Markovian Arrival Process Research Articles

재시도 준개방형 대기행렬네트워크 분석

A semi-open queueing network is considered. The network of an arbitrary topology consists of a finite number of nodes operation of which is modeled by single-server queueing systems with a buffer and an exponential service time distribution. Customers arrive at the network according to a Markovian arrival process. The total number of customers, which can be processed simultaneously, is restricted by a fixed in advance threshold. If the number of customers in the network at a new customer arrival moment is less than the threshold, this customer is admitted for service. Otherwise, the customer temporarily leaves the system (it is said that the customer joins the orbit) and retries to get access to the network after the random time intervals duration of which has an exponential distribution. The capacity of the orbit is infinite. After the admission to the network, the customer starts sequential service in a random number of nodes. The choice of the first and the subsequent nodes for service is made stochastically according to a fixed stochastic vector and a transition probability matrix. The behavior of the network is described by the continuous-time Markov chain that belongs to the class of asymptotically quasi-Toeplitz Markov chains. This allows to derive a transparent condition for the ergodicity of the chain and compute its steady-state distribution. Expressions for computation of various performance measures of the network are derived. Numerical results are presented. The model can be applied, e.g., for analysis and optimization of the operation of logistics and manufacturing systems, and the wireless telecommunication networks.

Fitting Markovian binary trees using global and individual demographic data

We consider a class of continuous-time branching processes called Markovian binary trees (MBTs), in which the individuals lifetime and reproduction epochs are modelled using a transient Markovian arrival process (TMAP). We develop methods for estimating the parameters of the TMAP by using either age-specific averages of reproduction and mortality rates, or age-specific individual demographic data. Depending on the degree of detail of the available information, we follow a weighted non-linear regression or a maximum likelihood approach. We discuss several criteria to determine the optimal number of states in the underlying TMAP. Our results improve the fit of an existing MBT model for human demography, and provide insights for the future conservation management of the threatened Chatham Island black robin population.

Fitting procedure for the two-state Batch Markov modulated Poisson process

The Batch Markov Modulated Poisson Process (BMMPP) is a subclass of the versatile Batch Markovian Arrival Process (BMAP) which has been proposed for the modeling of dependent events occurring in batches (such as group arrivals, failures or risk events). This paper focuses on exploring the possibilities of the BMMPP for the modeling of real phenomena involving point processes with group arrivals. The first result in this sense is the characterization of the two-state BMMPP with maximum batch size equal to K, the BMMPP2(K), by a set of moments related to the inter-event time and batch size distributions. This characterization leads to a sequential fitting approach via a moments matching method. The performance of the novel fitting approach is illustrated on both simulated and a real teletraffic data set, and compared to that of the EM algorithm. In addition, as an extension of the inference approach, the queue length distributions at departures in the queueing system BMMPP/M/1 is also estimated.

A stochastic inventory system with replacement of perishable items

This article presents a continuous review perishable inventory system in which the perished items will be replaced by the supplier at a later time. Demands occur according to a Markov arrival process. The items in the inventory have exponential life times and these perished items are stored in a place, called pool for replacement. The (s, S) ordering policy is adopted. At the time of placing an order, the ordering quantity is adjusted with number of items in the pool. The lead time is assumed to have phase type distribution. The joint probability distribution of the inventory level and the number of pooled items is obtained in the steady state case using the matrix-geometric methods. Various system performance measures in the steady state are derived and the total expected cost rate is calculated under a prefixed cost structure. The results derived in this work are numerically illustrated.

On a queueing-inventory system with impatient customers, advanced reservation, cancellation, overbooking and common life time

We consider a queueing-inventory problem with Markovian arrival process, phase type distributed service time. The life time of items is exponentially distributed; all items perish together which means they have a common life time. Reservation (purchase) and cancellation of purchased items is permitted as long as the common life time is not realized. In addition overbooking of items upto a certain maximum is also permitted. When system is fully overbooked waiting customers tend to leave the system. On realization of common life time, instantly S items are procured resulting in the commencement of next cycle. In the particular case of Poisson process and exponentially distributed service time we show that the system admits asymptotic product form solution. System state characteristics are computed which are then numerically illustrated for different sets of values for the input parameters.

