Multi-dimensional Markov Chain Research Articles

An analytical framework in LEO mobile satellite systems servicing batched Poisson traffic

The authors consider a low earth orbit (LEO) mobile satellite system (MSS) that accepts new and handover calls of multirate service-classes. New calls arrive in the system as batches, following the batched Poisson process. A batch has a generally distributed number of calls. Each call is treated separately from the others and its acceptance is decided according to the availability of the requested number of channels. Handover calls follow also a batched Poisson process. All calls compete for the available channels under the complete sharing policy. By considering the LEO-MSS as a multirate loss system with ‘satellite-fixed’ cells, it can be analysed via a multi-dimensional Markov chain, which yields to a product form solution (PFS) for the steady-state distribution. Based on the PFS, they propose a recursive and yet efficient formula for the determination of the channel occupancy distribution, and consequently, for the calculation of various performance measures including call blocking and handover failure probabilities. The latter are much higher compared to the corresponding probabilities in the case of the classical (and less bursty) Poisson process. Simulation results verify the accuracy of the proposed formulas. Furthermore, they discuss the applicability of the proposed model in software-defined LEO-MSS.

Spectrum access in cognitive smart-grid communication system with prioritized traffic

Smart grid (SG) comprises heterogeneous characteristics of traffic such as control commands (safety, sensing, smart-meter readings) among power generation, transmission and end users. An effective and reliable two-way communication infrastructure between users and utility providers is important for improving the performance of smart grid. To support such traffic types in the smart grid, we have studied cognitive radio communication infrastructure-based smart grid. In this paper, we proposed a prioritized spectrum access scheme for cognitive smart grid communication systems to address the quality of service (QoS) requirements of heterogeneous traffic. SG traffic is categorized into two priority classes (high and low based on service requirements). We model the proposed spectrum access scheme (by considering the spectrum sensing error) using a multidimensional Markov chain. We have measured the performance based on the blocking probability, the dropping probability, the interference probability and the call completion rate for both the traffic types (high and the low class smart grid traffic). By Considering an environment where the traffic is heavy tailed, we have modeled the arrivals to follow a Pareto distribution, as the Poisson process model may under estimate the traffic especially when traffic is heavy tailed. The results illustrate that the proposed priority based scheme is able to significantly improve the QoS of high-priority traffic. It is also observed that the prioritized system is preferable over the non-prioritized system where all traffic types are treated the same in terms of SG data delivery.

Effects of using Synthesized Driving Cycles on Vehicle Fuel Consumption

Creating a driving cycle (DC) for the design and validation of new vehicles is an important step that will influence the efficiency, functionality and performance of the final systems. In this work, a DC synthesis method is introduced, based on multi-dimensional Markov Chain, where both the velocity and road slope are investigated. Particularly, improvements on the DC synthesis method are proposed, to reach a more realistic slope profile and more accurate fuel consumption and CO2 emission estimates. The effects of using synthesized DCs on fuel consumption are investigated considering three different vehicle models: conventional ICE, and full hybrid and mild hybrid electric vehicles. Results show that short but representative synthetic DCs will results in more realistic fuel consumption estimates (e.g. in the 5%-10% range) and in much faster simulations. Using the results of this proposed method also eliminates the need to use very simplified DCs, as the New European Driving Cycle(NEDC), or long, measured DCs.

Performance Analysis of a MAC Protocol Consisting of EDCA on the CCH and a Reservation on the SCHs for the IEEE 802.11p/1609.4 WAVE

