Two-state Markov Modulated Poisson Process Research Articles

A bivariate two-state Markov modulated Poisson process for failure modeling

Motivated by a real failure dataset in a two-dimensional context, this paper presents an extension of the Markov modulated Poisson process (MMPP) to two dimensions. The one-dimensional MMPP has been proposed for the modeling of dependent and non-exponential inter-failure times (in contexts as queuing, risk or reliability, among others). The novel two-dimensional MMPP allows for dependence between the two sequences of inter-failure times, while at the same time preserves the MMPP properties, marginally. The generalization is based on the Marshall–Olkin exponential distribution. Inference is undertaken for the new model through a method combining a matching moments approach with an Approximate Bayesian Computation (ABC) algorithm. The performance of the method is shown on simulated and real datasets representing times and distances covered between consecutive failures in a public transport company. For the real dataset, some quantities of importance associated with the reliability of the system are estimated as the probabilities and expected number of failures at different times and distances covered by trains until the occurrence of a failure.

An Aggregated Process-Based Traffic Generator for Network Performance Evaluation

Nowadays, it is desirable to generate traffic which can reflect realistic network traffic environment for performance evaluation of network equipments. Existing traffic generation solutions mainly include special test equipments, software traffic generators and field programmable gate array (FPGA)-based traffic generators. However, special test equipments are generally too expensive, software traffic generators could not achieve high data rates and FPGA-based traffic generators mostly lack flexibility. This paper presents a novel traffic generation solution according to an aggregated process-based model to overcome the weakness of above methods. The traffic generator can generate real-time Poisson, two-state Markov-modulated Poisson process (MMPP-2) and self-similar traffic by hardware. The main structure of the traffic generator is presented and statistical properties of the generated traffic have been evaluated. Experiment results indicate that the proposed traffic generation solution can achieve better performance of the generated traffic compared with existing FPGA-based traffic generators while the required date rates can be up to Gbps line rate.

BURSE: A Bursty and Self-Similar Workload Generator for Cloud Computing

As two of the most important characteristics of workloads, burstiness and self-similarity are gaining more and more attention. Workload generation, which is a key technique for performance analysis and simulations, has also attracted an increasing interest in cloud community in recent years. Though a large number of methods for synthetically generating bursty or self-similar workloads have been proposed in the literature, none of them can deal with workload generation with both of the two characteristics. In this paper, a configurable and intelligible synthetic generator (BURSE) is proposed for bursty and self-similar workloads in cloud computing based on a superposition of two-state Markov Modulated Poisson Processes (MMPP2s). The proposed generator can produce workloads with both specified intension of burstiness and self-similarity. Detailed experimental evaluation demonstrates the accuracy, robustness and good applicability of BURSE.

Modeling Cross Correlation in Three-Moment Four-Parameter Decomposition Approximation of Queueing Networks

In two-moment decomposition approximations of queueing networks, the arrival process is modeled as a renewal process, and each station is approximated as a GI/G/1 queue whose mean waiting time is approximated based on the first two moments of the interarrival times and the service times. The departure process is also approximated as a renewal process even though the autocorrelation of this process may significantly affect the performance of the subsequent queue depending on the traffic intensity. When the departure process is split into substreams by Markovian random routing, the split processes typically are modeled as independent renewal processes even though they are correlated with each other. This cross correlation might also have a serious impact on the queueing performance. In this paper, we propose an approach for modeling both the cross correlation and the autocorrelation by a three-moment four-parameter decomposition approximation of queueing networks. The arrival process is modeled as a nonrenewal process by a two-state Markov-modulated Poisson process, viz., MMPP(2). The cross correlation between randomly split streams is accounted for in the second and third moments of the merged process by the innovations method. The main contribution of the present research is that both the cross correlation and the autocorrelation can be modeled in parametric decomposition approximations of queueing networks by integrating the MMPP(2) approximation of the arrival/departure process and the innovations method. We also present numerical results that strongly support our refinements.

