix...

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 ac...

At the time of the final publication of the paper, in December 2016, Yixin Zhao’s affiliation had changed.

The performance of service units may depend on various randomly changing environmental effects. It is quite often the case that these effects vary on different timescales. In this paper, we consider small and large scale (short and long term) service variability, where the short term variability affects the instantaneous service speed of the servic...

The paper presents a numerical analysis approach for the transient solution of a piecewise homogeneous QBD process. The proposed approach computes the transient probabilities based on the linear combination of matrix geometric series in Laplace transform domain, and builds on the availability of an efficient numerical inverse Laplace transformation...

Numerical inverse Z-transformation (NIZT) methods have been efficiently used in engineering practice for a long time. In this paper, we compare the abilities of the most widely used NIZT methods, and propose a new variant of a classic NIZT method based on contour integral approximation, which is efficient when the point of interest (at which the va...

This paper presents matrix-exponential (ME) distributions, whose squared coefficient of variation (SCV) is very low. Currently, there is no symbolic construction available to obtain the most concentrated ME distributions, and the numerical optimization-based approaches to construct them have many pitfalls. We present a numerical optimization-based...

In cellular multiuser multiple input multiple output (MU-MIMO) systems the quality of the available channel state information (CSI) has a large impact on the system performance. Specifically, reliable CSI at the transmitter is required to determine the appropriate modulation and coding scheme, transmit power and the precoder vector, while CSI at th...

This paper investigates the performance of the numerical inverse Laplace transformation (ILT) method based on concentrated matrix exponential (CME) distributions, referred to as the CME method.
The CME method does not generate overshoot and undershoot (i.e., avoids Gibbs oscillation), preserves monotonicity of functions, its accuracy is gradually i...

In multi-user multiple input multiple output (MU-MIMO) systems that employ pilot-symbol aided channel estimation, the pilot-to-data power ratio (PDPR) has a large impact on the system performance. In this paper we consider the problem of setting the PDPR in multi-cell MU-MIMO systems in the presence of channel estimation errors, intercell interfere...

We introduce and analyse the M∕G∕1 resampling queue with non-preemptive LIFO policy, then we use it to provide bounds on the performance characteristics of an M∕G∕1 processor sharing queue with inaccurate service time information.

The M/G/1 queueing system is similar to the M/M/1 queueing system and the only difference is that the service time is not exponential. First we mention some ideas, most of which were described in the previous chapter in connection with an M/M/1 system.

The previous chapter presented analysis methods for stochastic models where some of the distributions were different from exponential. In these cases the analysis of the models is more complex than the analysis of Markov models. In this chapter we introduce a methodology to extend the set of models which can be analyzed by Markov models while the d...

When considering technical, economic, ecological, or other problems, in several cases the quantities \(\left \{ X_{t},\;t\in \mathcal {T}\right \} \) being examined can be regarded as a collection of random variables. This collection describes the changes (usually in time and in space) of considered quantities. If the set \(\mathcal {T}\) is a subs...

The theory of queueing systems dates back to the seminal work of A.K. Erlang (1878–1929), who worked for the telecom company in Copenhagen and studied the telephone traffic in the early twentieth century. To this today the terminology of queueing theory is closely related to telecommunications (e.g., channel, call, idle/busy, queue length, and util...

Up to now, we have overviewed the main methods for the analysis of individual queueing systems. However, the analysis of large telecommunication systems or computer systems executing complex inter-related tasks (e.g., transaction processing systems and web server farms) requires the application of systems models which contain several servers (poten...

In the early twentieth century, Markov (1856–1922) introduced in Markov (Izvestiya Fiziko-matematicheskogo Obschestva pri Kazanskom Universitete 15:135–156, 1906) a new class of models called Markov chains, applying sequences of dependent random variables that enable one to capture dependencies over time.

In the previous chapters we have studied queueing systems with different interarrival and service time distributions. Chapter 7 is devoted to the analysis of queueing systems with exponential interarrival and service time distributions.

Traditional telephone networks were designed to implement a single type of communication service, i.e., the telephone service. Today’s telecommunication networks implement a wide range of communication services. In this section we introduce Markov models of communication services which compete for the bandwidth of a finite capacity communication li...

Queueing systems whose underlying stochastic process is a continuous-time Markov chain (CTMCs) are the simplest and most often used class of queueing systems. The analysis of these systems is based on the essential results available for the analysis of CTMCs. As a consequence, several interesting properties of these queueing systems can be describe...

Let {N(t), t ≥ 0} be a nonnegative-integer-valued stochastic process that counts the occurrences of a given event. That is, N(t) is the number of events in the time interval [0, t]. For example, N(t) can be the number of bulb replacements in a lamp that is continuously on, and the dead bulbs are immediately replaced.

