Analysis of MAP/PH/1 Queueing System with Degrading Service Rate and Phase Type Vacation

Degradation of services arises in practice due to a variety of reasons including wear-and-tear of machinery and fatigue. In this paper, we look at MAP/PH/1-type queueing models in which degradation is introduced. There are several ways to incorporate degradation into a service system. Here, we model the degradation in the form of the service rate declining (i.e., the service rate decreases with the number of services offered) until the degradation is addressed. The service rate is reset to the original rate either after a fixed number of services is offered or when the server becomes idle. We look at two models. In the first, we assume that the degradation is instantaneously fixed, and in the second model, there is a random time that is needed to address the degradation issue. These models are analyzed in steady state using the classical matrix-analytic methods. Illustrative numerical examples are provided. Comparisons of both the models are drawn.

Yaglom limit for stochastic fluid models

In this paper we analyse the limiting conditional distribution (Yaglom limit) for stochastic fluid models (SFMs), a key class of models in the theory of matrix-analytic methods. So far, only transient and stationary analyses of SFMs have been considered in the literature. The limiting conditional distribution gives useful insights into what happens when the process has been evolving for a long time, given that its busy period has not ended yet. We derive expressions for the Yaglom limit in terms of the singularity˜$s^*$ such that the key matrix of the SFM, ${\boldsymbol{\Psi}}(s)$, is finite (exists) for all $s\geq s^*$ and infinite for $s<s^*$. We show the uniqueness of the Yaglom limit and illustrate the application of the theory with simple examples.

Ruin problems for epidemic insurance

The paper discusses the risk of ruin in insurance coverage of an epidemic in a closed population. The model studied is an extended susceptible–infective–removed (SIR) epidemic model built by Lefèvre and Simon (Methodology Comput. Appl. Prob.22, 2020) as a block-structured Markov process. A fluid component is then introduced to describe the premium amounts received and the care costs reimbursed by the insurance. Our interest is in the risk of collapse of the corresponding reserves of the company. The use of matrix-analytic methods allows us to determine the distribution of ruin time, the probability of ruin, and the final amount of reserves. The case where the reserves are subjected to a Brownian noise is also studied. Finally, some of the results obtained are illustrated for two particular standard SIR epidemic models.

Queuing-Inventory Models with MAP Demands and Random Replenishment Opportunities

Combining the study of queuing with inventory is very common and such systems are referred to as queuing-inventory systems in the literature. These systems occur naturally in practice and have been studied extensively in the literature. The inventory systems considered in the literature generally include (s,S)-type. However, in this paper we look at opportunistic-type inventory replenishment in which there is an independent point process that is used to model events that are called opportunistic for replenishing inventory. When an opportunity (to replenish) occurs, a probabilistic rule that depends on the inventory level is used to determine whether to avail it or not. Assuming that the customers arrive according to a Markovian arrival process, the demands for inventory occur in batches of varying size, the demands require random service times that are modeled using a continuous-time phase-type distribution, and the point process for the opportunistic replenishment is a Poisson process, we apply matrix-analytic methods to study two of such models. In one of the models, the customers are lost when at arrivals there is no inventory and in the other model, the customers can enter into the system even if the inventory is zero but the server has to be busy at that moment. However, the customers are lost at arrivals when the server is idle with zero inventory or at service completion epochs that leave the inventory to be zero. Illustrative numerical examples are presented, and some possible future work is highlighted.

Optimizing a Multi-State Cold-Standby System with Multiple Vacations in the Repair and Loss of Units

A complex multi-state redundant system with preventive maintenance subject to multiple events is considered. The online unit can undergo several types of failure: both internal and those provoked by external shocks. Multiple degradation levels are assumed as both internal and external. Degradation levels are observed by random inspections and, if they are major, the unit goes to a repair facility where preventive maintenance is carried out. This repair facility is composed of a single repairperson governed by a multiple vacation policy. This policy is set up according to the operational number of units. Two types of task can be performed by the repairperson, corrective repair and preventive maintenance. The times embedded in the system are phase type distributed and the model is built by using Markovian Arrival Processes with marked arrivals. Multiple performance measures besides the transient and stationary distribution are worked out through matrix-analytic methods. This methodology enables us to express the main results and the global development in a matrix-algorithmic form. To optimize the model, costs and rewards are included. A numerical example shows the versatility of the model.

