Error bounds for last-column-block-augmented truncations of block-structured Markov chains

This paper discusses the error estimation of the last-column-block-augmented northwest-corner truncation (LC-block-augmented truncation, for short) of block-structured Markov chains (BSMCs) in continuous time. We first derive upper bounds for the absolute difference between the time-averaged functionals of a BSMC and its LC-block-augmented truncation, under the assumption that the BSMC satisfies the general $f$-modulated drift condition. We then establish computable bounds for a special case where the BSMC is exponentially ergodic. To derive such computable bounds for the general case, we propose a method that reduces BSMCs to be exponentially ergodic. We also apply the obtained bounds to level-dependent quasi-birth-and-death processes (LD-QBDs), and discuss the properties of the bounds through the numerical results on an M/M/$s$ retrial queue, which is a representative example of LD-QBDs. Finally, we present computable perturbation bounds for the stationary distribution vectors of BSMCs.Comment: This version has fixed the bugs for the positions of Figures 1 through

Retrial queue MMPP/M/N under heavy load condition

In the paper, a multi-server retrial queueing system with MMPP arrivals is considered. The service and retrial times are exponentially distributed. The two-dimension stochastic process of number of calls in the orbit and states of service unit is analyzed. The system of Kolmogorov differential equations is composed. The matrix form of the equations in steady-state regime for partial characteristic functions is written. The method of asymptotic analysis under the heavy load condition for its solving is proposed. It is proved that the asymptotic characteristic function of the number of calls in the orbit has the gamma distribution with obtained parameters. Some numerical examples of comparison asymptotic and simulate distributions are presented

Unreliable Retrial Queues in a Random Environment

This dissertation investigates stability conditions and approximate steady-state performance measures for unreliable, single-server retrial queues operating in a randomly evolving environment. In such systems, arriving customers that find the server busy or failed join a retrial queue from which they attempt to regain access to the server at random intervals. Such models are useful for the performance evaluation of communications and computer networks which are characterized by time-varying arrival, service and failure rates. To model this time-varying behavior, we study systems whose parameters are modulated by a finite Markov process. Two distinct cases are analyzed. The first considers systems with Markov-modulated arrival, service, retrial, failure and repair rates assuming all interevent and service times are exponentially distributed. The joint process of the orbit size, environment state, and server status is shown to be a tri-layered, level-dependent quasi-birth-and-death (LDQBD) process, and we provide a necessary and sufficient condition for the positive recurrence of LDQBDs using classical techniques. Moreover, we apply efficient numerical algorithms, designed to exploit the matrix-geometric structure of the model, to compute the approximate steady-state orbit size distribution and mean congestion and delay measures. The second case assumes that customers bring generally distributed service requirements while all other processes are identical to the first case. We show that the joint process of orbit size, environment state and server status is a level-dependent, M/G/1-type stochastic process. By employing regenerative theory, and exploiting the M/G/1-type structure, we derive a necessary and sufficient condition for stability of the system. Finally, for the exponential model, we illustrate how the main results may be used to simultaneously select mean time customers spend in orbit, subject to bound and stability constraints

Approximate Analysis of an Unreliable M/M/2 Retrial Queue

This thesis considers the performance evaluation of an M/M/2 retrial queue for which both servers are subject to active and idle breakdowns. Customers may abandon service requests if they are blocked from service upon arrival, or if their service is interrupted by a server failure. Customers choosing to remain in the system enter a retrial orbit for a random amount of time before attempting to re-access an available server. We assume that each server has its own dedicated repair person, and repairs begin immediately following a failure. Interfailure times, repair times and times between retrials are exponentially distributed, and all processes are assumed to be mutually independent. Modeling the number of customers in the orbit and status of the servers as a continuous-time Markov chain, we employ a phase-merging algorithm to approximately analyze the limiting behavior. Subsequently, we derive approximate expressions for several congestion and delay measures. Using a benchmark simulation model, we assess the accuracy of the approximations and show that, when the algorithm assumptions are met, the approximation procedure yields favorable results. However, as the rate of abandonment for blocked arrivals decreases, the performance declines while the results are insensitive to the rate of abandonment of customers preempted by a server failure

MAP/PH/1 systems with group service: performance analysis under different admission strategies