A Hybrid Token-CDMA MAC Protocol for Wireless Ad Hoc Networks

Token passing medium access control (MAC) protocols are gaining interest amongst wireless ad hoc network researchers as they provide unrivalled advantages over the existing IEEE 802.11 standards. This paper introduces a hybrid Token-CDMA MAC protocol that is based on a token passing scheme with the incorporation of code division multiple access (CDMA). With its unique CDMA feature, the proposed MAC is able to support multiple, simultaneous transmissions. The proposed protocol provides both Quality of Service (QoS) and high network resource utilization, while ensuring the stability of the network. This paper examines the performance of the proposed MAC scheme by simulation, and compares its performance against that of other MAC protocols that have appeared in the literature. Simulation results demonstrate that our proposed MAC scheme is effective in decreasing the packet delay and significantly shortens the length of the queue. The input traffic model used in the simulation is a two-state Markov Modulated Poisson Process (MMPP). The data rate QoS is enforced by implementing a modified leaky bucket mechanism in the proposed MAC scheme. The simulation also takes into account channel link errors caused by the wireless link by implementing a multi-layered Gilbert-Elliot model.

Path-wise performance in a tree-type network: Per-stream loss probability, delay, and delay variance analyses

This paper deals with path-wise performance analysis rather than a nodal one to enrich results previously obtained in the literature under simple but unsatisfactory assumptions, e.g., Poisson processes. First deriving the per-stream loss probability, delay, and delay variance of an isolated queue with multi-class input streams modeled by heterogeneous two-state Markov-modulated Poisson processes (MMPPs), we then propose simple and novel decomposition schemes working together with an input parameter modification scheme to (approximately) extract the per-stream output process for a lossy queue receiving MMPPs under a general service time distribution. The novelty of the decompositions is that they can be easily implemented based on a lossless queueing model. Through numerical experiments, we show that the accuracy in estimating the per-stream output process using such schemes is good. These decomposition schemes together with the input parameter modification scheme and a moment-based fitting algorithm used to fit the per-stream output as a two-state MMPP make analysis of path-wise performance viable by virtually treating each node in isolation along a path to get performance measures sequentially from the source node en route to the destination node.

A Performance Analysis Model of PC-based Software Router Supporting IPv6-IPv4 Translation for Residential Gateway

This paper presents a queuing analysis model of a PC-based software router supporting IPv6-IPv4 translation for residential gateway. The propose models are M/G/1/K or MMPP-2/G/1/K by arrival process of PC-based software router. In M/G/1/K, the arrival process is assumed to be Poisson process which is independent, identically distributed. In MMPP-2/G/1/K, The arrival process is assumed to be two-state Markov modulated Poisson process (MMPP). MMPP-2/G/1/K model is used to represent self-similar traffic. The service time distribution is arbitrary. The service discipline of server is processor sharing. The total number of packets that can be processed at one time is limited to K. We obtain PC-based software router performance metrics such as system sojourn time, blocking probability and throughput. The parameters are estimated by maximizing the log-like hood function of the measured average response time. The model has been validated through measurements. The performance metrics predicted by the model fit well to the experimental outcome.

Generalized Variance-Based Markovian Fitting for Self-Similar Traffic Modelling

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.

Performance analysis of an ATM MUX with a new space priority mechanism under ON-OFF arrival processes

We propose a new space priority mechanism, and analyze its performance in a single Constant Bit Rate (CBR) server. The arrival process is derived from the superposition of two types of traffics, each in turn results from the superposition of homogeneous ON-OFF sources that can be approximated by means of a two-state Markov Modulated Poisson Process (MMPP). The buffer mechanism enables the Asynchronous Transfer Mode (ATM) layer to adapt the quality of the cell transfer to the Quality of Service (QoS) requirements and to improve the utilization of network resources. This is achieved by “Selective-Delaying and Pushing-In” (SDPI) cells according to the class they belong to. The scheme is applicable to schedule delay-tolerant non-real time traffic and delay-sensitive real time traffic. Analytical expressions for various performance parameters and numerical results are obtained. Simulation results in term of cell loss probability conform with our numerical analysis.