Supply chain stochastic modeling

Στην παρούσα διατριβή προτείνονται αναλυτικά μοντέλα για την αριθμητική εκτίμηση της απόδοσης εφοδιαστικών συστημάτων που παρουσιάζουν πρακτικό ενδιαφέρον. Τα μοντέλα χρησιμοποιούνται για μια εκτεταμένη αριθμητική μελέτη των αντίστοιχων συστημάτων με στόχο την καλύτερη κατανόηση της συμπεριφοράς τους, και όπου είναι δυνατό, την εξαγωγή συμπερασμάτων χρήσιμων στην υποστήριξη λήψης αποφάσεων. Οι γενικές παραδοχές περιλαμβάνουν χαμένη πλεονάζουσα ζήτηση, στοχαστικούς χρόνους αναμονής, και στοχαστική εξωτερική ζήτηση. Οι αβεβαιότητες μοντελοποιούνται με χρήση εκθετικής κατανομής, ή όπου κρίνεται σκόπιμο με πιο γενικές κατανομές όπως η σύνθετη Poisson και η Coxian με δύο φάσεις. Τα συστήματα μοντελοποιούνται σαν Μαρκοβιανές αλυσίδες συνεχούς χρόνου – διακριτού χώρου και εφαρμόζονται τεχνικές βασισμένες στην ανάλυση της δομής των πινάκων μεταπήδησης (Matrix analytic methods). Προτείνεται ένας αλγόριθμος για την κατασκευή του πίνακα γεννήτορα για οποιονδήποτε συνδυασμό παραμέτρων, την κατασκευή και επίλυση του αντίστοιχου συστήματος γραμμικών εξισώσεων, και τον αλγοριθμικό υπολογισμό μέτρων απόδοσης μέσω του διανύσματος στάσιμων πιθανοτήτων. Η ορθότητα των μαθηματικών μοντέλων επιβεβαιώνεται με χρήση προσομοίωσης. Τα συμπεράσματα βασίζονται στην αριθμητική επίλυση ενός πλήθους σεναρίων, ενώ οι βέλτιστες πολιτικές προκύπτουν μετά από διεξοδική μελέτη όλων των πιθανών πολιτικών που βρίσκονται εντός των προκαθορισμένων ορίων. Εξετάζονται τρία διαφορετικά συστήματα: Ένα σειριακό, οριζόντια ολοκληρωμένο σύστημα ώθησης-έλξης (Push-Pull) με ανεξάρτητη μεταφορά; ένα σειριακό σύστημα τριών σταδίων όπου η διαχείριση των αποθεμάτων γίνεται από τον προμηθευτή (Vendor Managed Inventory - VMI); και ένα σύστημα έλξης (pull) τριών στοιβάδων με δενδροειδή δομή και ανεξάρτητους σταθμούς μεταφοράς.

SIR epidemics with stages of infection

In this paper we are concerned with a stochastic model for the spread of an epidemic in a closed homogeneously mixing population when an infective can go through several stages of infection before being removed. The transitions between stages are governed by either a Markov process or a semi-Markov process. An infective of any stage makes contacts amongst the population at the points of a Poisson process. Our main purpose is to derive the distribution of the final epidemic size and severity, as well as an approximation by branching, using simple matrix analytic methods. Some illustrations are given, including a model with treatment discussed by Gani (2006).

FINITE TWO LAYERED QUEUEING SYSTEMS

We study layered queueing systems comprised two interlacing finite M/M/• type queues, where users of each layer are the servers of the other layer. Examples can be found in file sharing programs, SETI@home project, etc. Let Li denote the number of users in layer i, i=1, 2. We consider the following operating modes: (i) All users present in layer i join forces together to form a single server for the users in layer j (j≠i), with overall service rate μjLi (that changes dynamically as a function of the state of layer i). (ii) Each of the users present in layer i individually acts as a server for the users in layer j, with service rate μj.These operating modes lead to three different models which we analyze by formulating them as finite level-dependent quasi birth-and-death processes. We derive a procedure based on Matrix Analytic methods to derive the steady state probabilities of the two dimensional system state. Numerical examples, including mean queue sizes, mean waiting times, covariances, and loss probabilities, are presented. The models are compared and their differences are discussed.