We propose an analytical model for the medium access control (MAC) protocol that consists of an enhanced distributed channel access (EDCA) on the control channel (CCH) and a reservation method on the service channels (SCHs) for the IEEE 802.11p/1609.4 wireless access in vehicular environments of intelligent transportation systems. Specifically, a safety packet of a high priority and a request for service (RFS) of a low priority for the SCH reservation are serviced on the CCH with a contention-based EDCA mechanism; meanwhile, nonsafety applications such as map download and commercial advertisements are serviced on an SCH with a contention-free scheme through the reservation of an SCH from the handshaking of an RFS packet on the CCH. To satisfy the safety packet quality-of-service (QoS) requirements regarding a within-100-ms delay and a successful delivery probability higher than 98%, we assume that each on-board unit of a vehicle is equipped with two transceivers, one of which always stays tuned on the CCH, while the other is tuned on the assigned SCH; furthermore, a roadside unit sends an acknowledgment message to the broadcasted safety packets to guarantee a high successful delivery probability. By constructing multidimensional Markov chains for the proposed MAC, we derive the intended successful delivery probability and the delay distribution of both the safety packet and the RFS packet. Numerical results show that the QoS requirements for the safety packet are satisfied, i.e., safety packets can be delivered within 100 ms with a successful delivery probability above 98%, even when the number of vehicles reaches up to 150. We compare the results of our proposed contention-free access scheme in the SCH selected by reservation with those of a contention-based access scheme in the SCH selected by randomness.

Retrial queue with discipline of adaptive permanent pooling

• A single-server retrial queue with novel customer service discipline is introduced and analysed. • Transparent ergodicity condition is derived. • Key performance measures of the system are computed. • Numerical results revealing advantages of the proposed discipline are presented. A novel customer service discipline for a single-server retrial queue is proposed and analysed. Arriving customers are accumulated in a pool of finite capacity. Customers arriving when the pool is full go into orbit and attempt to access the service later. It is assumed that customers access the service as a group. The size of the group is defined by the number of customers in the pool at the instant the service commences. All customers within a group finish receiving the service simultaneously. If the pool is full at the point the service finishes, a new service begins immediately and all customers from the pool begin to be served. Otherwise, the customer admission period starts. The duration of this period is random and depends on the number of customers in the pool when the admission period begins. However, if the pool becomes full before the admission period expires, this period is terminated and a new service begins. The system behaviour is described by a multi-dimensional Markov chain. The generator and the condition of ergodicity of this Markov chain are derived, and an algorithm for computing the stationary probability distribution of the states of the Markov chain is given. Formulas for computing various performance measures of the system are presented, and the results of numerical experiments show that these measures essentially depend on the capacity of the pool and the distribution of the duration of the admission period. The advantages of the proposed customer service discipline over the classical discipline and the discipline in which customers cannot enter the pool during the service period are illustrated numerically.

Analysis of an MAP/PH/1 Queue with Flexible Group Service

AbstractA novel customer batch service discipline for a single server queue is introduced and analyzed. Service to customers is offered in batches of a certain size. If the number of customers in the system at the service completion moment is less than this size, the server does not start the next service until the number of customers in the system reaches this size or a random limitation of the idle time of the server expires, whichever occurs first. Customers arrive according to a Markovian arrival process. An individual customer’s service time has a phase-type distribution. The service time of a batch is defined as the maximum of the individual service times of the customers which form the batch. The dynamics of such a system are described by a multi-dimensional Markov chain. An ergodicity condition for this Markov chain is derived, a stationary probability distribution of the states is computed, and formulas for the main performance measures of the system are provided. The Laplace–Stieltjes transform of the waiting time is obtained. Results are numerically illustrated.

Steady-state analysis of a multiclass MAP/PH/cqueue with acyclic PH retrials

AbstractA multiclassc-server retrial queueing system in which customers arrive according to a class-dependent Markovian arrival process (MAP) is considered. Service and retrial times follow class-dependent phase-type (PH) distributions with the further assumption that PH distributions of retrial times are acyclic. A necessary and sufficient condition for ergodicity is obtained from criteria based on drifts. The infinite state space of the model is truncated with an appropriately chosen Lyapunov function. The truncated model is described as a multidimensional Markov chain, and a Kronecker representation of its generator matrix is numerically analyzed.

Analysis of a Priority Queue with Phase-Type Service and Failures

We consider a single server queue with two types of customers. We propose a discipline of flexible priority in access that combines the features of randomization and the threshold type control. We introduce a new class of distributions, phase-type with failures (PHF) distribution, that generalizes the well-known phase-type (PH) distribution to the case when failures can occur during service of a customer. The arrival flow is described by the marked Markovian arrival process. The service time distribution is of PHF type with the parameters depending on the type of a customer. Customers of both types can be impatient. Behavior of the system is described by the multidimensional Markov chain. Problem of existence and computation of the stationary distribution of this Markov chain is discussed in brief as well as the problem of computation of the key performance measures of the system. Numerical examples are presented that give some insight into behavior of the system performance measures under different values of the parameters defining the strategy of customers access to service.

