Three-dimensional Markov Chain Research Articles

A Markovian Analysis of an IEEE-802.11 Station with Buffering

The purpose of this paper is to analyze the so-called backoff tech- nique of the IEEE 802.11 protocol with buffers. This protocol rules the trans- missions on a radio channel between nodes (or stations) of a network exchanging packets of information. In contrast to existing models, packets arriving at a sta- tion which in the backoff state are not discarded, but are stored in a buffer of in nite capacity. The backoff state corresponds to the number of time-intervals (mini-slot) that a node must wait before its packet is actually transmitted. As in previous studies, the key point of our analysis hinges on the assumption that the time on the radio channel is viewed as a random succession of trans- mission slots (whose duration corresponds to a packet transmission time) and mini-slots, which stand for the time intervals during which the backoff of the station is decremented. During these mini-slots the channel is idle, which im- plies that there is no packet transmission. These events occur independently with given probabilities, and the external arrivals of messages follow a Pois- son process. The state of a node is represented by a three-dimensional Markov chain in discrete-time, formed by the triple (backoff counter, number of packets at the station, number of transmission attempts). Stability (ergodicity) condi- tions are obtained for an arbitrary station and interpreted in terms of maximum throughput. Several approximations related to these models are also discussed.

Transient analysis of energy-saving strategy for cognitive radio networks using G-queue with heterogeneity

Energy efficiency, high data speeds, and effective spectrum utilization stand as the primary factors governing 5G and beyond networks. A base station (BS) in a cognitive radio network (CRN) that conserves energy during sleep periods is a promising contender for achieving more effective spectrum allocation. An efficient energy-saving strategy is offered in this study to achieve greener communication in wireless cellular networks. This study aims to propose and analyze an energy-saving scheme for the BS, taking into account the heterogeneity and reliability of networks. The entire system is modeled as a three-dimensional Markov chain by establishing a discrete-time preemptive priority queueing model with single vacation, heterogeneous packets, and negative packets such as viruses. Then, the transient analysis of the Markov chain is performed using the recursive method. Utilizing the transient probability distribution, we present numerical results derived from reliability analysis and queueing analysis to substantiate the validity of the proposed model. Furthermore, results based on the energy-saving degree are showcased to affirm the effectiveness of the proposed scheme. Finally, a reward cost function is obtained depending on the successful transmission of secondary packets, and the Quasi-Newton method is used to determine the optimal reward cost.

Equilibrium Analysis for Batch Service Queueing Systems with Strategic Choice of Batch Size

Various transportation services exist, such as ride-sharing or shared taxis, in which customers receive services in a batch of flexible sizes and share fees. In this study, we conducted an equilibrium analysis of a variable batch service model in which customers who observe no waiting customers in an incomplete batch can strategically select a batch size to maximize the individual utilities. We formulated this model as a three-dimensional Markov chain and created a book-type transition diagram. To consider the joining/balking dilemma of customers for this model, we proposed an effective algorithm to construct a necessary and sufficient size of state space for the Markov chain provided that all customers adopt the threshold-type equilibrium strategy. Moreover, we proved that the best batch size is a non-decreasing function for i if the reward for the completion of batch service with size l is an increasing function of l assuming that a tagged customer observes i complete batches in the system upon arrival; in other words, the fee decreases as the batch becomes larger. We then derive several performance measures, such as throughput, social welfare, and monopolist’s revenue. Throughout the numerical experiment, a comparison between the present variable batch service model and regular batch service model in which customers were served in a constant batch, was discussed. It was demonstrated that the three performance measures can be optimized simultaneously in the variable batch service model, as long as the fee was set relatively high.