Analysis of Queueing System with Non-Preemptive Time Limited Service and Impatient Customers

We consider a single-server queueing system with server vacations as the important component of the polling queueing model of a real-world system. Period of continuous operation of the server (the maximum server attendance time) is restricted, but the service of a customer cannot be interrupted when this period expires. Such features are inherent for many real-world systems with resource sharing. We assume that the customers arrival is described by the Markovian Arrival Process and service, vacation and maximum server attendance times have a phase-type distribution. We derive the stationary distributions of the system states and waiting time. Taking in mind the necessity of further application of the results to modeling the polling queueing systems, the distribution of the server visiting time is derived. Extensive numerical results are presented. They highlight that an account of the coefficient of variation of vacation and maximum attendance time is very important for exact evaluation of the key performance measures of the system, while only the results for the coefficient of variation equal to zero or one are known in the literature. Impact of the possible customers impatience, which is intuitively important because the time-limited service is considered, is confirmed by the results of the numerical experiments. Optimization problem of matching the durations of vacation and maximum attendance time is considered.

Discrete-Time Analysis of Cognitive Radio Networks with Nonsaturated Source of Secondary Users

Sensing is a fundamental aspect in cognitive radio networks and one of the most complex issues. In the design of sensing strategies, a number of tradeoffs arise between throughput, interference to primary users, and energy consumption. This paper provides several Markovian models that enable the analysis and evaluation of sensing strategies under a broad range of conditions. The occupation of a channel by primary users is modeled as alternating idle and busy intervals, which are represented by a Markov phase renewal process. The behavior of secondary users is represented mainly through the duration of transmissions, sensing periods, and idle intervals between consecutive sensing periods. These durations are modeled by phase-type distributions, which endow the model with a high degree of generality. Unlike our previous work, here the source of secondary users is nonsaturated, which is a more practical assumption. The arrival of secondary users is modeled by the versatile Markovian arrival process, and models for both finite and infinite queues are built. Furthermore, the proposed models also incorporate a quite general representation of the resumption policy of an SU transmission after being interrupted by PUs activity. A comprehensive analysis of the proposed models is carried out to derive several key performance indicators in cognitive radio networks. Finally, some numerical results are presented to show that, despite the generality and versatility of the proposed models, their numerical evaluation is perfectly feasible.

Threshold Control Policy Optimization for Real-Time Reverse Referral Decision of Chinese Comprehensive Hospitals

In recent years, imbalanced utilization of medical resources is widely concerned within the tiered Chinese hospital system. Reverse referral, as a measure of promoting patient flows from upper level hospitals (ULHs) to lower level hospitals (LLHs), has demonstrated its advantages on alleviating ULH workload and balancing resource utilization between ULHs and LLHs. Nevertheless, it remains unclear on how to control the reverse referral decision process at the operational level. In this paper, we consider an ULH-dominant setting at which the LLH must accept patient referrals from the ULH whenever it has available beds. We focus our attention on an easy-to-implement threshold policy for the ULH to make the reverse referral decision. To investigate, we first formulate an analytically tractable queueing model for a simplified reverse referral process. We then investigate a more general patient flow control model, for which we analyze the patient population dynamics with a Markov chain process, and apply the concept of state-dependent Markovian arrival process to generate an infinitesimal generator of the system. We use RG factorization to compute the system performance measures. We next formulate a threshold optimization problem with the objective of maximizing the ULH profit. Simulation experiments are performed, which conclude that the threshold control policy is insensitive to the service time distribution. Finally, we report real-world inspired numerical studies, from which we generate insights into effective adjustment of the control threshold in response to the system parameters and discuss potential hindrance from the LLH incorrectly informing its real-time resource availability to the ULH. Our work is the first that applies systems engineering to the real-time reverse referral decision problem in China. It provides the novel perspective of resource balancing to patient flow control studies in the care transition management literature. Note to Practitioners —It is recognized, in almost all countries with insufficient health care delivery infrastructure, that the care coordination is a key strategy to improve the effectiveness, safety, and efficiency of the health care system. Through communicating/sharing knowledge, multiple care organizations work together to create a proactive care plan for the patients to meet their needs and preferences, especially dealing with transitions of care. With the promotion of population care management, practitioners throughout the world have increasingly acknowledged the importance of patient flow streamlining, resource alignment within each population, and linking to community care resources from acute care facilities. However, applying care coordination in everyday routines of a medical practice can be overwhelming, even when it is obvious that the changes will improve patient care and provide efficiency. In this paper, we investigate an easy-to-implement patient flow control policy in making reverse referral decisions for patients receiving postdischarge care. We expect to improve provider’s profit with improved operational management. Through our analyses, we want to stress the importance of applying systems engineering and mathematical modeling to study care coordination and interorganization patient flow control. We also want to advocate knowledge sharing and accountability establishment, as a well-functioned partnership between hospitals at different tiers may very much rely on them.