Queueing System Operating in Random Environment as a Model of a Cell Operation

We consider a multi-server queueing system without buffer and with two types of customers as a model of operation of a mobile network cell. Customers arrive at the system in the marked Markovian arrival flow. The service times of customers are exponentially distributed with parameters depending on the type of customer. A part of the available servers is reserved exclusively for service of first type customers. Customers who do not receive service upon arrival, can make repeated attempts. The system operation is influenced by random factors, leading to a change of the system parameters, including the total number of servers and the number of reserved servers. The behavior of the system is described by the multi-dimensional Markov chain. The generator of this Markov chain is constructed and the ergodicity condition is derived. Formulas for computation of the main performance measures of the system based on the stationary distribution of the Markov chain are derived. Numerical examples are presented.

Synthesis of Realistic Driving Cycles With High Accuracy and Computational Speed, Including Slope Information

This paper describes a new method to synthesize driving cycles, where not only the velocity is considered but the road slope information of the real-world measured driving cycle as well. Driven by strict emission regulations and tight fuel targets, hybrid or electric vehicle manufacturers aim to develop new and more energy- and cost-efficient powertrains. To enable and facilitate this development, short, yet realistic, driving cycles need to be synthesized. The developed driving cycle should give a good representation of measured driving cycles in terms of velocity, slope, acceleration, and so on. Current methods use only velocity and acceleration and assume a zero road slope. The heavier the vehicle is, the more important the road slope becomes in powertrain prototyping (as with component sizing or control design); hence, neglecting it leads to unrealistic or limited designs. To include the slope, we extend existing methods and propose an approach based on multidimensional Markov chains. The validation of the synthesized driving cycle is based on a statistical analysis (as the average acceleration or maximum velocity) and a frequency analysis. This new method demonstrates the ability of capturing the measured road slope information in the synthesized driving cycle. Furthermore, results show that the proposed method outperforms current methods in terms of accuracy and speed.

Optimization of the service strategy in a queueing system with energy harvesting and customers’ impatience

Abstract A single-server queueing system with an infinite buffer is considered. The service of a customer is possible only in the presence of at least one unit of energy, and during the service the number of available units decreases by one. New units of energy arrive in the system at random instants of time if the finite buffer for maintenance of energy is not full. Customers are impatient and leave the system without service after a random amount of waiting time. Such a queueing system describes, e.g., the operation of a sensor node which harvests energy necessary for information transmission from the environment. Aiming to minimize the loss of customers due to their impatience (and maximize the throughput of the system), a new strategy of control by providing service is proposed. This strategy suggests that service temporarily stops if the number of customers or units of energy in the system becomes zero. The server is switched off (is in sleep mode) for some time. This time finishes (the server wakes up) if both the number of customers in the buffer and the number of energy units reach some fixed threshold values or when the number of energy units reaches some threshold value and there are customers in the buffer. Arrival flows of customers and energy units are assumed to be described by an independent Markovian arrival process. The service time has a phase-type distribution. The system behavior is described by a multi-dimensional Markov chain. The generator of this Markov chain is derived. The ergodicity condition is presented. Expressions for key performance measures are given. Numerical results illustrating the dependence of a customer’s loss probability on the thresholds defining the discipline of waking up the server are provided. The importance of the account of correlation in arrival processes is numerically illustrated.

