Markov Modulated Poisson Process Research Articles

Markov-Modulated Poisson Process Modeling for Machine-to-Machine Heterogeneous Traffic

Theoretical mathematics is a key evolution factor of artificial intelligence (AI). Nowadays, representing a smart system as a mathematical model helps to analyze any system under development and supports different case studies found in real life. Additionally, the Markov chain has shown itself to be an invaluable tool for decision-making systems, natural language processing, and predictive modeling. In an Internet of Things (IoT), Machine-to-Machine (M2M) traffic necessitates new traffic models due to its unique pattern and different goals. In this context, we have two types of modeling: (1) source traffic modeling, used to design stochastic processes so that they match the behavior of physical quantities of measured data traffic (e.g., video, data, voice), and (2) aggregated traffic modeling, which refers to the process of combining multiple small packets into a single packet in order to reduce the header overhead in the network. In IoT studies, balancing the accuracy of the model while managing a large number of M2M devices is a heavy challenge for academia. One the one hand, source traffic models are more competitive than aggregated traffic models because of their dependability. However, their complexity is expected to make managing the exponential growth of M2M devices difficult. In this paper, we propose to use a Markov-Modulated Poisson Process (MMPP) framework to explore Human-to-Human (H2H) traffic and M2M heterogeneous traffic effects. As a tool for stochastic processes, we employ Markov chains to characterize the coexistence of H2H and M2M traffic. Using the traditional evolved Node B (eNodeB), our simulation results show that the network’s service completion rate will suffer significantly. In the worst-case scenario, when an accumulative storm of M2M requests attempts to access the network simultaneously, the degradation reaches 8% as a completion task rate. However, using our “Coexistence of Heterogeneous traffic Analyzer and Network Architecture for Long term evolution” (CHANAL) solution, we can achieve a service completion rate of 96%.

Mist–fog-assisted real-time emotion recognition using deep transfer learning framework for smart city 4.0

Face emotion recognition (FER) has found several applications across numerous industries. One such application area lies in the field of human–computer interaction, where it can improve user experiences by allowing systems to adjust based on users’ emotional states by leveraging the capabilities of the Internet of Things (IoT) to facilitate more responsive interactions. The increasing demand for precise and immediate emotion analysis across various applications drives the need for FER utilizing deep transfer learning (DTL). In this study, a novel Mist-Fog framework for real-time FER is introduced, which employs a DTL algorithm called Mobile-Net for quick and easy image classification on mobile and embedded devices. It addresses the challenges of operating deep neural networks (DNN) under limits on power and processing resources. By utilizing depth-wise separable convolutions, the MobileNet algorithm reduces computational complexity while maintaining reasonable accuracy on the FER-2013 dataset. A Fog-based architecture, in which a mist layer comprising edge devices like Raspberry-Pi (RPi), an ARM-based single-board computer, was considered for a smart city scenario. Further, the M/M/1/K queuing system was considered, which follows the First-Come-First-Serve (FCFS) scheduling criteria and a Markov Modulated Poisson Process (MMPP) arrival rate to evaluate the system’s performance. The effectiveness of this novel approach towards industrial computer vision paradigms is assessed pertaining to different performance criteria.

Burst traffic: Congestion management and performance optimization strategies in heterogeneous cognitive radio networks

AbstractIn order to solve the network congestion problem caused by burst service flows in heterogeneous cognitive radio networks (Het‐CRN), a two‐stage dual‐threshold random early detection (TD‐RED) control mechanism is proposed here. Meanwhile, a three‐state Markov modulated Poisson process (MMPP‐3) is used to model the burst service to analyze its arrival in depth. Moreover, a performance evaluation system (PES) is proposed to quantify the interference of burst services to different classes of services and their quality of service, which can comprehensively evaluate the performance enhancement of the TD‐RED control mechanism for each service. The system examines performance metrics such as packet loss rate, average delay, average captain etc. PES simulation results show that TD‐RED significantly reduces the packet rejection rate and transmission delay compared to the no congestion control (NCC) and random early detection (RED) methods, thus providing superior quality of service to all services. thereby providing a superior quality of service to cognitive users. Its lower average captain fully proves that TD‐RED can effectively alleviate congestion.