<i>MAP/PH</i>(1), <i>PH</i>(2)/2 finite retrial inventory system with service facility, multiple vacations for servers

In this paper, we consider a retrial (s, S) inventory system with multiple server vacations for two heterogeneous servers. We have assumed that the customers arrive according to a Markovian arrival process and two parallel servers who provide heterogeneous phase type services to customers. The lead times for the orders are assumed to have independent and identical exponential distributions. The vacation times of both servers are assumed to be independent and identically distributed exponential random variables. The arriving customer who finds both servers are busy or both servers are on vacation, joins an orbit of finite size. These orbiting customers retry for their demand after a random time, which is assumed to be exponential distribution. Some important performance measures are obtained and the optimality of an expected total cost rate is shown through numerical illustration.

Performance Analysis of a Polling Model With BMAP and Across-Queue State-Dependent Service Discipline

As various video services become popular, video streaming will dominate the mobile data traffic. The H.264 standard has been widely used for video compression. As the successor to H.264, H.265 can compress video streaming better, hence it is gradually gaining market share. However, in the short term H.264 will not be completely replaced, and will co-exist with H.265. Using H.264 and H.265 standards, three types of frames are generated, and among different types of frames exist dependencies. Since the radio resources are limited, using dependencies and quantities of frames in buffers, an appropriate time division transmission policy can be applied to transmit different types of frames sequentially, in order to avoid the occurrence of video carton or decoding failure. Polling models with batch Markovian arrival process (BMAP) and across-queue state-dependent service discipline are considered to be effective means in the design and optimization of appropriate time division transmission policies. However, the BMAP and across-queue state-dependent service discipline of the polling models lead to the large state space and several coupled state transition processes, which complicate the performance analysis. There have been very few researches in this regard. In this paper, a polling model of this type is analyzed. By constructing a supplementary embedded Markov chain and applying the matrix-analytic method based on the semi-regenerative process, the expressions of important performance measures including the joint queue length distribution, the customer blocking probability and the customer mean waiting time are obtained. The analysis will provide inspiration for analyzing the polling models with BMAP and across-queue state-dependent service discipline, to guide the design and optimization of time division transmission policies for transmitting the video compressed by H.264 and H.265.

Two Extensions of Kingman's GI/G/1 Bound

A simple bound in GI/G/1 queues was obtained by Kingman using a discrete martingale transform. We extend this technique to 1) multiclass Σ\textrmGI/G/1 $ queues and 2) Markov Additive Processes (MAPs) whose background processes can be time-inhomogeneous or have an uncountable state-space. Both extensions are facilitated by a necessary and sufficient ordinary differential equation (ODE) condition for MAPs to admit continuous martingale transforms. Simulations show that the bounds on waiting time distributions are almost exact in heavy-traffic, including the cases of 1) heterogeneous input, e.g., mixing Weibull and Erlang-k classes and 2) Generalized Markovian Arrival Processes, a new class extending the Batch Markovian Arrival Processes to continuous batch sizes.

A discrete MMAP for analysing the behaviour of a multi-state complex dynamic system subject to multiple events

A complex multi-state system subject to different types of failures, repairable and/or non-repairable, external shocks and preventive maintenance is modelled by considering a discrete Markovian arrival process with marked arrivals (D-MMAP). The internal performance of the system is composed of several degradation states partitioned into minor and major damage states according to the risk of failure. Random external events can produce failures throughout the system. If an external shock occurs, there may be an aggravation of the internal degradation, cumulative external damage or extreme external failure. The internal performance and the cumulative external damage are observed by random inspection. If major degradation is observed, the unit goes to the repair facility for preventive maintenance. If a repairable failure occurs then the system goes to corrective repair with different time distributions depending on the failure state. Time distributions for corrective repair and preventive maintenance depend on the failure state. Rewards and costs depending on the state at which the device failed or was inspected are introduced. The system is modelled and several measures of interest are built into transient and stationary regimes. A preventive maintenance policy is shown to determine the effectiveness of preventive maintenance and the optimum state of internal and cumulative external damage at which preventive maintenance should be taken into account. A numerical example is presented, revealing the efficacy of the model. Correlations between the numbers of different events over time and in non-overlapping intervals are calculated. The results are expressed in algorithmic-matrix form and are implemented computationally with Matlab.

