Discrete-time Queueing System Research Articles

DELAY ANALYSIS OF A MULTIMEDIA ATM MULTIPLEXER WITH HOMOGENEOUS ARRIVALS

In this paper we present a delay analysis of an ATM multiplexer handling multimedia traffic of two classes: real time class, e.g., live audio and video and nonreal time class, e.g., file transfers. Packets arrive into the multiplexer in batches of general size and are output one at a time. The arrivals are homogeneous in the sense that either the arriving batch is all class-1 packets or all class-2 packets. The multiplexer is modeled as a priority discrete time queueing system with batch arrivals and geometric service. The homogeneity of the arrivals is reflected by the addition of an indicator random variable into the model. The analysis is comprehensive in that it covers the three main delay measures: the unfinished work, the busy period and the waiting time. For these measures, probability generating functions are derived, from which moments of arbitrary orders can be obtained. However, only the first moments are obtained in the article to produce expected values. Presented at the end is a special case when the arriving batches are of Poissonian size. Some numerical values are obtained and plotted for this case.

Complete analytic and computational analyses of the discrete-time bulk-arrival infinite-server system: GIX/ Geom/∞

In this paper, we consider the discrete-time queueing system with bulk arrivals, generally distributed interbatch times, geometrically distributed service times, and infinite number of servers: the GI X / Geom/∞ queue. Under the assumptions of early arrival system (EAS) and late arrival system (LAS), we derive the system size distributions at two different epochs—prearrival and random. In addition, simple relations among the binomial moments of the steady-state system-size distributions of the two systems are derived. Special cases, such as the GI/ Geom/∞ or GI/ D/∞ queues, are also presented. Some numerical results are included at the end.

Discrete-time multiserver queues with geometric service times

In this paper, a discrete-time multiserver queueing system with infinite buffer size and general independent arrivals is considered. The service times of packets are assumed to be independent and identically distributed according to a geometric distribution. Each packet gets service from only one server. In the paper, the behavior of the queueing system is studied analytically by means of a generating-functions approach. This results in closed-form expressions for the mean values, the variances and the tail distributions of the system contents and the packet delay. Some numerical examples are given to illustrate the analysis. Scope and purpose Discrete-time queueing models are widely used to study the behavior of various types of telecommunication and computer systems. Most of the existing studies of discrete-time multiserver queueing models assume that the service times of the customers are constant. Recently, however, there has been an increased interest in discrete-time models with non-deterministic service times, due to the more and more complicated and irregular service mechanisms in nowadays telecommunication networks. In view of this, this paper focuses on a discrete-time queueing system with multiple servers and geometrically distributed service times. We show that the queueing model can be analyzed by means of a generating-functions approach. The results of the analysis are important to evaluate performance measures such as packet loss and delay in communication and computer networks.

An extended combinatorial analysis framework for discrete-time queueing systems with general sources

The paper considers a general class of discrete time systems with batch arrivals and departures. Such models appear frequently in the teletraffic analysis of computer and communications networks. Our arrival models are assumed to be quite general. They could be independent and identically distributed (i.i.d.) in successive slots, periodic, Markovian, or described by the moving average time-series model, etc. Our solution framework is novel and unifying and it uses a combination of multidimensional generating functions and combinatorial analysis utilizing extensions of classical ballot theorems. In general, we provide an explicit analytical expression as an infinite sum to obtain the system stationary probability distribution avoiding classical root-finding methods, matrix analytical methodologies, and spectral decomposition approaches. We provide a number of analytical and numerical examples including: i.i.d. models with Poisson and Binomial arrivals; multiserver queueing systems fed by Markovian sources; queues fed with a discrete moving average source of the first and second order; an i.i.d. discrete Pareto batch arrival model. Closed-form analytical expressions are obtained for the stationary distribution of the system queue lengths and numerical examples are also provided when appropriate.