Analysis and optimization of Guard Channel Policy with buffering in cellular mobile networks

A multi-server retrial queue with two types of calls (handover and new calls) is analyzed. This queue models the operation of a cell of a mobile communication network. Calls of two types arrive at the system according to the marked Markovian arrival process. The service times of both types of calls have phase-type distributions with different service rates. Handover calls have priority over new calls. Priority is ensured by provision of a dedicated buffer and reservation of several servers of the system exclusively for the service of handover calls. A handover call is dropped and leaves the system if all servers are busy and the buffer is full at the arrival epoch. During the waiting time in the buffer, the handover calls are impatient and may leave the system without service. A new call is blocked if all servers available to new calls are busy. Such a call has options to balk (to leave the system without obtaining service) or to retry later. The behavior of the system is described by the multi-dimensional Markov chain belonging to the class of the asymptotically quasi-Toeplitz Markov chains (AQTMC). In this paper, the generator of this Markov chain is constructed, and it is shown that the chain is ergodic for any set of the system parameters. Formulas for computing various performance measures of the system based on the stationary distribution of the Markov chain are derived. The results of numerical experiments illustrating the behavior of key performance measures of the system depending on the number of reserved servers are presented. A highly positive effect of buffering the handover calls and the server’s reservation is demonstrated.

MultiVCRank With Applications to Image Retrieval.

In this paper, we propose and develop a multi-visual-concept ranking (MultiVCRank) scheme for image retrieval. The key idea is that an image can be represented by several visual concepts, and a hypergraph is built based on visual concepts as hyperedges, where each edge contains images as vertices to share a specific visual concept. In the constructed hypergraph, the weight between two vertices in a hyperedge is incorporated, and it can be measured by their affinity in the corresponding visual concept. A ranking scheme is designed to compute the association scores of images and the relevance scores of visual concepts by employing input query vectors to handle image retrieval. In the scheme, the association and relevance scores are determined by an iterative method to solve limiting probabilities of a multi-dimensional Markov chain arising from the constructed hypergraph. The convergence analysis of the iteration method is studied and analyzed. Moreover, a learning algorithm is also proposed to set the parameters in the scheme, which makes it simple to use. Experimental results on the MSRC, Corel, and Caltech256 data sets have demonstrated the effectiveness of the proposed method. In the comparison, we find that the retrieval performance of MultiVCRank is substantially better than those of HypergraphRank, ManifoldRank, TOPHITS, and RankSVM.

A Dynamic Hop-Aware Buffer Management Scheme for Multi-hop Ad Hoc Networks

In this letter, we propose a dynamic hop-aware buffering scheme to reduce the packets loss and delay in MANETs. In that scheme, the buffer is partitioned virtually based on the number of traversed hops and the significance of the packet. Furthermore, each virtual partition size changes dynamically according to the traffic load and the partition’s safeguard thresholds. We evaluate the performance of the proposed scheme through a multi-dimensional continuous-time Markov chain and simulation, where important performance metrics, i.e., end to end delay and blocking probability are derived.

Evaluation of Blocking Probability for Downlink in Poisson Networks

In recent years, wireless network modeling using stochastic geometry tools, like Poisson Point Process (PPP), has acquired the attention of researchers. Modeling of interference, coverage, and capacity for cellular networks using PPP is presented in literature. However, the blocking probability remains to be investigated. In this letter, we present an analytical framework for evaluation of network blocking probability for downlink. The services can be categorized using multiple classes based on the average resource units required by the service. The multi-class service scenario is modeled using multi-dimensional Markov Chain. The Markov chain is further approximated using the Erlang approximation method to evaluate the activity factor of a base station (BS). Moreover, an approximate closed form expression for coverage probability is derived, which is applicable to interference as well as noise limited scenario. Exploiting the coupling between coverage and activity factor, we determine the class-wise arrival rate. Further, the class-wise arrival rates are used to determine the networks blocking probability. We validate the proposed analytical framework through the extensive simulation.

Multiple networks modules identification by a multi-dimensional Markov chain method