Distributed Batteryless Access Control for Data and Energy Integrated Networks: Modeling and Performance Analysis

Radio-frequency (RF) signals are capable of simultaneously transferring data and energy from a hybrid access point (HAP) toward battery-powered and batteryless wireless devices. Battery-powered and batteryless wireless devices with the capability of RF energy harvesting need a distributed access control protocol with collision avoidance to achieve higher energy efficiency. We study the performance of a data and energy integrated network (DEIN) that adopts an enhanced carrier sensing multiple access with collision avoidance (CSMA/CA) protocol. Each device in this network can switch to RF energy harvesting mode or data reception mode according to HAP’s instruction, and freezes its backoff counter when energy storage is insufficient. By invoking a three-dimensional (3D) Markov chain, we model the operating behaviors of batteryless wireless devices and an HAP in a DEIN. Apart from backoff operations of devices, the 3D Markov chain also depicts their dynamic energy changes, including RF energy harvesting and energy consumption. Wireless devices consume energy harvested from the HAP’s downlink transmissions for powering their data upload and random backoff. With the aid of the 3D Markov chain, the upload throughput of devices can be obtained in semi-closed-form. Moreover, a decoupling method is proposed to approximate throughput performance with low complexity. The accuracy of our theoretical model is validated by simulation results. By characterizing the impact of various parameters on throughput performance, a design guideline for a DEIN with a distributed batteryless access protocol is provided.

An Outage-Probability-Aware MAC Protocol for UAV-Assisted Networks

Unmanned aerial vehicles (UAVs) are widely used in wireless communication, either base stations (BS) or relays. In some occasions of UAV-assisted communication such as low-altitude communication in an urban environment, the channel conditions can be poor due to obstacles. Thus, they will experience an outage and cannot receive the signal transmitted by the UAVs. In this paper, we propose an outage-probability-aware medium access control (MAC) protocol (OPM), based on the study of outage probabilities in different UAV-assisted environments. Specifically, we first analyze different outage scenarios in information transmission. Then, the physical-layer outage probability with channel conditions and ground users’ mobility taken into consideration is discussed, while an outage detection method is proposed, and different measures are taken according to the detection results. Thirdly, a three-dimensional (3D) Markov chain model is used to investigate per-user behavior for the proposed OPM. Finally, the simulation results on packet discard probability, average transmission delay, and system throughput are analyzed. The results show the effectiveness of the proposed protocol, which leads to a 30%-50% decrease in both the probability of packet discard and average transmission delay, along with an increase in throughput of more than 40%.

Geo/D/1/1 모형에서의 실시간 원격 추정값의 오차 분석

In this paper, we study real-time error in the context of monitoring a binary information source through a delay system. To derive the average real-time error, we model the delay system as a discrete time Geo/D/1/1 queueing model. Using a discrete time three-dimensional Markov chain with finite state space, we analyze the queueing model. We also perform some numerical analysis on various system parameters: state transition probabilities of binary information source; transmission times; and transmission frequencies. When the state changes of the information source are positively correlated and negatively correlated, we investigate the relationship between transmission time and transmission frequency.

A communication failure and repair mechanism with adjustable transmission rates for PU packets in CRNs

As one key technology of future radio communication networks, cognitive radio networks (CRNs) can effectively handle the shortage of network resources. Two classes of users exist in traditional CRNs, namely, primary users (PUs) with higher priority and secondary users (SUs) with cognitive ability. In CRNs, during the communication process, packets need to travel through various servers, such as switches and routers, and these devices may fail at any time. We consider this type of problem to be a communication failure. The occurrence of communication failures trigger system repairs, which make the system return to regular work. In this paper, we present a communication failure and repair mechanism with adjustable transmission rates for PU packets in CRNs. We assume that PU packets can maintain low-speed transmission during the system failure state and resume high-speed transmission after the failure is repaired. We establish a three-dimensional Markov chain (3DMC) and build a queueing model based on discrete time. Through numerical experiments, we analyze some indicators’ impact on the system capability. In addition, compared with the traditional communication failure and repair mechanism, our proposed mechanism can reduce the blocking rate while considerably increasing the throughput of data packets.

Retrial Queues with Unreliable Servers and Delayed Feedback

In this paper, models of unreliable multi-server retrial queues with delayed feedback are examined. The Bernoulli retrial is allowed upon the arrival of both primary (from outside) and feedback customers (from orbit), as well as the Bernoulli feedback that may occur after each service in this system. Servers can break down both during the service of customers and when they are idle. If a server breaks down during the service of a customer, then the interrupted customer, in accordance with the Bernoulli scheme, decides either to leave the system or join a common orbit of retrial and feedback customers. An approximate method, based on the space merging approach of three-dimensional Markov chains, is proposed for the calculation of the steady-state probabilities, as well as performance measures of the system. The results of the numerical experiments are demonstrated.

