A Discrete-Time Queueing System With Different Types Of Displacement

Abstract

The performance prediction in communication, jobs processing in computers, etc, are always influenced by the customers behavior and the provision of this additional information will be useful in upgrading the service. Our paper is concerned under a loss and trigger protocol where each customer has a service requirement which may depend on the arrival of a positive or negative customer. In our study we consider customer expulsions and different types of customers displacements taking into account or not its past time of service. The main purpose of this work is to spread the discrete-time queueing theory about expulsions and displacement. We provide a unified way to handle the combinations of different conditions such as positive arrival, negative arrival, trigger movements, past time in service, etc. INTRODUCTION An investigation of discrete-time queueing system is important due to their application to slotted systems such as communication systems and other related areas and therefore it has been found more appropriate than their continuous-time counterpart. The study of discrete-time queues was initiated by (Meisling 1958; Birdsall et al. 1962; Powell et al. 1967). Reference works and more detailed applications on discrete-time queueing theory include the monographs (Bruneel and Kim 1993; Takagi 1993). Further, a detailed treatment regarding this subject can be found in a two-volume book on applied probability (Hunter 1983). A rapid increase in the literature on queueing system with negative arrivals are analyzed extensively in continuous-time models but not so much in discrete-time. The arrival of a negative customer to a queueing system causes one ordinary customer to be removed or killed if any is present. The pioneer work on discrete-time considering negative arrivals without retrials can be found in (Atencia and Moreno 2004; Atencia and Moreno 2005) where the authors considered several killing strategies for the negative customers. For a survey on this topics the authors refer to (Gelenbe and Label 1998) and (Artalejo 2000), for applications on engineering to (Chao et al. 1999) and for application in communication networks we refer to (Harrison et al. 2000) and (Park et al. 2009). In many real problems it is also interesting to consider the movement of jobs, customers, etc., from one place to another. This mechanism is called a synchronized or triggered motion, see for example (Artalejo 2000) and (Gelenbe and Label 1998) and concerning with inverse order discipline we refer to (Pechinkin and Svischeva 2004), (Pechinkin and Shorgin 2008) and (Cascone et al. 2011). For service interruptions with expulsions we refer to (Atencia and Pechinkin 2012) and (Atencia et al. 2013). THE MATHEMATICAL MODEL We consider a discrete-time queueing system where the time axis is segmented into a sequence of equal time intervals (called slots). It is assumed that all queueing activities (arrivals, departures and retrials) occur at the slot boundaries, and therefore, they may take place at the same time. That is why we must detail the order in which the arrivals and departures occur in case of simultaneity in a discrete-time system. Basically, there are two rules: (i) If an arrival takes precedence over a departure, it is identified with Late Arrival System (LAS) (see Figure 1(a)); (ii) if a departure takes precedence over an arrival, it is recognized by Early Arrival System (EAS) (see Figure 1(b)). The former case is also known as Arrival First (AF) policy and the latter as Departure First (DF) policy. For more details on these and related concepts, see (Gravey and Hebuterne 1992) and (Hunter 1983). Let us note, that for mathematical convenience, we will follow the second policy, that is the departures occur at the moment immediately before the slot boundaries, Proceedings 27th European Conference on Modelling and Simulation ©ECMS Webjorn Rekdalsbakken, Robin T. Bye, Houxiang Zhang (Editors) ISBN: 978-0-9564944-6-7 / ISBN: 978-0-9564944-7-4 (CD) but arrivals occurs at the moment immediately after the slot boundaries.