Markovian Arrival Process Research Articles

Complete analysis of a discrete-time batch service queue with batch-size-dependent service time under correlated arrival process: D-MAP/Gn(a,b)/1

Discrete-time queueing models find a large number of applications as they are used in modeling queueing systems arising in digital platforms like telecommunication systems and computer networks. In this paper, we analyze an infinite-buffer queueing model with discrete Markovian arrival process. The units on arrival are served in batches by a single server according to the general bulk-service rule, and the service time follows general distribution with service rate depending on the size of the batch being served. We mathematically formulate the model using the supplementary variable technique and obtain the vector generating function at the departure epoch. The generating function is in turn used to extract the joint distribution of queue and server content in terms of the roots of the characteristic equation. Further, we develop the relationship between the distribution at the departure epoch and the distribution at arbitrary, pre-arrival and outside observer’s epochs, where the first is used to obtain the latter ones. We evaluate some essential performance measures of the system and also discuss the computing process extensively which is demonstrated by some numerical examples.

Optimizing a Multi-State Cold-Standby System with Multiple Vacations in the Repair and Loss of Units

A complex multi-state redundant system with preventive maintenance subject to multiple events is considered. The online unit can undergo several types of failure: both internal and those provoked by external shocks. Multiple degradation levels are assumed as both internal and external. Degradation levels are observed by random inspections and, if they are major, the unit goes to a repair facility where preventive maintenance is carried out. This repair facility is composed of a single repairperson governed by a multiple vacation policy. This policy is set up according to the operational number of units. Two types of task can be performed by the repairperson, corrective repair and preventive maintenance. The times embedded in the system are phase type distributed and the model is built by using Markovian Arrival Processes with marked arrivals. Multiple performance measures besides the transient and stationary distribution are worked out through matrix-analytic methods. This methodology enables us to express the main results and the global development in a matrix-algorithmic form. To optimize the model, costs and rewards are included. A numerical example shows the versatility of the model.

Modeling of Railway Stations Based on Queuing Networks

Among the micro-logistic transport systems, railway stations should be highlighted, such as one of the most important transport infrastructure elements. The efficiency of the transport industry as a whole depends on the quality of their operation. Such systems have a complex multi-level structure, and the incoming traffic flow often has a stochastic character. It is known that the most effective approach to study the operation of such systems is mathematical modeling. Earlier, we proposed an approach to transport hub modeling using multiphase queuing systems with a batch Markovian arrival process (BMAP) as an incoming flow. In this paper, we develop the method by applying more complex models based on queuing networks that allow us to describe in detail the route of requests within an object with a non-linear hierarchical structure. This allows us to increase the adequacy of modeling and explore a new class of objects—freight railway stations and marshalling yards. Here we present mathematical models of two railway stations, one of which is a freight railway station located in Russia, and the other is a marshalling yard in the USA. The models have the form of queuing networks with BMAP flow. They are implemented as simulation software, and a numerical experiment is carried out. Based on the numerical results, some “bottlenecks” in the structure of the studied stations are determined. Moreover, the risk of switching to an irregular mode of operation is assessed. The proposed method is suitable for describing a wide range of cargo and passenger transport systems, including river ports, seaports, airports, and multimodal transport hubs. It allows a primary analysis of the hub operation and does not need large statistical information for parametric identification.

Analysis of a k-Stage Bulk Service Queuing System with Accessible Batches for Service

In most of the service systems considered so far in queuing theory, no fresh customer is admitted to a batch undergoing service when the number in the batch is less than a threshold. However, a few researchers considered the case of customers accessing ongoing service batch, irrespective of how long service was provided to that batch. A queuing system with a different kind of accessibility that relates to a real situation is studied in the paper. Consider a single server queuing system in which the service process comprises of k stages. Customers can enter the system for service from a node at the beginning of any of these stages (provided the pre-determined maximum service batch size is not reached) but cannot leave the system after completion of service in any of the intermediate stages. The customer arrivals to the first node occur according to a Markovian Arrival Process (MAP). An infinite waiting room is provided at this node. At all other nodes, with finite waiting rooms (waiting capacity cj,2≤j≤k), customer arrivals occur according to distinct Poisson processes with rates λj,2≤j≤k. The service is provided according to a general bulk service rule, i.e., the service process is initiated only if at least a customers are present in the queue at node 1 and the maximum service batch size is b. Customers can join for service from any of the subsequent nodes, provided the number undergoing service is less than b. The service time distribution in each phase is exponential with service rate μjm, which depends on the service stage j,1≤j≤k, and the size of the batch m,a≤m≤b. The behavior of the system in steady-state is analyzed and some important system characteristics are derived. A numerical example is presented to illustrate the applicability of the results obtained.