Cache Update and Delivery of Dynamic Contents: A Stochastic Game Approach

In this paper, we propose a cache update policy for wireless networks considering dynamic popularity for file requests. Our network scenario is a wireless network comprised of some cache-equipped access points (APs) which are deployed densely in an area and have connections with core network servers. We model the dynamics of file requests as Markov-modulated Poisson processes. Considering the congestion-dependent delay of APs' services and their overlapping coverage areas, we model the cache service for the cache-missed requests, as a stochastic game in which a Vickrey–Clarke–Groves (VCG) mechanism is exploited at each stage of the game as our proposed file routing policy, and the APs decide independently how to update their caches regarding the routed files. To this end, the APs' utilities are defined based on long-term average file delivery delay including the queueing delay in APs. By finding the Nash equilibrium of the game, we propose policies for both delivery and cache update of dynamic contents. Finally, by comparing numerical results of our proposed scheme with some conventional caching schemes, we show significant improvements in network performance in terms of file delivery delay in different conditions.

Performance approximations of Markov Modulated Poisson Process arrival queue with Markovian Service using matrix geometric approach

Queue analysis of correlated work of the arrival and the service process are not available much in literature. This research paper applies matrix geometric approach to study the MMPP/MSP/1 queuing model under the quasi-birth death(QBD) process. Customers admitted in the system are having two different intensity of arrival followed by Markov Modulated Poisson Process(MMPP). MMPP is a versatile class of MAP and it is necessary to adapt the first characterization of the queuing model. In service mechanism Markov Service Process(MSP) is adopted. As like MMPP, MSP is also a versatile and can capture the correlation of the modulated arrivals. The methodology presented in this research work calculates the metric performances not depending on the series of infinite state of the system. The assumed model is formulated as QBD process to obtain the steady state probability of the system. The complexity model formulated involves structures matrices with appropriate dimension. Neut?s pioneered the matrix geometric approach to such complex models which involves the rate computation of rate matrix R. The rate matrix R is approximated to derive the performance metrics of the system and then compute the cost for the underlying Markov chain. To attain this approximation the sensitivity analysis is done with numerical results.

Markov-modulated marked Poisson processes for modeling disease dynamics based on medical claims data.

We explore Markov-modulated marked Poisson processes (MMMPPs) as a natural framework for modeling patients' disease dynamics over time based on medical claims data. In claims data, observations do not only occur at random points in time but are also informative, that is,driven by unobserved disease levels, as poor health conditions usually lead to more frequent interactions with the health care system. Therefore, we model the observation process as a Markov-modulated Poisson process, where the rate of health care interactions is governed by a continuous-time Markov chain. Its states serve as proxies for the patients' latent disease levels and further determine the distribution of additional data collected at each observation time, the so-called marks. Overall, MMMPPs jointly model observations and their informative time points by comprising two state-dependent processes: the observation process (corresponding to the event times) and the mark process (corresponding to event-specific information), which both depend on the underlying states. The approach is illustrated using claims data from patients diagnosed with chronic obstructive pulmonary disease by modeling their drug use and the interval lengths between consecutive physician consultations. The results indicate that MMMPPs are able to detect distinct patterns of health care utilization related to disease processes and reveal interindividual differences in the state-switching dynamics.

Spatiotemporal Analysis of the Background Seismicity Identified by Different Declustering Methods in Northern Algeria and Its Vicinity

The main purpose of this paper was to, for the first time, analyse the spatiotemporal features of the background seismicity of Northern Algeria and its vicinity, as identified by different declustering methods (specifically: the Gardner and Knopoff, Gruenthal, Uhrhammer, Reasenberg, Nearest Neighbour, and Stochastic Declustering methods). Each declustering method identifies a different declustered catalogue, namely a different subset of the earthquake catalogue that represents the background seismicity, which is usually expected to be a realisation of a homogeneous Poisson process over time, though not necessarily in space. In this study, a statistical analysis was performed to assess whether the background seismicity identified by each declustering method has the spatiotemporal properties typical of such a Poisson process. The main statistical tools of the analysis were the coefficient of variation, the Allan factor, the Markov-modulated Poisson process (also named switched Poisson process with multiple states), the Morisita index, and the L–function. The results obtained for Northern Algeria showed that, in all cases, temporal correlation and spatial clustering were reduced, but not totally eliminated in the declustered catalogues, especially at long time scales. We found that the Stochastic Declustering and Gruenthal methods were the most successful methods in reducing time correlation. For each declustered catalogue, the switched Poisson process with multiple states outperformed the uniform Poisson model, and it was selected as the best model to describe the background seismicity in time. Moreover, for all declustered catalogues, the spatially inhomogeneous Poisson process did not fit properly the spatial distribution of earthquake epicentres. Hence, the assumption of stationary and homogeneous Poisson process, widely used in seismic hazard assessment, was not met by the investigated catalogue, independently from the adopted declustering method. Accounting for the spatiotemporal features of the background seismicity identified in this study is, therefore, a key element towards effective seismic hazard assessment and earthquake forecasting in Algeria and the surrounding area.