Analysis of Multi-Server Queue with Self-Sustained Servers

A novel multi-server vacation queuing model is considered. The distinguishing feature of the model, compared to the standard queues, is the self-sufficiency of servers. A server can terminate service and go on vacation independently of the system manager and the overall situation in the system. The system manager can make decisions whether to allow the server to start work after vacation completion and when to try returning some server from a vacation to process customers. The arrival flow is defined by a general batch Markov arrival process. The problem of optimal choice of the total number of servers and the thresholds defining decisions of the manager arises. To solve this problem, the behavior of the system is described by the three-dimensional Markov chain with the special block structure of the generator. Conditions for the ergodicity of this chain are derived, the problem of computation of the steady-state distribution of the chain is discussed. Expressions for the key performance indicators of the system in terms of the distribution of the chain states are derived. An illustrative numerical result is presented.

Delayed mobile data offloading scheme for quality of service traffic: Design and analysis

As the world is entering the era of 5G, we are witnessing exponential growth in the volume of data traffic generated and consumed by mobile devices. Mobile data offloading handles the surge in mobile data traffic by offloading part of the traffic onto the Wi-Fi network. In this context, we design and analyse a quality of service (QoS) enhancement scheme for delayed mobile data offloading. We first consider prioritised queuing of traffic as a scheme for QoS provisioning. Using the concepts of virtual waiting time, renewal process, and level crossing arguments, we derive the average transmission delay of the offloaded and reneged packets of high priority. Next, we mathematically validate the benefit of balking all such packets whose expected virtual waiting time exceeds their deadline. Using a three-dimensional Markov chain model, we derive the required balking probability. Moreover, we merge the technique of balking with prioritisation for improving QoS in delayed offloading. Through an extensive simulation, we validate our analysis and demonstrate how the proposed scheme reduces the transmission delay without sacrificing offloading efficiency. Our investigation has the potential to be adopted in future mobile data offloading standards for improving QoS.

Analysis of Instantaneous Feedback Queue with Heterogeneous Servers

A system with heterogeneous servers, Markov Modulated Poisson flow and instantaneous feedback is studied. The primary call is serviced on a high-speed server, and after it is serviced, each call, according to the Bernoulli scheme, either leaves the system or requires re-servicing. After the completion of servicing of a call in a slow server, according to the Bernoulli scheme, it also either leaves the system or requires re-servicing. If upon arrival of a primary call the queue length of such calls exceeds a certain threshold value and the slow server is free, then the incoming primary call, according to the Bernoulli scheme, is either sent to the slow server or joins its own queue. A mathematical model of the studied system is constructed in the form of a three-dimensional Markov chain. Approximate algorithms for calculating the steady-state probabilities of the models with finite and infinite queues are proposed and their high accuracy is shown. The results of numerical experiments are presented.

Comprehensive performance analysis of satellite-to-ground FSO/QKD systems using key retransmission

We propose to investigate the performance of a satellite-to-ground quantum key distribution (QKD) system that uses key retransmission as a method for improving the system reliability. We develop analytical frameworks based on two three-dimensional Markov chain models allowing us to comprehensively analyze the proposed system’s performance in terms of key loss rate, link utilization, and delay outage rate. Our performance analysis takes into account the physical layer impairments induced by free-space optical channel and receiver noise. Numerical results quantitatively demonstrate that the performance of satellite-to-ground QKD system is significantly improved due to key retransmission. In addition, the appropriate selection of system parameters corresponding to different turbulence conditions to achieve the best performance improvement is also provided.