Performance analysis of a single-server ATM queue with a priority scheduling

In this paper, we consider a discrete-time queueing system with head-of-line 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. Scope and purpose Queueing 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 delay of the delay-sensitive traffic. This paper studies the impact of a priority scheduling on the buffer characteristics.

Performance analysis of a multi-rate mobile in a wireless packet network

A cell in wireless packet networks with a variety of mobiles servicing multi-rate traffic is considered. The network system incorporates distributed-queueing request update multiple access for demand assignment, and multi-code CDMA protocol for multiple transmission in parallel without self-interference. One multi-rate mobile in the network is modelled as a discrete-time queueing system with a set-up time and batch service where the service batch size is a random variable depending on the present queue size. The probability distribution of the queue size at steady state, and the waiting time distribution are obtained. Mobiles in are a cell classified into several classes according to their transmitter capability, traffic characteristic and QoS requirement, and then the set-up time distribution of a mobile is obtained in each class. Finally, the analytic results are shown to match quite well with simulations.

On the Exact Analysis of a Discrete-Time Queueing System with Autoregressive Inputs

In this paper, we provide an exact analysis of a discrete-time queueing system driven by a discrete autoregressive model of order 1 (DAR(1)) characterized by an arbitrary marginal batch size distribution and a correlation coefficient. Closed-form expressions for the probability generating function and mean queue length are derived. It is shown that the system performance is quite sensitive to the correlation of the arrival process. In addition, a comparison with traditional Markovian processes shows that arrival processes of DAR(1) type exhibit larger queue length as compared with the traditional Markovian processes when the marginal densities and correlation coefficients are matched.

NONPARAMETRIC STATISTICAL ANALYSIS OF DISCRETE-TIME QUEUES, WITH APPLICATIONS TO ATM TELETRAFFIC DATA

In this paper nonparametric statistical analysis of a discrete-time queueing system is considered. Estimation of performance measures of the system is studied. The attention is first focused on the estimation of the waiting time probability distribution, as well as of functionals of interest (mean waiting time, variance of the waiting time, etc.). The approach is based on the estimation of the corresponding generating function. Attention is paid to the estimation of the probability of a “long delay”, in view of its importance for applications. Results for possibly unstable models are also obtained. Finally, an application to ATM teletraffic data is provided.

Analysis of a discrete-time GI–G–1 queueing model subjected to bursty interruptions

In this contribution, we investigate a discrete-time single-server queue subjected to server interruptions generated by a 2-state Markov process. The model under consideration assumes customers with multiple-slot service times, which leads to the introduction of two different service strategies depending on whether service of an interrupted customer continues or restarts after an interruption. For both alternatives, we establish expressions for the steady-state probability generating functions of the buffer contents, the unfinished work and the customer delay in terms of the effective customer service times. From these results, closed-form expressions for various performance measures, such as the moments of these quantities, can be established. After dealing with some stability issues, we illustrate the impact of both service strategies on the buffer performance with some numerical examples. Scope and purpose Discrete-time queueing theory is distinguished from its more developed continuous-time counterpart by the synchronous nature of its service. Time is divided into fixed-length slots and service of a customer typically takes an integer number of slots. Discrete-time queueing models are particularly appropriate to describe the various queueing related phenomena in digital computer and communication systems including mobile and B-ISDN networks based on asynchronous transfer mode (ATM) technology, due to the packetized nature of these transport protocols. This work focuses on a discrete-time queueing system with server interruptions. Interruptions are an abstraction of temporary server unavailability. The latter can be caused by temporary occupancy of the server by other customers — regulated by a scheduling discipline such as round robin or priority scheduling — or temporary failure of the server (e.g. due to transmission errors, maintenance or repair). Typically, interruptions occurring in these systems have a bursty nature, i.e., they occur in bursts rather than being randomly spread over time. The presented model allows to study the effects of the burstiness of the server interruptions on various performance measures such as mean queue contents and mean customer delay. Two different operation strategies are hereby considered: after an interruption, service may either continue where it left off or it may completely restart.