Performance Modelling and Quantitative Analysis of Vehicular Edge Computing With Bursty Task Arrivals

The quantitative performance analysis plays a critical role in assessing the capability of vehicular edge computing (VEC) systems to meet the requirements of vehicular applications. However, developing accurate analytical models for VEC systems is extremely challenging due to the unique features of intelligent vehicular applications. Specifically, recent work revealed that the tasks generated by intelligent vehicular applications exhibit a high degree of burstiness, rendering the existing models that were designed based on the assumption of the non-bursty Poisson process unsuitable for VEC systems. To fill this gap, we developed an original analytical model to investigate the performance of VEC systems with bursty task arrivals. To facilitate vehicle cooperation, a new priority-based resource allocation scheme is exploited to schedule the tasks of vehicular applications, which are modelled by a Markov Modulated Poisson Process (MMPP). Next, a multi-state Markov chain is established to investigate the impact of load sharing strategy on the performance of VEC systems. Then, the end-to-end transmission latency is derived based on the proposed model. Comprehensive experiments are conducted to validate the accuracy of this analytical model under various system configurations. Furthermore, the developed model is used as a cost-effective tool to investigate the performance bottleneck of VEC systems.

Evaluation of the Waiting Time in a Finite Capacity Queue with Bursty Input and a Generalized Push-Out Strategy

In this paper, we study a finite capacity queue where the arrival process is a special case of the discrete time Markov modulated Poisson process, the service times are generally distributed, and the server takes repeated vacations when the system is empty. The buffer acceptance strategy is based on a generalized push-out scheme: when the buffer is full, an arriving customer pushes out the Nth customer in the queue, where N takes values between 2 and the capacity of the system, and the arriving customer joins the end of the queue. Such a strategy is important when, as well as short waiting times for served customers, the time a pushed-out customer occupies a buffer space is also an important performance measure. The Laplace transform of the waiting time of a served customer is determined. Numerical examples show the influence of the bustiness of the input process and also the trade-off between the average waiting time of served customers and the occupancy of the buffer space of pushed-out customers.

A New Upper Bound on Cache Hit Probability for Non-Anticipative Caching Policies

Caching systems have long been crucial for improving the performance of a wide variety of network and web-based online applications. In such systems, end-to-end application performance heavily depends on the fraction of objects transferred from the cache, also known as the cache hit probability . Many caching policies have been proposed and implemented to improve the hit probability. In this work, we propose a new method to compute an upper bound on hit probability for all non-anticipative caching policies and for policies that have no knowledge of future requests. Our key insight is to order the objects according to the ratio of their Hazard Rate (HR) function values to their sizes, and place in the cache the objects with the largest ratios till the cache capacity is exhausted. When object request processes are conditionally independent, we prove that this cache allocation based on the HR-to-size ratio rule guarantees the maximum achievable expected number of object hits across all non-anticipative caching policies. Further, the HR ordering rule serves as an upper bound on cache hit probability when object request processes follow either independent delayed renewal process or a Markov modulated Poisson process. We also derive closed form expressions for the upper bound under some specific object request arrival processes. We provide simulation results to validate its correctness and to compare it to the state-of-the-art upper bounds, such as produced by Bélády’s algorithm. We find it to be tighter than state-of-the-art upper bounds for some specific object request arrival processes such as independent renewal, Markov modulated, and shot noise processes.

Infinite-Server Resource Queueing Systems with Different Types of Markov-Modulated Poisson Process and Renewal Arrivals

In this paper, we propose models that significantly expand the scope of practical applications, namely, queueing systems with various nodes for processing heterogeneous data that require arbitrary resource capacities for their service. When a customer arrives in the system, the customer typeis randomly selected according to a set of probabilities. Then the customer goes to the server of the corresponding device type, where its service is performed during a random time period with a distribution function depending on the type of customer. Moreover, each customer requires a random amount of resources, of which the distribution function also depends on the customer type, but is independent of its service time. The aim of this research was to develop a heterogeneous queueing resource system with an unlimited number of servers and an arrival process in the form of a Markov-modulated Poisson process or stationary renewal process, and with requests for a random number of heterogeneous resources. We have performed analysis under conditions of growing intensity of the arrival process. Here we formulate the theorems and prove that under high-load conditions, the joint asymptotic probability distribution of the n-dimensional process of the total amounts of the occupied resources in the system is a multidimensional Gaussian distribution with parameters that are dependent on the type of arrival process. As a result of numerical and simulation experiments, conclusions are drawn on the limits of the applicability of the obtained asymptotic results. The dependence of the convergence of experimental results on the type of distribution of the system parameters (including the distributions of the service time and of the customer capacity) are also studied. The results of the approximations may be applied to estimating the optimal total number of resources for a system with a limited amount of resources.