As a general approach to study interactions among small biological molecules such as genes and proteins, network analysis has aroused great interest of people from various research disciplines. However, the construction of network is usually quite sensitive to noise which is unavoidable in real data. Besides, the parameter selections for network construction can also affect the result significantly. These two factors largely decrease the consistency of results generated in network analysis. In particular, we consider detecting closely connected subgraphs named module structure. As an important common property of biological networks, this module structure is often destroyed corrupted by both noise and poor parameter selections in network construction. To conquer these two disadvantages to improve the consistency of module structure identified, we propose to process multiple networks for same set of biological molecules simultaneously for common module structure. More specifically, we combine multiple networks together by building an order 3 tensor data with each layer as one of the multiple networks. Then given any molecule(s) as prior information, a novel tensor-based Markov chain algorithm is proposed to iteratively detect the module that includes the prior node. Moreover, the proposed algorithm is capable of evaluating the contribution scores of each network to the detected module structure. The contribution scores from multiple networks can be not only useful criteria to measure the consistency of module structure, but also valid indicator of corruption in networks. To demonstrate the effectiveness and efficiency of the proposed tensor-based Markov chain algorithm, experimental results on synthetic data set as well as two real gene co-expression data sets of human beings are reported. We also validate that the identified common modules are biologically meaningful.

Bounded truncation error for long-run averages in infinite Markov chains

We consider long-run averages of additive functionals on infinite discrete-state Markov chains, either continuous or discrete in time. Special cases include long-run average costs or rewards, stationary moments of the components of ergodic multi-dimensional Markov chains, queueing network performance measures, and many others. By exploiting Foster-Lyapunov-type criteria involving drift conditions for the finiteness of long-run averages we determine suitable finite subsets of the state space such that the truncation error is bounded. Illustrative examples demonstrate the application of this method.

Bounded truncation error for long-run averages in infinite Markov chains

We consider long-run averages of additive functionals on infinite discrete-state Markov chains, either continuous or discrete in time. Special cases include long-run average costs or rewards, stationary moments of the components of ergodic multi-dimensional Markov chains, queueing network performance measures, and many others. By exploiting Foster-Lyapunov-type criteria involving drift conditions for the finiteness of long-run averages we determine suitable finite subsets of the state space such that the truncation error is bounded. Illustrative examples demonstrate the application of this method.

Priority retrial queueing model operating in random environment with varying number and reservation of servers

We consider a multi-server queueing system with two types of customers operating in the Markovian random environment. Under the fixed state of the random environment, the arrival flow is described by the marked Markovian arrival process. Type 1 customers have preemptive priority over type 2 customers. Type 1 customer is lost only if, at its arrival moment, all servers are busy by type 1 customers. The service times of both types of customers have an exponential distribution with the rate depending on the type of a customer. Type 2 customer is accepted for service if the number of busy servers at its arrival epoch does not exceed a fixed threshold. Otherwise, the arriving customer makes a randomized choice to leave the system permanently (to balk) or to join so called orbit and try to obtain service later. The inter-retrial times have an exponential distribution. Customers in orbit can be impatient and may leave the system after an exponentially distributed amount of time. Due to preemptive priority of type 1 customers, service of type 2 customer can be interrupted. In this case, the interrupted customer makes a randomized choice to leave the system permanently or go into orbit. When the random environment changes its state, immediately the following parameters of the system change their value: the total number of servers, the number of servers available for type 2 customers, the matrices defining the arrival process, the rates of service of customers, the rate of retrials, the impatience intensity, the probability of balking the system at the arrival moment or the moment of termination of service of type 2 customer. Behavior of the system is described by the level dependent multi-dimensional Markov chain that belongs to the class of asymptotically quasi-Toeplitz Markov chains. This allows us to derive the ergodicity condition for this Markov chain and compute its stationary distribution. The main performance measures of the system are expressed via the stationary state probabilities. Numerical illustrations are presented.

Modeling Study on Dynamic Spectrum Sharing System Under Interference Temperature Constraints in Underground Coal Mines

This paper proposes a dynamic spectrum access system for underground wireless communication——a dynamic spectrum sharing system under interference temperature constrains. It can make the best of spectrum resources and improve the utilization efficiency. Then, a multi-dimensional Markov chain is used to model the system. On the basis, the secondary users’ performance under interference temperature constrains is obtained. As two important performance indexes to measure secondary users’ performance, cognitive users’ interrupting probability and blocking probability are calculated. Finally, cognitive users’ performance under different users’ access is analyzed, and the performances in dynamic spectrum access system and overlay access system are compared. Simulation results indicate that the dynamic spectrum sharing access system under interference temperature constrains is superior to the overlay access system and helpful to improve the spectrum sharing system in coal mines.