A performance analysis of a discrete-time queueing system with server interruption for modeling wireless ATM multiplexer

In this paper, we study the effect of channel errors on wireless ATM. The salient characteristic of the wireless channel is that it is time varying. This variability is modeled by a Markov chain with two states corresponding to high and low error states, respectively. The channel is modeled as being synchronous with the basic time unit being the slot. The transitions from high to low error states, and vice versa occur at the slot boundaries. It is assumed that no transmission over the channel is possible during the high-error state. The information source feeding into the channel is modeled as the superposition of independent sources. Each source is controlled by a two-state Markov chain which operates in synchronism with, but statistically independent of the channel. In the source Off state, no data is emitted while, in the On state, the source generates a random number of packets in a slot each of which fits into a channel slot. It is assumed that the traffic generated by all the sources will form a single global queue which will be served by the two-state Markov channel. A discrete-time queuing analysis derives the probability generating function (PGF) of the queue length under the assumption of an infinite buffer. From the PGF, we determine mean queue length and mean delay. Finally, numerical results are presented to demonstrate the effect of wireless channel error characteristics on performance.

The waiting time analysis of a discrete-time queue with arrivals as a discrete autoregressive process of order 1

We consider a discrete-time queueing system with the discrete autoregressive process of order 1 (DAR(1)) as an input process and obtain the actual waiting time distribution and the virtual waiting time distribution. As shown in the analysis, our approach provides a natural numerical algorithm to compute the waiting time distributions, based on the theory of the GI/G/1 queue, and consequently we can easily investigate the effect of the parameters of the DAR(1) on the waiting time distributions. We also derive a simple approximation of the asymptotic decay rate of the tail probabilities for the virtual waiting time in the heavy traffic case.

The waiting time analysis of a discrete-time queue with arrivals as a discrete autoregressive process of order 1

We consider a discrete-time queueing system with the discrete autoregressive process of order 1 (DAR(1)) as an input process and obtain the actual waiting time distribution and the virtual waiting time distribution. As shown in the analysis, our approach provides a natural numerical algorithm to compute the waiting time distributions, based on the theory of the GI/G/1 queue, and consequently we can easily investigate the effect of the parameters of the DAR(1) on the waiting time distributions. We also derive a simple approximation of the asymptotic decay rate of the tail probabilities for the virtual waiting time in the heavy traffic case.

Delay characteristics in discrete-time GI-G-1 queues with non-preemptive priority queueing discipline

Priority scheduling for packets is becoming a hot topic, as attempts are being made to integrate voice services in existing data networks. In this paper, we consider a discrete-time queueing system with head-of-line (HOL) non-preemptive priority scheduling. Two classes of traffic will be considered, i.e., high-priority and low-priority traffic, which both generate variable-length packets. We will derive expressions for the probability generating function of the packet delay of the high-priority traffic and the low-priority traffic. From these, some performance measures (such as the mean value) will be derived. These will be used to illustrate the significance of priority scheduling and will be applied to an output queueing switch.

The delay distribution of a type k customer in a first-come-first-served MMAP[K]/PH[K]/1 queue

This paper presents an algorithmic procedure to calculate the delay distribution of a type k customer in a first-come-first-served (FCFS) discrete-time queueing system with multiple types of customers, where each type has different service requirements (the MMAP[K]/PH[K]/1 queue). First, we develop a procedure, using matrix analytical methods, to handle arrival processes that do not allow batch arrivals to occur. Next, we show that this technique can be generalized to arrival processes that do allow batch arrivals to occur. We end the paper by presenting some numerical examples.