Stationary Markovian arrival processes: Results and open problems

We consider two classes of irreducible Markovian arrival processes specified by the matrices \(C\) and \(D\): the Markov-modulated Poisson process (MMPP) and the Markov-switched Poisson process (MSPP). The former exhibits a diagonal matrix D while the latter exhibits a diagonal matrix C. For these two classes we consider the following four statements: (I) the counting process is overdispersed; (II) the hazard rate of the event-stationary interarrival time is nonincreasing; (III) the squared coefficient of variation of the event-stationary process is greater than or equal to one; (IV) there is a stochastic order showing that the time-stationary interarrival time dominates the event-stationary interarrival time. For general MSPPs and order two MMPPs, we show that (I)–(IV) hold. Then for general MMPPs, it is easy to establish (I), while (II) is shown to be false by a counter-example. For general simple point processes, (III) follows from (IV). For MMPPs, we conjecture that (IV) and thus (III) hold. We also carry out some numerical experiments that fail to disprove this conjecture. Importantly, modelling folklore has often treated MMPPs as “bursty”, and implicitly assumed that (III) holds. However, to the best of our knowledge, proving this is still an open problem. doi: https://doi.org/10.1017/S1446181122000013

Modeling Poisson Error Process on Wireless Channels

Burst error modeling has seen extensive research and progress over several decades evolving into ever more complex modeling techniques used today. This paper analyzed usefulness of the most prominent generative and descriptive (analytical) methods. Data containing error bits and packets from real wireless transmission has been used to obtain statistical information about error burst and gap behavior in the channel and various generative and descriptive modeling techniques were applied to model the error process with the goal of establishing advantages and disadvantages of each technique. Generative methods were represented by the commonly implemented Elliot’s model with parameters calculated using a generalized algebraic form and descriptive methods were represented by one of the most flexible exponentially shaped distributions with regard to parameterization and heavy-tailed function modeling - gamma distribution, and lastly a technique represented by Markov modulated Poisson process (MMPP-2) producing second-order hyper-exponentially distributed characteristics. Results of the experiments were highly in favor of Elliot’s and MMPP-2 model demonstrating possible application of MMPP-2 model in application to commonly observed exponentially-shaped error process on the wireless channel.

A Learning-Based Fast Uplink Grant for Massive IoT via Support Vector Machines and Long Short-Term Memory

The current random access (RA) allocation techniques suffer from congestion and high signaling overhead while serving massive machine type communication (mMTC) applications. To this end, 3GPP introduced the need to use fast uplink grant (FUG) allocation in order to reduce latency and increase reliability for smart internet-of-things (IoT) applications with strict QoS constraints. We propose a novel FUG allocation based on support vector machine (SVM), First, MTC devices are prioritized using SVM classifier. Second, LSTM architecture is used for traffic prediction and correction techniques to overcome prediction errors. Both results are used to achieve an efficient resource scheduler in terms of the average latency and total throughput. A Coupled Markov Modulated Poisson Process (CMMPP) traffic model with mixed alarm and regular traffic is applied to compare the proposed FUG allocation to other existing allocation techniques. In addition, an extended traffic model based CMMPP is used to evaluate the proposed algorithm in a more dense network. We test the proposed scheme using real-time measurement data collected from the Numenta Anomaly Benchmark (NAB) database. Our simulation results show the proposed model outperforms the existing RA allocation schemes by achieving the highest throughput and the lowest access delay of the order of 1 ms by achieving prediction accuracy of 98 $\%$ when serving the target massive and critical MTC applications with a limited number of resources.

Exact and computationally efficient Bayesian inference for generalized Markov modulated Poisson processes.

Statistical modeling of temporal point patterns is an important problem in several areas. The Cox process, a Poisson process where the intensity function is stochastic, is a common model for such data. We present a new class of unidimensional Cox process models in which the intensity function assumes parametric functional forms that switch according to a continuous-time Markov chain. A novel methodology is introduced to perform exact (up to Monte Carlo error) Bayesian inference based on MCMC algorithms. The reliability of the algorithms depends on a variety of specifications which are carefully addressed, resulting in a computationally efficient (in terms of computing time) algorithm and enabling its use with large data sets. Simulated and real examples are presented to illustrate the efficiency and applicability of the methodology. A specific model to fit epidemic curves is proposed and used to analyze data from Dengue Fever in Brazil and COVID-19 in some countries.

