Mathematical modelling and analysis of communication networks: Transient characteristics of traffic processes and models for end-to-end delay and delay- jitter (original) (raw)
Rejoinder on: Queueing models for the analysis of communication systems
TOP, 2014
Queueing models can be used to model and analyze the performance of various subsystems in telecommunication networks, for instance to estimate the packet loss and packet delay in network routers. Since time is usually synchronized, discretetime models come natural. We start this paper with a review of suitable discretetime queueing models for communication systems. We pay special attention to two important characteristics of communication systems. First, traffic usually arrives in bursts, making the classic modeling of the arrival streams by Poisson processes inadequate and requiring the use of more advanced correlated arrival models. Second, different applications have different quality-of-service requirements (packet loss, packet delay, jitter,. . .). Consequently, the common first-come-first-served (FCFS) scheduling is not satisfactory and more elaborate scheduling disciplines are required. Both properties make common memoryless queueing models (M/M/1-type models) inadequate. After the review, we therefore concentrate on a discrete-time queueing analysis with two traffic classes, heterogeneous train arrivals and a priority scheduling discipline, as an example analysis where both time correlation and heterogeneity in the arrival process as well as non-FCFS scheduling are taken into account. Focus is on delay performance measures, such as the mean delay experienced by both types of packets and probability tails of these delays.
Transient Behaviour of Queueing Systems with Correlated Traffic
Performance Evaluation, 1996
In this paper, we present the time-dependent solutions of various stochastic processes associated with a finite Quasi-Birth-Death queueing system. These include the transient queueing solutions, the transient departure and loss intensity processes and certain transient cumulative measures associated with the queueing system. The focus of our study is the effect of the arrival process correlation on the queueing system before it reaches steady-state. With the aid of numerous examples, we investigate the strong relationship between the time scales of variation of the arrival process and those of the transient queueing, loss and departure processes. These time-dependent solutions require the computation of the exponential of the stochastic generator matrix G which may be of very large order. This precludes the use of known techniques to solve the matrix exponential such as the eigenvalue decomposition of G. We present a numerical technique based on the computation of the Laplace Transform of the matrix exponential which may then be numerically inverted to obtain the time-dependent solutions. In this paper, we also propose new QoS metrics based on transient measures and efficient techniques for their computation.
Comments on: Queueing models for the analysis of communication systems
TOP, 2014
In the article, authors restrict their attention to the class of discrete-time single-server queueing models useful for performance analysis of communication systems. This is quite natural because the leader of the author's group professor Herwig Bruneel has the great experience in analysis of discrete-time queues, see his book published twenty years ago and an impressive amount of journal and conference papers published by the authors after this book. As the authors mention, queueing models in discrete-time are very appropriate to describe traffic and congestion phenomena in digital communication systems, since these models reflect in a natural way the synchronous nature of modern transmission systems, whereby time is segmented into intervals ("slots") of fixed length and information packets are transmitted at slot boundaries only, i.e., at a discrete sequence of time points. Because it is well recognized that correlation is a typical feature of the traffic in modern telecommunication networks; authors concentrate their efforts in this paper on analysis of queues with correlation in the arrival process. Taking into account popularity of so-called BMAP (Batch Markov Arrival Process), see , for modeling correlated traffic in continuous-time queueing systems, see, e.g., ; , the authors might use the discrete counterpart of the BMAP, so-called DBMAP (Discrete Batch Markov Arrival Process) for description of the arrival process in their models. However, as authors mention in the text, in the case of the DBMAP usually it is possible to develop
Queueing models for the analysis of communication systems
Top, 2014
Queueing models can be used to model and analyze the performance of various subsystems in telecommunication networks; for instance, to estimate the packet loss and packet delay in network routers. Since time is usually synchronized, discretetime models come natural. We start this paper with a review of suitable discrete-time queueing models for communication systems. We pay special attention to two important characteristics of communication systems. First, traffic usually arrives in bursts, making the classic modeling of the arrival streams by Poisson processes inadequate and requiring the use of more advanced correlated arrival models. Second, different applications have different quality-of-service requirements (packet loss, packet delay, jitter, etc.). Consequently, the common first-come-first-served (FCFS) scheduling is not satisfactory and more elaborate scheduling disciplines are required. Both properties make common memoryless queueing models (M/M/1-type models) inadequate. After the review, we therefore concentrate on a discrete-time queueing analysis with two traffic classes, heterogeneous train arrivals and a priority scheduling discipline, as an example analysis where both time correlation and heterogeneity in the arrival This invited paper is discussed in the comments available
Heavy-traffic asymptotic formulas for the multiclass M^X/G/1 queue
This paper studies the heavy-traffic asymptotics for the multiclass FIFO M X /G/1 queue. We first derive the probability generating function of the joint queue length distribution. Using the probability generating function, we then present heavy-traffic asymptotic formulas for the joint queue length distribution and its joint moments (i.e., the joint queue length moments). These formulas are proved under weaker conditions on the service time distributions, compared to the ones reported in the literature. This fact leads us to conjectures that some of the conditions made in the literature are relaxed.