The delay distribution of a type k customer in a first-come-first-served MMAP[K]/PH[K]/1 queue

This paper presents an algorithmic procedure to calculate the delay distribution of a type k customer in a first-come-first-served (FCFS) discrete-time queueing system with multiple types of customers, where each type has different service requirements (the MMAP[K]/PH[K]/1 queue). First, we develop a procedure, using matrix analytical methods, to handle arrival processes that do not allow batch arrivals to occur. Next, we show that this technique can be generalized to arrival processes that do allow batch arrivals to occur. We end the paper by presenting some numerical examples.

A note on the discretization of Little's result

By considering discrete-time queueing systems as special cases of continuous-time queueing systems, we derive a discrete-time equivalent of Little's result. Our result is general in the sense that no assumptions are made regarding the exact details of arrival and departure processes within a discrete-time unit.

Analysis of a Discrete-Time Queueing System with a Single Server and Heterogeneous Markovian Arrivals

We consider a discrete-time queueing system with a single deterministic server, heterogeneous Markovian arrivals and finite capacity. Most existing techniques model the queueing system using a direct bivariate Markov chain which requires a state space that grows rapidly as the number of customer types increases. In this paper, we define renewal cycles in terms of the input process and model the system occupancy level on each renewal cycle using a one-dimensional Markov chain. We derive the exact joint steady-state probability distribution of both states of input and system occupancy with a considerably reduced state space, which leads to the efficient calculation of overall/individual performance measures such as loss probability and average delay.

Analysis of a Discrete-Time Queueing System with Timed Vacations

We consider a discrete-time GI-G-1 queueing system with server vacations. Vacations occur whenever the queue becomes empty or whenever a timer expires. When the timer expires one of the following four actions are considered: the server completes transmission of the present packet before leaving for a vacation, the service immediately leaves for a vacation and the interrupted packet's service is either continued, repeated or resampled and repeated after the vacation. Using a probability generating functions approach, we derive various performance measures such as moments of the buffer contents at various time epochs in equilibrium and moments of the packet delay in equilibrium. By means of an example, we then compare the operation modes under consideration.

Randomly Interrupted GI-G-1 Queues: Service Strategies and Stability Issues

We consider a discrete-time queueing system subjected to random server interruptions. As customers arriving in the queue require generally distributed service times, the server can be interrupted during a customer's service. Therefore, nine different service strategies are proposed and analyzed using a probability generating functions approach. Performance measures under investigation include moments of steady-state buffer contents at random slot boundaries in equilibrium and moments of the customer delay. In particular we focus on the stability requirements for the strategies under consideration.

Performance evaluation of CAI and RAI transmission modes in a GI-G-1 queue

In this paper, a discrete-time single-server queue is considered with an uncorrelated message arrival process and an uncorrelated server interruption process. The length of the messages is modeled as a series of positive i.i.d. random variables and two operation modes are considered. In continue after interruption mode (CAI) the processing of an interrupted message resumes with the next packet of this message. In repeat after interruption mode (RAI), the complete message is reprocessed after a server interruption. Using a generating-functions approach, explicit expressions for the mean and variance of buffer occupancy and message delay are derived for both operation modes. For illustration purposes, a discrete-time Pois-Geo-1 system is then evaluated. Scope and purpose Discrete-time queueing theory has gained a lot of interest recently to describe queueing related phenomena in an environment where time is slotted, such as various digital computer and communication systems including mobile and B-ISDN networks based on asynchronous transfer mode technology. This work focuses on a discrete-time queueing system with server interruptions, which can occur when several users have to share a common resource, or when this resource is temporarily unavailable due to external causes such as maintenance or system failures. The contribution of this paper mainly lies in the model that is considered, the solution method that is applied, as well as the results that are generated. First, we want to introduce a model with an interrupted service process that allows us to compare an environment where interrupted messages must be retransmitted with the situation where this is not the case. Second, we will show that a generating-functions approach is extremely suitable for solving such a queueing model. And finally, we want to establish closed-form formulas in terms of the system parameters that enable us to assess the system performance for the two transmission modes that are considered.