Efficient computational analysis of non-exhaustive service vacation queues: BMAP/R/1/N(∞) under gated-limited discipline

• A finite-buffer vacation queue with batch Markovian arrival process has been analyzed. • The server is subjected to serve under gated-limited service discipline. • Proposed analysis is based on the successive substitution and the supplementary variable method. • The results have been matched with the corresponding infinite-buffer queue. • Numerical results are presented for different service- and vacation-time distributions. This paper analyzes the finite-buffer single server queue with vacation(s). It is assumed that the arrivals follow a batch Markovian arrival process ( BMAP ) and the server serves customers according to a non-exhaustive type gated-limited service discipline. It has been also considered that the service and vacation distributions possess rational Laplace-Stieltjes transformation (LST) as these types of distributions may approximate many other distributions appeared in queueing literature. Among several batch acceptance/rejection strategies, the partial batch acceptance strategy is discussed in this paper. The service limit L (1 ≤ L ≤ N ) is considered to be fixed, where N is the buffer-capacity excluding the one in service. It is assumed that in each busy period the server continues to serve until either L customers out of those that were waiting at the start of the busy period are served or the queue empties, whichever occurs first. The queue-length distribution at vacation termination/service completion epochs is determined by solving a set of linear simultaneous equations . The successive substitution method is used in the steady-state equations embedded at vacation termination/service completion epochs. The distribution of the queue-length at an arbitrary epoch has been obtained using the supplementary variable technique. The queue-length distributions at pre-arrival and post-departure epoch are also obtained. The results of the corresponding infinite-buffer queueing model have been analyzed briefly and matched with the previous model. Net profit function per unit of time is derived and an optimal service limit and buffer-capacity are obtained from a maximal expected profit. Some numerical results are presented in tabular and graphical forms.

A Queueing Model and Analysis for Autonomous Vehicles on Highways

We investigate the effects of autonomous vehicles (AVs) on highway congestion. AVs have the potential to significantly reduce highway congestion, since these vehicles are able to maintain smaller inter-vehicle gaps and travel together in larger platoons (or batches) than human-driven vehicles (HVs). Various policies have been proposed to regulate AVs on highways, yet no in-depth comparison of these policies exists. To address this shortcoming, we develop a queueing model for a multi-lane highway, and analyze two policies for a mixed fleet of HVs and AVs: the designated-lane policy (“D policy”) under which one lane is designated to AVs, and the integrated policy (“I policy”) under which AVs travel together with HVs in all lanes. Our analysis connects the service rate of the queueing system to congestion on the highway, as well as inter-vehicle gaps using a Markovian arrival process (MAP). We measure the performance of these policies using mean travel time and throughput as metrics. We show that although the I policy performs at least as well as a benchmark case with no AVs, the D policy outperforms the benchmark only when the highway is heavily congested and AVs constitute the majority of vehicles; in such a case this policy may outperform the I policy as well. These findings caution against recent industry and government proposals that the D policy should be employed at the beginning of the mass appearance of AVs. Finally, we calibrate our model to data; our numerical analysis shows that for highly congested highways, a moderate number of AVs can make substantial improvement (e.g., 22% AVs can improve throughput by 30%), and when all vehicles are AVs, throughput can be increased over 400%.

Modelling and analysis of the impact of correlated inter-event data on production control using Markovian arrival processes

Empirical studies show that the inter-event times of a production system are correlated. However, most of the analytical studies for the analysis and control of production systems ignore correlation. In this study, we show that real-time data collected from a manufacturing system can be used to build a Markovian arrival processes (MAP) model that captures correlation in inter-event times. The obtained MAP model can then be used to control production in an effective way. We first present a comprehensive review on MAP modeling and MAP fitting methods applicable to manufacturing systems. Then we present results on the effectiveness of these fitting methods and discuss how the collected inter-event data can be used to represent the flow dynamics of a production system accurately. In order to study the impact of capturing the flow dynamics accurately on the performance of a production control system, we analyze a manufacturing system that is controlled by using a base-stock policy. We study the impact of correlation in inter-event times on the optimal base-stock level of the system numerically by employing the structural properties of the MAP. We show that ignoring correlated arrival or service process can lead to overestimation of the optimal base-stock level for negatively correlated processes, and underestimation for the positively correlated processes. We conclude that MAPs can be used to develop data-driven models and control manufacturing systems more effectively by using shop-floor inter-event data.