2015 - 2016Recent advances in wireless communication networks led to possibility of multi-rate transmission of information. The queueing theory represents a valid tool to study how the performances of such communication systems can be improved, and to give proper solutions. Modeling a multi-rate transmission system, in terms of queueing theory, means that a particular discipline has to be considered: a group of requests from users can be processed simultaneously in parallel and processing of the whole group is supposed finished if processing of all individual requests belonging to this group is over. In order to model this typology of telecommunication systems, some particular assumption can be made on arrivals, which occur by a Markovian arrival process, and on service time and length of admission period, which are regulated by phase type distributions. Thus, in this thesis MAP/PH/1 queueing systems have been considered, with and without retrial to take into account all possible behaviours of the customers. The main goal of the research activity presented in this work is to introduce novel admission strategies for the described systems, in order to give a major contribute to the current performance analysys, in particular as regard the choice of the optimal length of admission period and optimal size of the groups. Dynamics of such systems are described by multidimensional Markov chains. Ergodicity condition for these Markov chains have been derived, stationary probability distribution of the states have been computed, formulas for the main performance measures of the system have been attained. Essential advantages of the proposed customer’s service disciplines have been numerically illustrated. [edited by author]I recenti progressi ottenuti per le reti di comunicazione wireless, permettono la trasmissione multi-frequenza delle informazioni. La teoria delle code rappresenta un valido strumento per studiare come le performance di tali sistemi di comunicazione possano essere migliorate, e individuare opportune soluzioni. In termini di teoria delle code, modellare un sistema di trasmissione multi-frequenza significa considerare una determinata disciplina: un gruppo di richieste da parte di utenti possono essere processate simultaneamente in parallelo, e il processo dell’intero gruppo risulta completato se tutte le richieste appartenenti a tale gruppo sono espletate. Al fine di modellare tale tipologia di sistemi di telecomunicazione, si possono definire particolari assunzioni sugli arrivi, determinati da processi di arrivo Markoviani, e sul tempo di servizio e lunghezza del periodo di ammissione, regolati da distribuzioni di tipo a fasi. Pertanto, in tale lavoro di tesi sono stati considerati sistemi a coda di tipo MAP/PH/1, con e senza retrial per considerare tutti i possibili comportamenti degli utenti. Il principale obiettivo dell’attivita` di ricerca presentata in tale lavoro `e introdurre nuove strategie di ammissione per i sistemi descritti, al fine di fornire un maggior contributo alle attuali analisi sulle performance, in particolare relativamente alla scelta della lunghezza ottimale del periodo di ammissione e la dimensione ottimale dei gruppi. Le dinamiche di tali sistemi sono descritte da catene di Markov multidimensionali. `E stata ricavata la condizione di ergodicit`a per tali catene di Markov, `e stata calcolata la distribuzione delle probabilita` stazionarie degli stati, e sono state ottenute le formule per le misure dei principali parametri prestazionali del sistema. I principali vantaggi delle discipline di servizio proposte sono state illustrate numericamente. [a cura dell'autore]XXIX n.s

A Retrial Queueing Model With Thresholds and Phase Type Retrial Times

There is an extensive literature on retrial queueing models. While a majority of the literature on retrial queueing models focuses on the retrial times to be exponentially distributed (so as to keep the state space to be of a reasonable size), a few papers deal with nonexponential retrial times but with some additional restrictions such as constant retrial rate, only the customer at the head of the retrial queue will attempt to capture a free server, 2-state phase type distribution, and finite retrial orbit. Generally, the retrial queueing models are analyzed as level-dependent queues and hence one has to use some type of a truncation method in performing the analysis of the model. In this paper we study a retrial queueing model with threshold-type policy for orbiting customers in the context of nonexponential retrial times. Using matrix-analytic methods we analyze the model and compare with the classical retrial queueing model through a few illustrative numerical examples. We also compare numerically our threshold retrial queueing model with a previously published retrial queueing model that uses a truncation method

Markovian arrivals in stochastic modelling: a survey and some new results

This paper aims to provide a comprehensive review on Markovian arrival processes (MAPs), which constitute a rich class of point processes used extensively in stochastic modelling. Our starting point is the versatile process introduced by Neuts (1979) which, under some simplified notation, was coined as the batch Markovian arrival process (BMAP). On the one hand, a general point process can be approximated by appropriate MAPs and, on the other hand, the MAPs provide a versatile, yet tractable option for modelling a bursty flow by preserving the Markovian formalism. While a number of well-known arrival processes are subsumed under a BMAP as special cases, the literature also shows generalizations to model arrival streams with marks, nonhomogeneous settings or even spatial arrivals. We survey on the main aspects of the BMAP, discuss on some of its variants and generalizations, and give a few new results in the context of a recent state-dependent extension.Peer Reviewe