Decomposition of general tandem queueing networks with MMPP input

For tandem queueing networks with generally distributed service times, decomposition often is the only feasible solution method besides simulation. The network is partitioned into individual nodes which are analyzed in isolation. In most existing decomposition algorithms for continuous-time networks, the output of a queue is usually approximated as a renewal process, which becomes the arrival process to the next queue. In this paper, the internal traffic processes are described as semi-Markov processes (SMPs) and Markov modulated Poisson processes (MMPPs). Thus, correlations in the traffic streams, which are known to have a considerable impact on performance, are taken into account to some extent. A two-state MMPP, which arises frequently in communications modeling, serves as input to the first queue of the tandem network. Furthermore, the single nodes may have infinite or finite buffers. Customers who arrive at a full buffer will get lost. In principle, the analysis of an individual queue as an MMPP/G/1(/ K) system delivers a wide range of performance measures. For different examples of tandem networks, stationary mean queue lengths at arbitrary time are compared to simulation data. The relative errors of the results, which are computed promptly by the decomposition component of the tool TimeNET, remain within a reasonable range.

On the handoff arrival process in cellular communications

Handoffs in cellular communication systems cause interactions among cells that can be modeled using multi-dimensional birth–death process approaches and the concept of system state. However, exact numerical calculation of traffic performance characteristics is hindered by unmanageably large system state spaces even for systems of modest size. Previous analytical models get around the difficulty by isolating a cell of interestand invoking a Poisson process assumption for handoff arrivals to the cell. Interactions among cells are characterized by relating the mean handoff and departure rates from cells. The current paper seeks to explore the interactions in more detail. Two additional approximate analytical models are developed for this purpose. Each of these is more complicated than the simple Poisson process model, but is analytically tractable – at least for small system sizes. One model isolates a cluster of cells (rather than just the cell of interest) from the system and invokes a Poisson process assumption for cells on the cluster periphery. Performance is calculated for the central cell. The second model also isolates a cluster of cells surrounding the cell of interest, but uses an equivalent two-state Markov Modulated Poisson Process (MMPP) to characterize handoff arrival processes to the cell of interest from each of the neighboring cells. Poisson handoff arrivals to cells on the cluster periphery are assumed. This approach has fewer states than the cluster approach. Finally we present the exact solution for a regional coverage area consisting of a single seven-cell cluster. Teletraffic performance characteristics are computed for each modeling technique and are compared. It was found that all are in close agreement with the original “single isolated cell, Poisson handoff arrival model,” which requires the least states.

Performance of a priority-based dynamic capacity allocation scheme for wireless ATM systems

The performance of a priority-based dynamic capacity allocation suitable for wireless ATM systems is presented. The scheduling of ATM cell transmission in each uplink TDMA frame is based on a priority scheme with priority given to real-time traffic over nonreal-time traffic. Real-time traffic exceeding the uplink capacity is lost while nonreal-time traffic that cannot be served is stored in a first-in first-out (FIFO) queue. An analytical model is developed to evaluate the cell loss ratio (CLR) of both real-time and nonreal-time traffic. Aggregate voice, video, and data traffic is modeled by three two-state Markov-modulated Poisson processes (MMPPs). Analytical results for different system capacities and various traffic loads and scenarios are discussed. Simulation results with on-off sources and approximating MMPP sources are also presented.

Practical Time-Scale Fitting of Self-Similar Traffic with Markov-Modulated Poisson Process

Recent measurements of packet/cell streams in multimedia communication networks have revealed that they have the self-similar property and are of different characteristics from traditional traffic streams. In this paper, we first give some definitions of self-similarity. Then, we propose a fitting method for the self-similar traffic in terms of Markov-modulated Poisson process (MMPP). We construct an MMPP as the superposition of two-state MMPPs and fit it so as to match the variance function over several time-scales. Numerical examples show that the variance function of the self-similar process can be well represented by that of resulting MMPPs. We also examine the queueing behavior of the resulting MMPP/D/1 queueing systems. We compare the analytical results of MMPP/D/1 with the simulation ones of the queueing system with self-similar input.

A CAC scheme based on real-time cell loss estimation for ATM multiplexers

For the superposition of on/off sources, we propose a real-time computable two-state Markov modulated Poisson process [MMPP(2)] modeling method characterizing the aggregate cell arrival rate and the variance-time curve for cell counts. Numerical examples show that the proposed method yields a good estimation of the cell loss ratio (CLR) in asynchronous transfer mode (ATM) multiplexers. Then, we propose an approximation method for calculating the workload for the MMPP(2)/D/1 system. By considering the workload as an upper bound of CLR of ATM multiplexers, we propose a connection admission control (CAC) algorithm utilizing peak cell rate, sustainable cell rate, and maximum burst size as user traffic descriptors. Using the proposed CAC algorithm, an acceptance decision is made with computational complexities of less than 3000 floating point operations. It is also observed from numerical evaluations that the proposed CAC algorithm yields a high multiplexing gain, compared with other methods in the literature.