Telecommunication traffic, queueing models, and subexponential distributions
Queueing Systems - Theory and Applications, 1999
This article reviews various models within the queueing framework which have been suggested for teletraffic data. Such models aim to capture certain stylised features of the data, such as variability of arrival rates, heavy-tailedness of on- and off-periods and long-range dependence in teletraffic transmission. Subexponential distributions constitute a large class of heavy-tailed distributions, and we investigate their (sometimes disastrous) influence within teletraffic models. We demonstrate some of the above effects in an explorative data analysis of Munich Universities’ intranet data.
… and Evaluation of ATM & IP …, 1999
We consider a statistical multiplexer which is modeled as a discrete-time single-server queueing system. Messages consisting of a variable number of fixed-length packets arrive to the multiplexer at the rate of one packet per slot ("train arrivals"), which results in what we call a primary correlation in the packet arrival process. The distribution of the message lengths (in terms of packets) is general. Previous work put forward an analytic technique, based on the use of generating functions and an infinite-dimensional state description, for the analysis of the system in case the numbers of new messages generated by the user population in different slots are independent and identically distributed. Here we adapt this technique to handle the case where the arrival process contains an additional secondary correlation, which results from the fact that the distribution of the number of leading packet arrivals in a slot depends on some environment variable. We assume this environment to have two possible states, each with geometric sojourn times. Closed-form expressions are derived for the mean system contents and the mean packet delay. By means of some numerical examples we illustrate the effect of both primary and secondary correlation on the multiplexer performance.
Simple Approximations for the Batch-Arrival Mx/G/1 Queue
Operations Research, 1990
In this paper we consider the MX/G/I queueing system with batch arrivals. We give simple approximations for the waiting-time probabilities of individual customers. These approximations are checked numerically and they are found to perform very well for a wide variety of batch-size and service-time distributions. Batch-arrival queueing models can be used in 1) many practical situations, such as the analysis of message packetization in data communication systems. In general it is difficult, if not impossible, to find tractable expressions for the waiting-time probabilities of individual customers. It is, therefore, useful to have easily computable approximations for these probabilities. This paper gives such approximations for the single server MX/G/1 model. Exact methods for the computation of the waitingtime distribution in the MX/G/1 queue are discussed in Eikeboom and Tijms (1987), cf. also Chaudhry and Templeton (1983), Neuts (1981) and Tijms (1986). However, these methods apply only for special servicetime distributions and are, in general, not suited for routine calculations in practice. A simple approximation for the tail probabilities of the waiting time was given in Eikeboom and Tijms by using interpolation of the asymptotic expansions for the particular cases of deterministic and exponential services. This approximate approach uses only the first two moments of the service time. This paper presents an alternative approach that uses the actual service-time distribution rather than just its first two moments. This alternative approach starts with the asymptotic expansion of the waitingtime distribution. In Van Ommeren (1988) it is shown that the complementary waiting-time probability of an arbitrary customer in the MX/G/1 queue has an exponentially fast decreasing tail under some mild assumptions. By calculating the decay parameter and the amplitude factor, we get the asymptotic expansion of the waiting-time distribution. Such asymptotic expansions already provide a very powerful tool in practical queueing analysis, cf. Cromie, Chaudhry and Grassman (1979) and Tijms. It turns out that for Subject classifications: Queues, approximations: approximations based on asymptotic analysis. Queues, batch/bulk: approximations.