A discrete time queueing analysis of ATM systems with phase-type distributed talk spurts traffic sources

In multi-state life insurance, an adequate balance between analytic tractability, computational efficiency, and statistical flexibility is of great importance. This might explain the popularity of Markov chain modelling, where matrix analytic methods allow for a comprehensive treatment. Unfortunately, Markov chain modelling is unable to capture duration effects, so this paper presents aggregate Markov models as an alternative. Aggregate Markov models retain most of the analytical tractability of Markov chains, yet are non-Markovian and thus more flexible. Based on an explicit characterization of the fundamental martingales, matrix representations of the expected accumulated cash flows and corresponding prospective reserves are derived for duration-dependent payments with and without incidental policyholder behaviour. Throughout, special attention is given to a semi-Markovian case. Finally, the methods and results are illustrated in a numerical example.

In this paper we analyze a simple model to study the convenience of grouping ATM cells into frames before transmission over satellite channels for onboard switching. We consider a switch where the input/output ports are mapped to the spot beams of the satellite, with several Bernoulli users within each spot beam. We obtain exact results for the delay in the terrestrial multiplexing system, based on the grouping of states of a Markov chain into equivalent cost classes. Then we obtain an approximate solution for the onboard switch delay, based on a search algorithm of the most probable states in an infinitesate Markov chain. By using this kind of framing we reduce the number of switching operation per unit time, so we can use smaller and lighter onboard switches. However, although we also reduce the contention for accessing the uplink channel, this reduction cannot always compensate for the additional delay we incur during the framing process.

We study a single server queue with finite buffer that may represent an ATM multiplexer. The input traffic is a superposition of independent heterogeneous batch Markovian arrival processes. Service durations are supposed to be exponential. We derive the loss probability and the delay and jitter distributions for each individual process, as well as for the superposed one. We show that there is a significant difference between performances of individual and global flows, justifying our work. The approach is based on martingale techniques. >

Life history data are generally incomplete. Usually, they do not cover for each individual in the study the entire life span or the life segment of interest. If data are collected retrospectively, observation ends at interview date, and no information is available on events and experiences after the date. Data collected prospectively are incomplete because events and other experiences are recorded during a limited period of time only. To deal with data limitations, models are introduced. The model that is considered in this chapter describes life histories. The model is based on the premise that life histories are realisations of a continuous-time Markov process. A Markov process is a stochastic process that describes a system with multiple states and transitions between the states. The time at which a transition occurs is random but the distribution of the time to transition is known. In the continuous-time Markov process, the transition time has an exponential distribution. The rate of transition out of the current state (exit rate) is the parameter of the exponential distribution. It depends on the current state only and is independent of the history of the stochastic process. In a system with multiple states, an individual who leaves the current state may enter one of several states. In competing risks models, states in the state space are viewed as competing destinations and transition rates are destination-specific. The Markov process is a first-order process: the destination state depends on the current state only and is independent of states occupied previously.

We consider a single-hop, random-access, wireless sensor network (WSN) performing sequential distributed detection under a bandwidth constraint. At each time slot, the sensor nodes will make their decisions on whether the observed event is happening. The sensor censoring strategy is applied such that only the local decisions equal to one (i.e., the event happens) will be sent to the fusion center (FC). Because only one transmission channel between the FC and the sensor nodes is assumed, if two or more local decisions are sent at a time slot, a packet collision happens. We design a sequential probability ratio test (SPRT) at the FC that is aware of the collisions. The performance measures of the collision-aware SPRT are analyzed based on a Markov chain. In addition, we derive the asymptotic relative efficiency of the collision-aware SPRT relative to another SPRT. Numerical results show some interesting characteristics of the collision-aware SPRT. For example, the collision-aware SPRT prefers the scenario that many local decisions are likely to be sent per time slot.

We consider single-server queue, which models a telecommunication node. Packets arrive in the system according to the Markovian arrival process (MAP). The packets bring some amount of information, which is stored in a continuous-type unlimited buffer before it will be transmitted out of the system. We consider the system with fluid manner of service, which means that the server takes the information from the buffer with constant speed. Our aim is to derive the probability distribution of the real waiting time in the system.

This paper considers single-server queues with several customer classes. Arrivals of customers are governed by the underlying continuous-time Markov chain with finite states. The distribution of the amount of work brought into the system on arrival is assumed to be general, which may differ with different classes. Further, the service speed depends on the state of the underlying Markov chain. We first show that given such a queue, we can construct the corresponding new queue with constant service speed by means of a change of time scale, and the time-average quantities of interest in the original queue are given in terms of those in the new queue. Next we characterize the joint distribution of the length of a busy period and the number of customers served during the busy period in the original queue. Finally, assuming the FIFO service discipline, we derive the Laplace--Stieltjes transform of the actual waiting time distribution in the original queue.