In this chapter we summarize the most important notions and facts of probability theory that are necessary for elaboration of our topic. In the present summary, we will apply the more specific mathematical concept and facts—mainly measure theory and analysis—only to a necessary extent while, however, maintaining mathematical precision. Readers inte...

We propose a resource sharing scheme that takes into account the traffic history over several predefined time scales and provides fair resource sharing considering the traffic history. Our concept builds on a simplified version of core-stateless resource sharing, where we only use a few Drop Precedences (DPs). For packet marking we introduce Multi...

The steepest increase property of phase-type (PH) distributions was first proposed in O’Cinneide (1999) and proved in O’Cinneide (1999) and Yao (2002), but since then has received little attention in the research community. In this work we demonstrate that the steepest increase property can be applied for proving previously unknown moment bounds of...

Highly concentrated functions play an important role in many research fields including control system analysis and physics, and they turned out to be the key idea behind inverse Laplace transform methods as well. This paper uses the matrix-exponential family of functions to create highly concentrated functions, whose squared coefficient of variatio...

BuTools 2 is a collection of computational methods that are useful for Markovian and non-Markovian matrix-analytic performance analysis. It consists of various packages. There are packages to obtain, analyze, transform, and minimize discrete and continuous time phase-type (PH) distributions and Markovian arrival processes (MAP); to fit empirical me...

The book is the extended and revised version of the 1st edition and is composed of two main parts: mathematical background and queueing systems with applications. The mathematical background is a self-containing introduction to the stochastic processes of the later studied queueing systems. It starts with a quick introduction to probability theory...

The performance of the uplink of multiuser multiple input multiple output systems depends critically on the receiver architecture and on the quality of the acquired channel state information. A popular approach is to design linear receivers that minimize the mean squared error (MSE) of the received data symbols. Unfortunately, most of the literatur...

In this paper we provide an analysis for fluid polling models with Markov modulated load and gated discipline. The fluid arrival to the stations is modulated by a common continuous-time Markov chain. The fluid is removed at the stations during the service period by a station dependent constant rate.

The performance of service units might depend on various randomly changing environmental effects. It is quite often the case that these effects varies on different time scales. In this paper we consider short and long scale service variability, where the short scale variability affects the instantaneous service speed of the service unit and the lar...

We propose a numerical inverse Laplace transformation method without overshoot which is derived from matrix exponential (ME) distributions with minimal coefficient of variation. We discuss the properties of the method through an integral based interpretation of numerical inverse Laplace transformation methods belonging to the Abate–Whitt framework....

The fitting of Markov arrival processes (MAPs) with the expectation–maximization (EM) algorithm is a computationally demanding task. There are attempts in the literature to reduce the computational complexity by introducing special MAP structures instead of the general representation. Another possibility to improve the efficiency of MAP fitting is...

Processor sharing queues are often used to study the performance of time-sharing systems. In such systems the total service rate ?(m) depends on the number of jobs m present in the system and there is a limit implemented, called the multi-programming level (MPL), on the number of jobs k that can be served simultaneously. Prior work showed that unde...

Timeliness and throughput critical applications require a framework offering predictable temporal characteristics. The best practice for estimating a prediction of the system dynamics relies on benchmarking. Each novel middleware solution needs such an evaluation as part of the development process to assure an appropriate throughput in the future u...

This paper presents a matrix-analytic solution for second-order Markov fluid models (also known as Markov-modulated Brownian motion) with level-dependent behavior. A set of thresholds is given that divide the fluid buffer into homogeneous regimes. The generator matrix of the background Markov chain, the fluid rates (drifts) and the variances can be...

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 i...

Resequencing of customers during the service process results in hard to analyze delay distributions. A set of models with various service and resequencing policies have been analyzed already for memoryless arrival, service and resequencing processes with an intensive use of transform domain descriptions. In case of Markov modulated arrival, service...

Similar to other processes that are modulated by background Markov chains the matrix representation of a transient Markov arrival process is not unique and the use of a convenient unique canonical form is essential for practical computations.
The paper presents a set of 5 Markovian forms which provide a unique and minimal representation for all mem...

We consider equivalence relations for Fluid Stochastic Petri Nets (FSPNs). Based on equivalence relations for Stochastic Petri Nets (SPNs), which are derived from lumpability for Markov Chains, and from lumpability for certain classes of differential equations, we define an equivalence relation for FSPNs. Lumpability for the differential equations...

