Decomposition of General Tandem Queueing Networks with MMPP Input

Most of the proposed self-similar traffic models are asymptotic in nature. Hence, they are less effective in queueing-based performance evaluation when the buffer sizes are small. In this paper, we propose a short range dependent (SRD) process modelling by a generalized variance-based Markovian fitting to provide effective queueing-based performance measures when buffer sizes are small. The proposed method is to match the variance of the exact second-order self-similar processes. The fitting procedure determines the related parameters in an exact and straightforward way. The resultant traffic model essentially consists of a superposition of several two-state Markov-modulated Poisson processes (MMPPs) with distinct modulating parameters. We present how well the resultant MMPP could emulate the variance of original self-similar traffic in the range of the specified time scale, and could provide more accurate bounds for the queueing-based performance measures, namely tail probability, mean waiting time and loss probability. Numerical results show that both the second-order statistics and queueing-based performance measures when buffer capacity is small are more accurate than that of the variance-based fitting where the modulating parameters of each superposed two-state MMPP are equal. We then investigate the relationship between time scale and the number of superposed two-state MMPPs. We found that when the performance measures pertaining to larger time scales are not better than that of smaller ones, we need to increase the number of superposed two-state MMPPs to maintain the accurate and reliable queueing-based performance measures. We then conclude from the extensive numerical examples that an exact second-order self-similar traffic can be well represented by the proposed model.

The characteristics and dynamics of wide area network (WAN) traffic are discussed. The real-time experimental prototype system for monitoring and analyzing traffic allows the monitoring of different parts of the network for extended periods of time without a need for storing and processing the voluminous amounts of data that are captured during the monitoring periods. The results of the analysis show that the inter-site traffic follows a certain pattern that depends on the time of the day and the day of the week. The traffic between two sites is found to be reasonably symmetric, except for those sites designated as backup sites. WAN traffic is highly dynamic and up to 25:1 peak-to-low ratios are observed over a one-week monitoring period. Using the coefficient of variation as a measure for the traffic burstiness, a 1.5 degree of burstiness is maintained during the working hours. This moderate degree of burstiness can be attributed to two factors: the smoothing effect that results from aggregating a number of bursty sources, and the use of 1.5 Mb/s T1 links between sites that usually reshapes the traffic and changes its characteristics (i.e., reduces its burstiness). A two-state Markov modulated Poisson process (MMPP) model is developed to represent the correlation between packets, as well as a benchmark for the design and performance tuning of future wide area networks. >

It is proved that artificial neural networks are adequate tools for obtaining superposition models of multiplexed ATM traffic sources. It is shown how complex mathematical models can be replaced by a modular, adaptive and parallel architecture capable of developing complicated algorithms. In particular, the authors approximate a superposition of individual ATM sources by a two-state Markov modulated Poisson process (MMPP). This approximation is performed using a neural system, matching four statistics of the aggregate traffic to those of the MMPP. The approach is evaluated using numerical examples, showing that it is adequate for estimating delay attributes and cell-level congestion.

This paper presents a queuing analysis model of a PC-based software router supporting IPv6-IPv4 translation for residential gateway. The proposed models are M/G/1/K or MMPP-2/G/1/K by arrival process of the software PC router. M/G/1/K is a model of normal traffic and MMPP-2/G/1/K is a model of burst traffic. In M/G/1/K, the arriving process is assumed to be a Poisson process, which is independent and identically distributed. In MMPP-2/G/1/K, the arriving process is assumed to be two-state Markov Modulated Poisson Process (MMPP) which is changed from one state to another state with intensity. The service time distribution is general distribution and the service discipline of the server is processor sharing. Also, the total number of packets that can be processed at one time is limited to K. We obtain performance metrics of PC-based software router for residential gateway such as system sojourn time blocking probability and throughput based on the proposed model. Compared to other models, our model is simpler and it is easier to estimate model parameters. Validation results show that the model estimates the performance of the target system.

Three different approximation techniques are examined. The performance models studied differ primarily in the manner in which the superposition of the voice sources (i.e., the arrival process) is modeled. The first approach models the superimposed voice sources as a renewal process, and performance calculations are based only on the first two moments of the renewal process. The second approach is based on modeling the superimposed voice sources as a Markov modulated Poisson process (MMPP). The choice of parameters for the MMPP attempts to capture aspects of the arrival process in a more intuitive manner than previously proposed approaches for determining the MMPP parameters and is shown to compute loss more accurately. Finally, a fluid flow approximation for computing packet loss is evaluated. For all three approaches, a unifying example, the case of multiplexing voice sources over a T1-rate link is considered. The main conclusion is that both the MMPP model and the fluid flow approximation can provide accurate loss predictions for parameter ranges of practical interest.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The statistics of cell loss at both burst and cell levels are studied by simulation based on the configuration of the Singapore One ATM network. A virtual connection through the Singapore One network is modelled as a tandem queueing network to which mixtures of homogeneous discrete-time burst–silence traffic with different burst lengths are offered. Attention is focused on a designated traffic stream as it goes through the tandem queueing network. In the simulation, the authors have not introduced the ‘independence assumption’ between the adjacent queues, i.e. the actual tandem queueing network is used to obtain the performance measures for the designated traffic stream. Cell loss probability and high-order cell loss distributions such as cell loss-free run distribution and cell loss pattern distribution are obtained. These results provide valuable information for the estimation of end-to-end overall quality of service (QoS), design of the cell policing mechanism and flow control schemes in the ATM networks.