STATIONARY MARKOVIAN ARRIVAL PROCESSES: RESULTS AND OPEN PROBLEMS

AbstractWe consider two classes of irreducible Markovian arrival processes specified by the matrices C and D: the Markov-modulated Poisson process (MMPP) and the Markov-switched Poisson process (MSPP). The former exhibits a diagonal matrix D while the latter exhibits a diagonal matrix C. For these two classes we consider the following four statements: (I) the counting process is overdispersed; (II) the hazard rate of the event-stationary interarrival time is nonincreasing; (III) the squared coefficient of variation of the event-stationary process is greater than or equal to one; (IV) there is a stochastic order showing that the time-stationary interarrival time dominates the event-stationary interarrival time. For general MSPPs and order two MMPPs, we show that (I)–(IV) hold. Then for general MMPPs, it is easy to establish (I), while (II) is shown to be false by a counter-example. For general simple point processes, (III) follows from (IV). For MMPPs, we conjecture that (IV) and thus (III) hold. We also carry out some numerical experiments that fail to disprove this conjecture. Importantly, modelling folklore has often treated MMPPs as “bursty”, and implicitly assumed that (III) holds. However, to the best of our knowledge, proving this is still an open problem.

Bounds on the mean and squared coefficient of variation of phase-type distributions

AbstractWe consider a class of phase-type distributions (PH-distributions), to be called the MMPP class of PH-distributions, and find bounds of their mean and squared coefficient of variation (SCV). As an application, we have shown that the SCV of the event-stationary inter-event time for Markov modulated Poisson processes (MMPPs) is greater than or equal to unity, which answers an open problem for MMPPs. The results are useful for selecting proper PH-distributions and counting processes in stochastic modeling.

Optimal Open-Loop Routing and Threshold-Based Allocation in TWO Parallel QUEUEING Systems with Heterogeneous Servers

In this paper, we study the problem of optimal routing for the pair of two-server heterogeneous queues operating in parallel and subsequent optimal allocation of customers between the servers in each queue. Heterogeneity implies different servers in terms of speed of service. An open-loop control assumes the static resource allocation when a router has no information about the state of the system. We discuss here the algorithm to calculate the optimal routing policy based on specially constructed Markov-modulated Poisson processes. As an alternative static policy, we consider an optimal Bernoulli splitting which prescribes the optimal allocation probabilities. Then, we show that the optimal allocation policy between the servers within each queue is of threshold type with threshold levels depending on the queue length and phase of an arrival process. This dependence can be neglected by using a heuristic threshold policy. A number of illustrative examples show interesting properties of the systems operating under the introduced policies and their performance characteristics.

Dynamic resource allocation for service in mobile cloud computing with Markov modulated arrivals

Mobile Cloud Computing (MCC) is a modern architecture that brings together cloud computing, mobile computing and wireless networks to assist mobile devices in storage, computing and communication. A cloud environment is developed to support a wide range of users that request the cloud resources in a dynamic environment with possible constraints. Burstiness in users service requests causes drastic and unpredictable increases in the resource requests that have a crucial impact on policies of resource allocation. How can the cloud system efficiently handle such burstiness when the cloud resources are limited? This problem becomes a hot issue in the MCC research area. In this paper, we develop a system model for the resource allocation based on the Semi-Markovian Decision Process (SMDP), able of dynamically assigning the mobile service requests to a set of cloud resources, to optimize the usage of cloud resources and maximize the total long-term expected system reward when the arrival process is a finite-state Markov-Modulated Poisson Process (MMPP). Numerical results show that our proposed model performs much better than the Greedy algorithm in terms of achieving higher system rewards and lower service requests blocking probabilities, especially when the burstiness degree is high, and the cloud resources are limited.

Performance evaluation for Robotic Mobile Fulfillment Systems with time-varying arrivals

This paper analyzes the performance of Robotic Mobile Fulfillment Systems with a semi-open queueing network (SOQN) with time-varying arrivals. We use a Markov-modulated Poisson process to characterize the variability in the order arrival rate and offer a matrix-geometric method (MGM)-based approach to solve the SOQN. Simulations are used to validate the analytical model. We verify that assuming time-homogeneous order arrivals will underestimate order throughput time. Numerical experiments are also conducted to explore the effect of peak order arrival rates, peak order arrival durations, and average order arrival rates on system performance. The results show that peak order arrival rates and peak order arrival durations have the most significant impact on order throughput time. Moreover, we compare the order throughput time of various workstation layouts with different width-to-length ratios.