Multi-threshold Control of the BMAP/SM/1/K Queue with Group Services

We consider a finite capacity queue in which arrivals occur according to a batch Markovian arrival process (BMAP). The customers are served in groups of varying sizes. The services are governed by a controlled semi-Markovian process according to a multithreshold strategy. We perform the steady-state analysis of this model by computing (a) the queue length distributions at departure and arbitrary epochs, (b) the Laplace-Stieltjes transform of the sojourn time distribution of an admitted customer, and (c) some selected system performance measures. An optimization problem of interest is presented and some numerical examples are illustrated

Оптимальное многопороговое управление потоком в системе массового обслуживания G/MSP/1 с MAP-потоком сбоев

Рассмотрена система массового обслуживания G/MSP/1 с марковским потоком катастрофических сбоев и управляемым потоком запросов. Найдено стационарное распределение вероятностей состояний вложенной цепи Маркова. Разработан алгоритм нахождения оптимальной многопороговой стратегии управления потоком запросов. Приведен численный пример.G/MSP/1 queuing system with a MAP-input of disasters, causing all customers to leave the system instantaneously, and controlled input of customers is considered. The stationary queue length distribution has been derived and the optimal multithreshold strategy for customers input control has been determined. Obtained results are illustrated by a numerical example.Розглянуто систему масового обслуговування G/MSP/1 з марковським потоком катастрофічних збоїв та керованим потоком заявок. Знайдено стаціонарний розподіл станів укладеного ланцюга Маркова. Розроблено алгоритм пошуку оптимальної багатопорогової стратегії керування потоком заявок. Наведено чисельний приклад

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

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

Stochastic Clearing Systems With Markovian Inputs: Performance Evaluation and Optimal Policies

This thesis studies the stochastic clearing systems which are characterized by a non-decreasing stochastic input process {Y(t), t > 0}, where Y(t) is the cumulative quantity entering the system in [0, t], and an output mechanism that intermittently and instantaneously clears the system. Examples of such systems can be found in shipment consolidation, inventory backlog, lot sizing, shuttle bus dispatch, bulk service queues, and other stochastic service and storage systems. In our model, the input process is governed by an underlying discrete-time Markov chain such that, the distribution of the input in any given period depends on the underlying state in that period. The outstanding inputs in the system are recorded in strings to keep track of the ages, i.e., the time elapsed since their arrival, of each input. The decision of when to clear the system depends on a \clearing policy" which itself depends on the input quantities, ages, and the underlying state. Clearing the system will incur a fixed cost and a variable cost depending on the quantities cleared; a penalty is charged to the outstanding inputs in every period, and such penalty is non-decreasing in both the quantities and the ages of the inputs. We model the system as a tree structured Markov chain with Markovian input processes and evaluate the clearing policies with respect to the expected total costs over a finite horizon, the expected total discounted cost over an infinite horizon, as well as the expected average total cost per period over an infinite horizon. Relying on theories of Markov Decision Processes and stochastic dynamic programming, we then proceed to show some properties unique to the optimal clearing policies, and prove that a state-dependent threshold policy can be optimal under special conditions. We develop algorithms or heuristics to evaluate a given clearing policy and find the optimal clearing policy. We also use Matrix Analytic Methods to evaluate a given clearing policy and develop an efficient heuristic to find near-optimal clearing policies. Finally, we conduct extensive numerical analyses to verify the correctness, complexity, and optimality gap of our algorithms and heuristics. Our numerical examples successfully demonstrate the analytical results we proved

Safety and Reliability - Safe Societies in a Changing World

The contributions cover a wide range of methodologies and application areas for safety and reliability that contribute to safe societies in a changing world. These methodologies and applications include: - foundations of risk and reliability assessment and management - mathematical methods in reliability and safety - risk assessment - risk management - system reliability - uncertainty analysis - digitalization and big data - prognostics and system health management - occupational safety - accident and incident modeling - maintenance modeling and applications - simulation for safety and reliability analysis - dynamic risk and barrier management - organizational factors and safety culture - human factors and human reliability - resilience engineering - structural reliability - natural hazards - security - economic analysis in risk managemen

Optimal control for a BMAP/SM/1 queue with MAP-input of disasters and two operation modes

A single-server queueing system with a batch Markovian arrival process (BMAP) and MAP-input of disasters causing all customers to leave the system instantaneously is considered. The system has two operation modes, which depend on the current queue length. The embedded and arbitrary time stationary queue length distribution has been derived and the optimal control threshold strategy has been determined