Bayesian Analysis of a Two-State Markov Modulated Poisson Process

Bayesian Analysis of a Two-State Markov Modulated Poisson Process

Bayesian Analysis of a Two-State Markov Modulated Poisson Process

We postulate observations from a Poisson process whose rate parameter modulates between two values determined by an unobserved Markov chain. The theory switches from continuous to discrete time by considering the intervals between observations as a sequence of dependent random variables. A result from hidden Markov models allows us to sample from the posterior distribution of the model parameters given the observed event times using a Gibbs sampler with only two steps per iteration.

Analysing packetized voice and video traffic in an ATM multiplexer

This paper considers packetized voice and video traffic incoming to an ATM multiplexer. Each voice and video source is approximated by a two-state Markov–modulated Poisson process (MMPP(2)). We propose a method based on the asymptotic results of a renewal process to obtain the equivalent MMPP(2) parameters. The integrated voice and video sources are modelled by the superposition of two MMPP(2) processes, denoted MMPP(4). The analysis of the superposed traffic in terms of buffer occupancy, cell loss probability and delay is performed with an MMPP(4)/Er(k)/1/B queuing model considering different buffer sizes and numbers of sources. Results of the proposed method are also compared with those of Heffes and Lucantoni (IEEE J. Select. Areas Commun., SAC-4, 856–868 (1986)) and with a Poisson approximation. © 1998 John Wiley & Sons, Ltd.

A Markovian approach for modeling packet traffic with long-range dependence

We present a simple Markovian framework for modeling packet traffic with variability over several time scales. We present a fitting procedure for matching second-order properties of counts to that of a second-order self-similar process. Our models essentially consist of superpositions of two-state Markov modulated Poisson processes (MMPPs). We illustrate that a superposition of four two-state MMPPs suffices to model second-order self-similar behavior over several time scales. Our modeling approach allows us to fit to additional descriptors while maintaining the second-order behavior of the counting process. We use this to match interarrival time correlations.

Stochastic Bounds for Queueing Systems with Multiple On–Off Sources

Consider a queueing system where the input traffic consists of background traffic, modeled by a Markov Arrival Process, and foreground traffic modeled by N ≥ 1 homogeneous on–off sources. The queueing system has an increasing and concave service rate, which includes as a particular case multiserver queueing systems. Both the infinite-capacity and the finite-capacity buffer cases are analyzed. We show that the queue length in the infinite-capacity buffer system (respectively, the number of losses in the finite-capacity buffer system) is larger in the increasing convex order sense (respectively, the strong stochastic order sense) than the queue length (respectively, the number of losses) of the queueing system with the same background traffic and M N homogeneous on–off sources of the same total intensity as the foreground traffic, where M is an arbitrary integer. As a consequence, the queue length and the loss with a foreground traffic of multiple homogeneous on–off sources is upper bounded by that with a single on–off source and lower bounded by a Poisson source, where the bounds are obtained in the increasing convex order (respectively, the strong stochastic order). We also compare N ≥ 1 homogeneous arbitrary two-state Markov Modulated Poisson Process sources. We prove the monotonicity of the queue length in the transition rates and its convexity in the arrival rates. Standard techniques could not be used due to the different state spaces that we compare. We propose a new approach for the stochastic comparison of queues using dynamic programming which involves initially stationary arrival processes.

On variations of queue response for inputs with the same mean and autocorrelation function

This paper explores the variations in mean queue length for stationary arrival processes with the same mean and autocorrelation functions, or equivalently, the same mean and power spectrum. Three types of processes, namely, two-state Markov-modulated Poisson processes, periodic-sequence modulated Poisson processes and processes generated by randomly filtering a white noise process, are investigated. Results show that the mean queue length can vary substantially for the first type of process, and can vary moderately for the second type of process, as the parameters of the processes are varied, subject to a specified mean and autocorrelation function. However, the mean queue lengths for the third type of arrival processes are determined by the input mean and autocorrelation functions. The results suggest that the queueing performance can be hard to predict from spectral data alone when the power in low frequencies is large.