On reliability prediction of repairable redundant digital structures : J. C. Laprie. IEEE Trans. Reliab.R-25 (4), 256 (October 1976)

We study the performance of a statistical multiplexer whose inputs consist of a superposition of packetized voice sources and data. The performance analysis predicts voice packet delay distributions, which usually have a stringent requirement, as well as data packet delay distributions. The superposition is approximated by a correlated Markov modulated Poisson process (MMPP), which is chosen such that several of its statistical characteristics identically match those of the superposition. Matrix analytic methods are then used to evaluate system performance measures. In particular, we obtain moments of voice and data delay distributions and queue length distributions. We also obtain Laplace-Stieitjes transforms of the voice and data packet delay distributions, which are numerically inverted to evaluate tails of delay distributions. It is shown how the matrix analytic methodology can incorporate practical system considerations such as finite buffers and a class of overload control mechanisms discussed in the literature. Comparisons with simulation show the methods to be accurate. The numerical results for the tails of the voice packet delay distribution show the dramatic effect of traffic variability and correlations on performance.

The Markovian Arrival Process (MAP), which contains the Markov Modulated Poisson Process (MMPP) and the Phase-Type (PH) renewal processes as special cases, is a convenient traffic model for use in the performance analysis of Asynchronous Transfer Mode (ATM) networks. In ATM networks, packets are of fixed length and the buffering memory in switching nodes is limited to a finite numberK of cells. These motivate us to study the MAP/D/1/K queue. We present an algorithm to compute the stationary virtual waiting time distribution for the MAP/D/1/K queue via rational approximations for the deterministic service time distribution in transform domain. These approximations include the well-known Erlang distributions and the Padé approximations that we propose. Using these approximations, the solution for the queueing system is shown to reduce to the solution of a linear differential equation with suitable boundary conditions. The proposed algorithm has a computational complexity independent of the queue storage capacityK. We show through numerical examples that, the idea of using Padé approximations for the MAP/D/1/K queue can yield very high accuracy with tractable computational load even in the case of large queue capacities.

A note on approximate iterative solution of open tandem networks with blocking

We introduce a versatile class of point processes on the real line, which are closely related to finite-state Markov processes. Many relevant probability distributions, moment and correlation formulas are given in forms which are computationally tractable. Several point processes, such as renewal processes of phase type, Markov-modulated Poisson processes and certain semi-Markov point processes appear as particular cases. The treatment of a substantial number of existing probability models can be generalized in a systematic manner to arrival processes of the type discussed in this paper. Several qualitative features of point processes, such as certain types of fluctuations, grouping, interruptions and the inhibition of arrivals by bunch inputs can be modelled in a way which remains computationally tractable.

Studies the traffic shaping issues for VBR video services in ATM networks with stress put on the source clock frequency recovery aspect. The authors propose a "three-level traffic shaper" (TLTS) which is able to limit the minimum as well as the maximum cell rate of input VBR traffic. For a VBR video source modeled after a two-state Markov modulated Poisson process, they determine, as performance measures, the cell loss probability, the underflow probability, and the maximum delay that a cell can experience in the proposed TLTS. Then they apply the TLTS for source clock frequency recovery in VBR video services. In extending the existing CBR-based synchronous residual time stamp (SRTS) method for VBR video services, it is necessary to restrict the minimum, as well as the maximum, number of cells generated in a fixed period of time to ensure the transmission of timing information, and the proposed TLTS is tailored to meet this necessity by adopting an optimal dimensioning procedure based on the genetic algorithm (GA). Finally the authors provide some numerical examples to demonstrate the optimal dimensioning procedure of TLTS and the related performances of statistical multiplexing and source clock frequency recovery.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

6 Dependence notions in reliability theory

Performance-analysis of a switching system using priority-based dynamic capacity allocation is presented. The system provides connectivity between a number of point-to-multipoint communication access clusters. The scheduling of traffic transmission at each access cluster is based on a priority scheme with priority given to real-time (rt) traffic over nonreal-time (nrt) traffic. An analytical model is developed to evaluate the covariance functions of both rt and nrt traffic arriving at the output ports. Aggregate traffic arrival streams are approximated to 2-state Markov-modulated Poisson processes (MMPPs) by matching their statistical characteristics. Analytical and simulation results on performance of an example 4 /spl times/ 4 switch for different traffic loads are discussed.

On a tandem queueing network with breakdowns