Classical control technology has by and large focused on continuous-time systems???-???including their digitized and sampled equivalents. The unequivocal success in this arena has resulted in a broadening of interests and in explorations of the application of control concepts to other problems, even those that are not readily amenable to the techniques of traditional control. This chapter focuses on one such topic: discrete-event systems. Whereas control technology is mostly concerned with systems with internal dynamics that can be mediated by continuous-valued inputs, discrete-event systems (DES) exhibit dynamics that evolve in accordance with external events???-???the state of the system changes only when an event occurs. Many problems in communication networks, manufacturing systems, transportation and traffic, and numerous other domains can be seen as DES applications. In the first case, for example, events of interest can be the arrival of packets of information at a node in the network. (Connections between DES and communication networks are further elaborated in Chapter 19.) Modeling approaches for DES include state machines and automata, Markov chains, and timed models using event clocks. Simple examples are shown for each, drawn from computer systems with on, off, and failed states and single-server queues. Some of these models are also discussed in Chapter 7 which deals with systems that combine discrete-event and continuous-time dynamics. This chapter also discusses the topic of optimization of DESs: how control parameters of a DES (for example, the mean service time for jobs in a queue) can be selected to optimize some performance measure. To use gradient-based optimization methods for discrete-event systems, gradient information must be estimated; since the systems and their representations are not continuous, gradients cannot be analytically calculated. Stochastic approximation algorithms are an effective option and can even allow DES optimization to be performed on-line, while the system is operating.

In an opportunistic spectrum sharing (OSS) wireless network there are two types of users: primary users and secondary users. The primary users own the license for the system bandwidth, while the secondary users opportunistically share the spectrum resources with the primary users. When a secondary user detects a call arrival from a primary user in its current channel, the secondary user leaves the channel immediately and switches to an idle channel, if one is available, to continue the call. Otherwise, the secondary user is preempted. Call arrivals from primary users and secondary users in the OSS system are modeled by a Markovian arrival process (MAP) which captures correlation in the aggregate arrival process consisting of the two types of call arrivals. We derive the stationary probability vector using matrix-analytic methods and obtain expressions for a set of key performance measures. We present numerical results for a sample scenario.

We consider a single server retrial queueing system in which arrivals occur according to a Markovian arrival process. An arriving customer finding an idle server will get service immediately; otherwise the customer enters into a retrial orbit. The system is subject to catastrophic failures at which time all customers in the system (the one in service as well as the customers, if any, in the orbit) are lost. The system undergoes a repair and after completion of a repair the server in the system will be available for service. The customers in the orbit try to reach the server by sending a signal at random times and get service if the server is idle at those times. While customers are waiting in the orbit, they may become impatient and leave the system after a random amount of time. Any arrival finding the server unavailable (due to busy or under repair) and cannot enter the orbit due to the buffer being full (in the case of finite buffer only) is considered lost. Under exponential assumption for all except for the arrivals, the queueing model (for both finite and infinite orbit sizes) is studied using matrix-analytic methods and the qualitative nature of the model is brought out through some illustrative numerical examples.

We study a discrete-time, single-server queue with partial buffer sharing. There are two priority classes of jobs. Though class 1 jobs in the queue have higher priority for the next service than any of class 2 jobs, class 2 jobs are allowed to occupy their own part of buffer when the shared part of buffer is full. We characterize a bursty arrival process using bursts which consist of the same class of jobs. Once the first job of a burst arrives at the queue, the successive jobs will arrive on every time slot until the last job of the burst arrives. The numbers of jobs of a burst and the inter-arrival times of bursts are assumed to be i.i.d., respectively, and the service time is assumed to be equal to one slot. This model targets the buffer management to meet the quality of service requirments of different traffic types as video, voice and data in ATM multiplexer. In particular, class 1 jobs may correspond to cells with the strict delay requirments. On the other hand, class 2 jobs may correspond to cells with the strict cell loss requirments. We propose an efficient numerical method to exactly obtain the job loss probability, the waiting time distribution and the mean queue length. Some numerical examples are also given.

EAPRAD: A MAC protocol for Enhancing Access Probability and Reducing Access Delay in VANETs

In ATM networks, the time axis is divided into equal intervals called time slots, so most references assume that the Markov chain makes a transition from state i to state j in the boundary of time slots in view of Markov-modulated Bernoulli process (MMBP). This paper considers a more general situation, i.e., the Markov chain makes a transition from state i to state j in the boundary of frames whose length are an integral multiple of length of one time slot. Under this MMBP model, the paper develops a four-dimensional discrete-time Markov chain, and obtains results of the performance analysis of leaky bucket algorithms.

We propose a system approach to the asymptotic analysis of stochastic systems in the scheme of series with averaging and diffusion approximation. Stochastic systems are defined by Markov processes with locally independent increments in a Euclidean space with random switchings that are described by jump Markov and semi-Markov processes. We use the asymptotic analysis of Markov and semi-Markov random evolutions. We construct the diffusion approximation using the asymptotic decomposition of generating operators and solutions of problems of singular perturbation for reducibly inverse operators.

This paper presents a unifying algorithmic analysis for a general class of single server queueing systems with a state dependent Markovian input process and a phase-type service time distribution including single server queues with random and quasirandom input. Using regenerative analysis we develop numerically stable and efficient recursion schemes to compute the state probabilities. The computation of the waiting times is based on the state probabilities.