Reliability and Delay Trade-Off Analysis of Unslotted IEEE 802.15.4 Sensor Network for Shipboard Environment

Wireless sensor network technology has been widely studied and applied in different fields. The applications in shipborne wireless sensor networks face serious physical-layer packet loss problems due to the wireless signals blocking by steel structure and wireless channel interference. The MAC-layer retransmission mechanism can reduce the packet loss rate but increases latency. So it's necessary to analyze the performance of reliability and latency for different applications in shipborne wireless sensor networks. In this article, first, we analyze the distribution of the physical-layer packet loss rate for IEEE 802.15.4 sensor network in ship indoor environment. We also present an analysis model based on three-dimensional Markov chain for unslotted CSMA/CA of IEEE 802.15.4 sensor network considering physical-layer packet loss rate and MAC-layer retransmission. We evaluate the effects of three MAC parameters, macMinBE, macMaxCSMABackoffs, and macMaxFrameRetries, on average delay and reliability. The percentage error of the analytical model for delay and reliability are 2.09% and 1.37%, respectively. Our work provides trade-off analysis between average delay and average reliability using the maximum number of retransmission based on the measured data of physical-layer packet loss rate in shipboard environment for different QoS requirements.

Performance Analysis and Optimization for the MAC Protocol in UAV-Based IoT Network

Unmanned aerial vehicles (UAVs) have played an important role in air-ground integration network. Especially in Internet of Things (IoT) services, UAV equipped with communication equipments is widely adopted as a mobile base station (BS) for data collection from IoT devices on the ground. In this paper, we consider an air-ground network in which the UAV flies straightly to collect information from the IoT devices in a 2-D plane based on the CSMA/CA protocol. Due to UAV's continuous mobility, the communication durations of devices in different locations with UAV are not only time-limited, but also vary from each other. To analyze the throughput performance of uplink multiple access control (MAC) protocol, we propose a new analysis model to deal with the communications heterogeneity in the network. Firstly, we divide the devices in the coverage into different clusters according to their communication durations. Then, a quitting probability indicating the probability that a device quits the UAV's coverage at each time slot is clarified. A modified three-dimensional Markov chain model adopting the quitting probability and cluster division is developed for the performance analysis. Besides, we also propose a modified CSMA/CA protocol which fully considers the heterogeneity of the access time and adaptively allocates the time resource among the devices in different clusters. Finally, the effects of retry limit, initial contention window size, the density of the devices, UAVs speed and coverage area are discussed in the simulation section.

Performance analysis of secondary users under heterogeneous licensed spectrum environment in cognitive radio ad hoc networks

Cognitive radio (CR) is a hopeful technology to sort out spectrum scarcity and underutilization problem in ad hoc networks. With the help of cognitive radio technology, unlicensed users can efficiently utilize the unused part of heterogeneous licensed spectrum. In this article, we present a three-dimensional (3D) Markov chain analysis for spectrum management scheme under heterogeneous licensed bands of two different licensed spectrum pools in cognitive radio ad hoc networks. We present the concept of interpool and intrapool spectrum handoff in the proposed model and derive blocking probability, dropping probability, non-completion probability, and throughput to estimate the performance of the secondary users under heterogeneous licensed spectrum environment. The impact of secondary users dynamic along with the primary users’ activity model on the performance measuring metrics in terms of blocking probability, dropping probability, non-completion probability, and throughput for three different cases is also investigated. The proposed model offers significant improvement in the performance of secondary users under heterogeneous licensed spectrum environment in a CR ad hoc network.

Analysis of Queueing System MMPP/M/K/K with Delayed Feedback

The model of multi-channel queuing system with Markov modulated Poisson process (MMPP) flow and delayed feedback is considered. After the customer is served completely, they will decide either to join the retrial group again for another service (feedback) with some state-dependent probability or to leave the system forever with complimentary probability. Feedback calls organize an orbit of repeated calls (r-calls). If upon arrival of an r-call all the channels of the system are busy, then it either leaves the system with some state-dependent probability or with a complementary probability returns to orbit. Methods to calculate the steady-state probabilities of the appropriate three-dimensional Markov chain as well as performance measures of investigated system are developed. Results of numerical experiments are demonstrated.