Estimating the earthquake occurrence rates in Corinth Gulf (Greece) through Markovian arrival process modeling

ABSTRACTThe Markovian Arrival Process (MAP) is applied as a candidate model to describe the time-varying earthquake activity in Corinth Gulf, Greece. To the best of our knowledge, this is the first attempt to study the earthquake temporal evolution with the specific class of MAPs. A complete catalogue is used for the earthquake temporal distribution investigation, along with data sets of different magnitude cutoffs. The study area is divided into its western and eastern subareas, and possible variations in the earthquake occurrence times were sought. Hidden states of MAPs correspond to different levels of seismicity, and hence various numbers of states are examined. Akaike and Bayes information criteria are implemented for identifying the best model, and comparison to the most known and broadly accepted theoretical interevent time distributions is provided. In all cases, the fitted MAPs with phase type distributed intearrival times outperform the models with other distributions. Important indicators of the underlying Markov process are computed, and the earthquake frequency is approximated by the counting process. The analysis demonstrates high index of burstiness for the earthquake generation in the eastern part, i.e. long quiescent periods alternate with short ones of intense seismic activity.

Performance Analysis of Shock Models Governed by a Discrete-Time Markovian Arrival Process

This paper considers a system subject to discrete-time non-renewal process of shocks’ arrival, where only the impact of fatal shock is taken into account. The device survives any non fatal shock and instantaneously replaces the failure device by a new and identical one only after a fatal shock. Any shock to an operating device of the system having certain strength of damage leads to its failure only with some probability. We construct the transition probability matrix to compute the stationary probability vector of number of shocks. The system is analysed for both the restricted and unrestricted number of shocks. We provide several key performance measures such as the number of renewals, availability and occurrence of failure. Computational experiences with a variety of numerical results are discussed on system performance measures. This renders relevant information to the system analysts, researchers, and industry people who are interested to investigate congestion problems with the mechanism generating shocks.

Retrial multi-server queuing system with PHF service time distribution as a model of a channel with unreliable transmission of information

• A multi-server retrial queueing system with unreliable service is analyzed. • The ergodicity condition is derived. • New effective algorithm for computing the stationary distribution of level-dependent AQTMC is developed. • Performance measures of the system are computed. • Numerical results showing advantages of the proposed algorithm are presented. A multi-server retrial queue with a Markovian arrival process , as well as possible customer non-persistency and unreliable operation of servers, is analysed. The service of a customer (transmission of information) consists of a random number of phases. During a transmission, a failure can occur. The failure does not cause server breakdown or its subsequent recovery. The failure occurrence during service of a customer implies either that the departure of the customer from the server (the customer joins an orbit and retries later), the customer re-starts complete service immediately, or the customer resumes the service from the phase at which the failure occurred. The total occupation time of a server by a customer has a so-called PHF (phase-type with failures) distribution. The dynamics of the system are described by a multi-dimensional level-dependent Markov chain . An infinitesimal generator of the chain is derived. To obtain the ergodicity condition, we prove that the Markov chain belongs to the class of asymptotically quasi-Toeplitz Markov chains. Because we consider a multi-server queuing system, the dimensions of the blocks of the generator can be large. Thus, the use of a standard algorithm to find a stationary distribution of an asymptotically quasi-Toeplitz Markov chain can face problems in the stage of computer implementation. We develop a new efficient iterative algorithm for approximate computation of the stationary distribution that computes many times faster and requires much fewer computer resources than the standard algorithm. The advantages of the new algorithm and some features of the model under consideration are numerically demonstrated.

On Markovian modelling of arrival processes

Markovian arrival process (MAP) is a popular tool for modeling arrival processes of stochastic systems such as queueing systems, reliability systems and telecommunications networks. In this paper we show how properties of Markovian Arrival Processes can be derived from the general theory of Markov processes with a homogeneous second component. We also present a series of results on queueing systems with MAP arrivals which were published in RUDN University before 1990.

Delay Analysis of Network Coding in Multicast Networks With Markovian Arrival Processes: A Practical Framework in Cache-Enabled Networks

We develop an analytical framework for queueing and delay analysis in the case of a number of distinct flows arriving at a network node, where asynchronous partial network coding is applied for an efficient packet transmission. In order to perfectly model and analyze the practical communication networks, the arriving flows are assumed to be general Markovian arrival processes. As a key example, we apply the proposed model to the problem of coded caching in single bottleneck caching networks. In addition, we verify the accuracy of the proposed model through simulations and real trace-driven experiments.