Scope and purposeQueueing theory is an important subject in computers and operations research. Buffers/queues are used, e.g. in telecommunication networks, to store information that cannot be transmitted instantaneously. The study of the buffer behavior is important since network performance is directly related to it. Queues with a priority scheduling discipline are an important subject in queueing theory. As a result, these type of queues are thoroughly studied in the past, especially in continuous time. In discrete-time queueing models on the other hand, this type of queues is not as widely studied. Discrete-time queueing models are suitable for the performance evaluation of Asynchronous Transfer Mode (ATM) switches. In ATM, different types of traffic need different Quality of Service (QoS) standards. The delay characteristics of delay-sensitive traffic (e.g., voice) are more stringent than those of delay-insensitive traffic (e.g., data). We can thus give priority to delay-sensitive traffic over delay-insensitive traffic, thus trying to reduce the * Corresponding author 1 delay of the delay-sensitive traffic. This paper studies the impact of a priority scheduling on the buffer characteristics. AbstractIn this paper, we consider a discrete-time queueing system with head-of-line (HOL) priority. First, we will give some general results on a GI-1-1 queue with priority scheduling. In particular, we will derive expressions for the Probability Generating Function of the system contents and the cell delay. Some performance measures (such as mean, variance and approximate tail distributions) of these quantities will be derived, and used to illustrate the impact and significance of priority scheduling in an ATM output queueing switch.
Abstract:We develop power series approximations for a discrete-time queueing system with two parallel queues and one processor. If both queues are non-empty, a customer of queue 1 is served with probability β and a customer of queue 2 is served with probability 1 − β. If one of the queues is empty, a customer of the other queue is served with probability 1. We first describe the generating function U (z 1 , z 2 ) of the stationary queue lengths in terms of a functional equation, and show how to solve this using the theory of boundary value problems. Then, we propose to use the same functional equation to obtain a power series for U (z 1 , z 2 ) in β. The first coefficient of this power series corresponds to the priority case β = 0, which allows for an explicit solution. All higher coefficients are expressed in terms of the priority case. Accurate approximations for the mean stationary queue lengths are obtained from combining truncated power series and Padé approximation.
In this paper, we investigate a simplified head-of-the-line with priority jumps (HOL-PJ) scheduling discipline. Therefore, we consider a discrete-time single-server queueing system with two priority queues of infinite capacity and with a newly introduced HOL-PJ priority scheme. We derive expressions for the probability generating function of the system contents and the packet delay. Some performance measures (such as mean and variance) of these quantities are derived and are used to illustrate the impact and significance of the HOL-PJ priority scheduling discipline in an output queueing switch. We compare this dynamic priority scheduling discipline with a FIFO scheduling and a static priority scheduling (HOL) and we investigate the influence of the different parameters of the simplified HOL-PJ scheduling discipline.
In the past, many researchers have analysed queueing models with batch service. In such models, the server typically postpones service until the number of present customers reaches a service threshold, whereupon service is initiated of a batch consisting of several customers. In addition, correlation in the customer arrival process has been studied for many different queueing models. However, correlated arrivals in batch-service models has attracted only modest attention. In this paper, we analyse a discrete-time D-BMAP/G l,c /1 queue, whereby the service time of a batch is dependent on the number of customers within it. In addition, a timing mechanism is included, to avoid that customers suffer excessive waiting times because their service is postponed until the amount of customers reaches the service threshold. We deduce various useful performance measures related to the buffer content and we investigate the impact of the traffic parameters on the system performance through some numerical examples. We show that correlation merely has a small impact on the service threshold that minimizes the mean system content, and consequently, that the existing results of the corresponding independent system can be applied to determine a near-optimal service threshold policy, which is an important finding for practitioners. On the other hand, we demonstrate that for other purposes, such as performance evaluation and buffer management, correlation in the arrival process cannot be ignored, a conclusion that runs along the same lines as in queueing models without batch service.
This paper considers a simple discrete-time queueing model with two types (classes) of customers (types 1 and 2) each having their own dedicated server (server A and B resp.) New customers enter the system according to a general independent arrival process, i.e., the total numbers of arrivals during consecutive time slots are i.i.d. random variables with arbitrary distribution. Service times are deterministically equal to 1 slot each. The system uses a "global FCFS" service discipline, i.e., all arriving customers are accommodated in one single FCFS queue, regardless of their types. As a consequence of the "global FCFS" rule, customers of one type may be blocked by customers of the other type, in that they may be unable to reach their dedicated server even at times when this server is idle, i.e., the system is basically non-workconserving. One major aim of the paper is to estimate the negative impact of this phenomenon on the queueing performance of the system, in terms of the achievable throughput, the system occupancy, the idle probability of each server and the delay. As it is clear that customers of different types hinder each other more as they tend to arrive in the system more clustered according to class, the degree of "class clustering" in the arrival process is explicitly modelled in the paper and its very direct impact on the performance measures is revealed. The motivation of our work are systems where this kind of blocking is encountered, such as input-queueing network switches or road splits.
In this paper, we consider several discrete-time priority queues with priority jumps. In a priority scheduling scheme with priority jumps, real-time and non-real-time packets arrive in separate queues, i.e., the high-and low-priority queue respectively. In order to deal with possibly excessive delays however, non-real-time packets in the low-priority queue can in the course of time jump to the high-priority queue. These packets are then treated in the high-priority queue as if they were real-time packets. Many criteria can be used to decide when packets of the low-priority queue jump to the high-priority queue. Some criteria have already been introduced in the literature, and we first overview this literature. Secondly, we propose and analyse a new priority scheme with priority jumps. Finally, we extensively compare all cited schemes. The schemes all differ in their jumping mechanism, based on a certain jumping criterion, and thus all have a different performance. We show the pros and cons of each jumping scheme.
In this paper, we develop a simple method to approximate the transient behavior of queueing systems. In particular, it is shown how singularity analysis of a known generating function of a transient sequence of some performance measure leads to an approximation of this sequence. To illustrate our approach, several specific transient sequences are investigated in detail. By means of some numerical examples, we validate our approximations and demonstrate the usefulness of the technique.
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