Hybrid ARQ-CQI Feedback-Based Access Scheme in Cognitive Radio Networks

In this paper, we consider a cognitive radio (CR) network where the primary network’s feedback information is utilized to develop an access scheme for the secondary network to exploit the underutilized primary spectrum resources. Secondary users (SUs) identify the spectrum opportunities by sensing the spectrum for primary users (PUs) activities and by listening to the PUs feedback. The feedback signals monitored in this research work are the channel quality indicator (CQI) and automatic repeat request (ARQ) available in the PUs network. For detecting the PUs activities, SUs employ soft energy sensing, where SUs access the PUs’ channel with access probabilities that are based on the sensed PUs’ energy level. The access probabilities are optimized to maximize the SUs service rate while maintaining the PUs queues’ stability. The system is modeled as a three-dimensional Markov chain (MC) that captures the number of packets in the PUs queues and the state of the two observed PUs’ feedback signals. The performance of the system is evaluated by deriving the SUs service rate and the average PUs packet delay. We compare the performance of the proposed system with other baseline systems utilizing different types of PUs’ feedback signals. Results reveal the improvement in the SUs service rate and the PUs’ delay of the proposed system compared to the baseline systems. This improvement is mainly due to the fact that in our proposed system SUs have access to extra information, in terms of PUs feedback, as compared to other systems. Therefore, SUs in our proposed system can have better inference on the PUs’ activities; thus more collisions between the PUs and the SUs can be avoided, resulting in significant performance gains in terms of SUs’ throughput and PUs’ average delay.Part of this work was published in [1] .

Performance evaluation and Nash equilibrium of a cloud architecture with a sleeping mechanism and an enrollment service

Cloud computing makes it possible for application providers to provide services seamlessly and application users to receive services adaptively. By offering services that give users an initial experience, application providers can usually attract more users. This research proposes a type of sleeping mechanism-based cloud architecture where an experience service and an enrollment service are provided on one virtual machine (VM). Accordingly, we model the cloud architecture as a queue with an asynchronous multi-vacation and a selectable extra service. We also analyze the queueing model in the steady state by constructing a three-dimensional Markov chain. Following this, we evaluate the system performance of the proposed cloud architecture based on the energy conservation level of the system and the mean delay of the visitors who select the enrollment service. Moreover, we study the Nash equilibrium strategy of visitors by building an individual welfare function, and develop an improved intelligent search algorithm to investigate the socially optimal strategy of visitors. Aiming to achieve a social optimum, we formulate a pricing policy with a reasonable enrollment fee.

The distribution of the line length in a discrete time [formula omitted] queue

We consider methods to find the distribution of the line length in a discrete-time GI∕G∕1 queue with bounded inter-arrival and service times. Here, the term line is meant to include all elements in the system. The methods in question can be divided into two groups: direct and indirect methods. The direct methods formulate the problem as a three-dimensional Markov chain, with one dimension indicating the line length, one the arrival process, and the third one the departure process. The indirect methods find first the waiting time distribution or the roots defining this distribution, and then use this information to determine the distribution of the line length. As it turns out, the indirect methods are faster, often by many orders of magnitude. All indirect methods discussed in literature use characteristic roots. However, we present a new indirect method that does not use roots. We also present a new indirect method based on roots. This method leads to much simpler formulas. Numerical tests show that the indirect methods we developed are indeed much faster than any direct method described in literature.

Performance Analysis of IEEE 802.15.6-Based Coexisting Mobile WBANs With Prioritized Traffic and Dynamic Interference

Intelligent wireless body area networks (WBANs) have entered into an incredible explosive popularization stage. WBAN technologies facilitate real-time and reliable health monitoring in e-healthcare and creative applications in other fields. However, due to the limited space and medical resources, deeply deployed WBANs are suffering severe interference problems. The interference affects the reliability and timeliness of data transmissions, and the impacts of interference become more serious in mobile WBANs because of the uncertainty of human movement. In this paper, we analyze the dynamic interference taking human mobility into consideration. The dynamic interference is investigated in different situations for WBANs coexistence. To guarantee the performance of different traffic types, a health critical index is proposed to ensure the transmission privilege of emergency data for intra- and inter-WBANs. Furthermore, the performance of the target WBAN, i.e., normalized throughput and average access delay, under different interference intensity are evaluated using a developed three-dimensional Markov chain model. Extensive numerical results show that the interference generated by mobile neighbor WBANs results in 70% throughput decrease for general medical data and doubles the packet delay experienced by the target WBAN for emergency data compared with single WBAN. The evaluation results greatly benefit the network design and management as well as the interference mitigation protocols design.