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

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

Modelling and Optimisation of GSM and UMTS Radio Access Networks

The size and complexity of mobile communication networks have increased in the last years making network management a very complicated task. GSM/EDGE Radio Access Network (GERAN) systems are in a mature state now. Thus, non-optimal performance does not come from typical network start-up problems, but, more likely, from the mismatching between traffic, network or propagation models used for network planning, and their real counterparts. Such differences cause network congestion problems both in signalling and data channels. With the aim of maximising the financial benefits on their mature networks, operators do not solve anymore congestion problems by adding new radio resources, as they usually did. Alternatively, two main strategies can be adopted, a) a better assignment of radio resources through a re-planning approach, and/or b) the automatic configuration (optimisation, in a wide sense) of network parameters. Both techniques aim to adapt the network to the actual traffic and propagation conditions. Moreover, a new heterogenous scenario, where several services and Radio Access Technologies (RATs) coexist in the same area, is now common, causing new unbalanced traffic scenarios and congestion problems. In this thesis, several optimisation and modelling methods are proposed to solve congestion problems in data and signalling channels for single- and multi-RAT scenarios

Energieeffiziente und rechtzeitige Ereignismeldung mittels drahtloser Sensornetze

This thesis investigates the suitability of state-of-the-art protocols for large-scale and long-term environmental event monitoring using wireless sensor networks based on the application scenario of early forest fire detection. By suitable combination of energy-efficient protocol mechanisms a novel communication protocol, referred to as cross-layer message-merging protocol (XLMMP), is developed. Qualitative and quantitative protocol analyses are carried out to confirm that XLMMP is particularly suitable for this application area. The quantitative analysis is mainly based on finite-source retrial queues with multiple unreliable servers. While this queueing model is widely applicable in various research areas even beyond communication networks, this thesis is the first to determine the distribution of the response time in this model. The model evaluation is mainly carried out using Markovian analysis and the method of phases. The obtained quantitative results show that XLMMP is a feasible basis to design scalable wireless sensor networks that (1) may comprise hundreds of thousands of tiny sensor nodes with reduced node complexity, (2) are suitable to monitor an area of tens of square kilometers, (3) achieve a lifetime of several years. The deduced quantifiable relationships between key network parameters — e.g., node size, node density, size of the monitored area, aspired lifetime, and the maximum end-to-end communication delay — enable application-specific optimization of the protocol

Monotonicity and error bounds for networks of Erlang loss queues

Networks of Erlang loss queues naturally arise when modelling finite communication systems without delays, among which, most notably\ud (i) classical circuit switch telephone networks (loss networks) and\ud (ii) present-day wireless mobile networks.\ud \ud Performance measures of interest such as loss probabilities or throughputs can be obtained from the steady state distribution. However, while this steady state distribution has a closed product form expression in the first case (loss networks), it has not in the second case due to blocked (and lost) handovers. Product form approximations are therefore suggested. These approximations are obtained by a combined modification of both the state space (by a hyper cubic expansion) and the transition rates (by extra redial rates). It will be shown that these product form approximations lead to\ud \ud - secure upper bounds for loss probabilities and\ud - analytic error bounds for the accuracy of the approximation for various performance measures.\ud \ud The proofs of these results rely upon both monotonicity results and an analytic error bound method as based on Markov reward theory. This combination and its technicalities are of interest by themselves. The technical conditions are worked out and verified for two specific applications:\ud \ud - pure loss networks as under (i)\ud - GSM-networks with fixed channel allocation as under (ii).\ud \ud The results are of practical interest for computational simplifications and, particularly, to guarantee blocking probabilities not to exceed a given threshold such as for network dimensioning.\u

Stability Problems for Stochastic Models: Theory and Applications II

Most papers published in this Special Issue of Mathematics are written by the participants of the XXXVI International Seminar on Stability Problems for Stochastic Models, 21­25 June, 2021, Petrozavodsk, Russia. The scope of the seminar embraces the following topics: Limit theorems and stability problems; Asymptotic theory of stochastic processes; Stable distributions and processes; Asymptotic statistics; Discrete probability models; Characterization of probability distributions; Insurance and financial mathematics; Applied statistics; Queueing theory; and other fields. This Special Issue contains 12 papers by specialists who represent 6 countries: Belarus, France, Hungary, India, Italy, and Russia

Statistical distributions for service times

Queueing models have been used extensively in the design of call centres. In particular, a queueing model will be used to describe a help desk which is a form of a call centre. The design of the queueing model involves modelling the arrival an service processes of the system.Conventionally, the arrival process is assumed to be Poisson and service times are assumed to be exponentially distributed. But it has been proposed that practically these are seldom the case. Past research reveals that the log-normal distribution can be used to model the service times in call centres. Also, services may involve stages/tasks before completion. This motivates the use of a phase-type distribution to model the underlying stages of service.This research work focuses on developing statistical models for the overall service times and the service times by job types in a particular help desk. The assumption of exponential service times was investigated and a log-normal distribution was fitted to service times of this help desk. Each stage of the service in this help desk was modelled as a phase in the phase-type distribution.Results from the analysis carried out in this work confirmed the irrelevance of the assumption of exponential service times to this help desk and it was apparent that log-normal distributions provided a reasonable fit to the service times. A phase-type distribution with three phases fitted the overall service times and the service times of administrative and miscellaneous jobs very well. For the service times of e-mail and network jobs, a phase-type distribution with two phases served as a good model.Finally, log-normal models of service times in this help desk were approximated using an order three phase-type distribution

Queues in series with blocking

This piece of work describes a hospital's Critical Care Unit and uses different mathematical techniques to model the behaviour seen there. The main factor that is included in these models is the problem of bed blocking in the Unit. Blocking is defined as patients who are well enough not to be in the Critical Care Unit, but remain there, for any number of reasons. These patients are using up an expensive and limited resource. The mathematical techniques that the models are built on are extensively reviewed and analysed. These are the Coxian Phase Type Distribution and Networks of Queues with Blocking Equations. Both techniques are described in detail and their distributions analysed under different circumstances. The final chapter shows how the two distributions can be used to model a complex situation such as the one found in the Critical Care Unit. The models are tested and compared. Finally, the models are tested under a number of 'what if scenarios to predict the effect of changing certain factors on the actual Unit