Blockchain technologies target domains where strict performance guarantees are required and formal Service Level Agreements are commonplace. Designing for performance targets in a trustworthy manner requires performance models; we present a performance characterization approach that addresses the complexity of Blockhain technologies. We apply the m...

The problem of optimizing Markovian models with infinitely or finite but infeasible large state space is considered. In several practically interesting cases the state space of the model is finite and extremely large or infinite, and the transition and decision structures have some regular property which can be exploited for efficient analysis and...

The mean-field limit of stochastic models with exponential and deterministic delays has been proved for the case when the deterministic delays cannot be interrupted by an exponential one.
In this paper we extend the mean-field limit for the class of stochastic models with exponential and deterministic delays where the activities with exponential de...

We revisit earlier attempts for finding matrix exponential (ME) distributions of a given order with low coefficient of variation (\(\hbox {cv}\)). While there is a long standing conjecture that for the first non-trivial order, which is order 3, the \(\hbox {cv}\) cannot be less than 0.200902 but the proof of this conjecture is still missing.
In pre...

This paper presents a more general class of MAP/MAP/1 exhaustive vacation queue, in which the Markov modulated arrival and service processes are dependent. This model class requires the evaluation of the busy period of quasi birth death process with arbitrary initial level, which is a new analysis element.
The model is analyzed by applying matrix a...

There are real life applications (e.g., requests of http sessions in web browsing) with a finite number of events and correlated inter-arrival times. Terminating point processes can be used to model such behavior. Transient Markov arrival processes (TMAPs) are computationally appealing terminating point processes which are terminating versions of M...

Approximating various real-world observations with stochastic processes is an essential modelling step in several fields of applied sciences. In this chapter, we focus on the family of Markov-modulated point processes, and propose some fitting methods. The core of these methods is the computation of the distance between elements of the model family...

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...

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 (...

Stochastic aspects of complex systems require more and more involved analysis approaches. Answering reachability and related analysis questions can often be reduced to steady-state, transient, reward or sensitivity value analysis of stochastic models. In this paper we introduce a configurable stochastic analysis framework which supports the user to...

Since their introduction, properties of Phase Type (PH) distributions have been analyzed and many interesting theoretical results found. Thanks to these results, PH distributions have been profitably used in many modeling contexts where non-exponentially distributed behavior is present. Matrix Exponential (ME) distributions are distributions whose...

In this paper we analyze stable fluid vacation models with exhaustive discipline, in which the fluid source is modulated by a background continuous-time Markov chain and the fluid is removed at constant rate during the service period. Due to the continuous nature of the fluid the state space of the model becomes continuous, which is the major novel...

The minimal Markovian representation of PH distributions is an open research problem, which was actively investigated during the last two decades. We present a numerical method for finding small Markovian representation of PH distributions and investigate the general quality of the method by comparing the size of the obtained representation with th...

In spite of the fact that discrete phase type (DPH) distributions are used almost as often as continuous phase type (CPH) distributions canonical representation is not available for general (cyclic) order 3 DPH distributions yet. In this paper we investigate the canonical representation of DPH distributions of order 3. During the course of this inv...

Using an elegant transform domain iterative operator approach exhaustive fluid vacation models with strictly negative fluid rate during service have been analyzed, recently. Unfortunately, the potential presence of positive fluid rate during service (when the fluid input rate is larger than the fluid service rate) inhibits the use of all previously...

We consider the uplink of a multiuser multiple input multiple output (MU MIMO) system, in which the base station acquires channel state information (CSI) for which the estimation error depends on the resources assigned to the uplink pilot symbols. For this system, we first derive the receiver that minimizes the mean square error (MSE) of the uplink...

Canonical forms of Markovian distributions and processes provide an efficient way of describing these structures by eliminating the redundancy of the general description. Canonical forms of order-2 stationary Markov arrival processes (MAPs) have already been established for both continuous and discrete time. In this paper we prove that the canonica...

The paper reports an effort made for understanding the effect of task delegation policy in a peer-to-peer volunteer computing platform. This effort includes the implementation of a simulation environment and the development of associated analytical models for the analysis of task delegation policies in peer-to-peer computing platforms. Based on the...

There are simple service disciplines where the system time of a tagged customer depends only on the customers arriving in the system earlier (for example first-in-first-out (FIFO)) or later (for example LIFO) than the tagged one. In this paper we consider a single-server queueing system with two infinite queues in which the system time of a tagged...

For pilot sequence based multiple input multiple output (MIMO) channel estimation, the arrangements of pilot symbols, such as the block or comb type arrangement, is known to play an important role. In this paper we compare the performance of block and comb pilot symbol patterns in terms of uplink mean square error (MSE) and spectral efficiency when...