Computing stationary distributions of the D-MAP/D-MSP$$^{(a,b) }/1$$ queueing system

The present study investigates a single-server batch service queueing model with arrival process as discrete-time Markovian arrival process and service process as discrete-time Markovian service process. The server serves the customers in batches followed by the general bulk-service rule. We first determine the random epoch probabilities using the matrix-geometric method, where the rate matrix $${\mathbf{R}}$$ is determined by an efficient approach based on the eigenvalues and the corresponding eigenvectors of the associated characteristic equation. Next we obtain the explicit closed-form expressions for the pre-arrival, intermediate, outside observer’s and post-departure epoch probabilities by developing the relations among them in equilibrium state. Further, we provide an analytically simple approach to carry out the waiting-time distribution in the queue measured in slots as well as the distribution of the size of a service batch of an arriving customer. We also demonstrate a cost function to evaluate the optimum value of the minimum service batch size and the corresponding expected cost of the system. Finally, an adequate variety of numerical experiments are performed for validation purpose of our analytical results and they are conferred in the form of tables and graphs.

On a queueing inventory problem with necessary and optional inventories

Queueing inventory models are extensively analysed since 1992. Very few among these discuss multi-commodity system. In this paper, we present a multi-commodity queueing inventory problem involving one essential and a set of m optional item(s). Immediately after the service of an essential item, the customer either leaves the system with probability p or with probability 1-p he goes for optional item(s). However, in the absence of an essential item, service will not be provided. More than one optional item can be demanded by the customer. The i th optional item or i th and j th optional items or i th, j th and k th and so on or all the optional items together, could be demanded by a customer, with probabilities $$p_{i}$$ , $$p_{ij}$$ , $$p_{ijk}$$ $$\ldots $$ $$p_{12\ldots m}$$ respectively. If the demanded optional item(s) is(are) not available, the customer leaves the system after purchasing the essential item. With the arrival of customers forming Markovian Arrival Process (MAP), service time of essential item Phase type distributed and that for optional items exponentially distributed( depending on the type(s) of item(s)), all given by the same (single) server, we analyse the system. Then we obtain the system state probability distribution. In-order to get a picture of how the system performs, we derive several characteristics of the system. With control policies for essential and optional items determined respectively, by (s, S) and ( $$s_{i}$$ , $$S_{i}$$ ), $$i=1,2,3,$$ ..., m, we investigate the optimal values of $$s,S,s_{i}$$ and $$S_{i}$$ s’. To this end, we set up a cost function, involving these control variables.

Analysis of a Batch Arrival, Batch Service Queuing-Inventory System with Processing of Inventory While on Vacation

A single-server queuing-inventory system in which arrivals are governed by a batch Markovian arrival process and successive arrival batch sizes form a finite first-order Markov chain is considered in this paper. Service is provided in batches according to a batch Markovian service process, with consecutive service batch sizes forming a finite first-order Markov chain. A service starts for the next batch on completion of the current service, provided that inventory is available at that epoch; otherwise, there will be a delay in starting the next service. When the service of a batch is completed, the inventory decreases by 1 unit, irrespective of batch size. A control policy in which the server goes on vacation when a service process is frozen until a quorum can initiate the next batch service is proposed to ensure idle-time utilization. During the vacation, the server produces inventory (items) for future services until it hits a specified level L or until the number of customers in the system reaches a maximum service batch size N, with whichever occurring first. In the former case, a server stays idle once the processed inventory level reaches L until the number of customers reaches (or even exceeds because of batch arrival) a maximum service batch size N. The time required for processing one unit of inventory follows a phase-type distribution. In this paper, the steady-state probability vector of this infinite system is computed. The distributions of inventory processing time in a vacation cycle, idle time in a vacation cycle, and vacation cycle length are found. The effect of correlation in successive inter-arrival times and service times on performance measures for such a queuing system is illustrated with a numerical example. An optimization problem is considered. The proposed system is then compared with a queuing-inventory system without the Markov-dependent assumption on successive arrivals as well as service batch sizes using numerical examples.

Supervised learning-based approximation method for single-server open queueing networks with correlated interarrival and service times

Efficient performance evaluation methods are needed to design and control production systems. We propose a method to analyse single-server open queueing network models of manufacturing systems composed of delay, batching, merge and split blocks with correlated interarrival and service times. Our method (SLQNA) is based on using a supervised learning approach to determine the mean, the coefficient of variation, and the first-lag autocorrelation of the inter-departure time process as functions of the mean, coefficient of variation and first-lag autocorrelations of the interarrival and service times for each block, and then using the predicted inter-departure time process as the input to the next block in the network. The training data for the supervised learning algorithm is obtained by simulating the systems for a wide range of parameters. Gaussian Process Regression is used as a supervised learning algorithm. The algorithm is trained once for each block. SLQNA does not require generating additional training data for each unique network. The results are compared with simulation and also with the approximations that are based on Markov Arrival Process modelling, robust queueing, and G/G/1 approximations. Our results show that SLQNA is flexible, computationally efficient, and significantly more accurate and faster compared to the other methods.

Ιdentification of spatiotemporal seismicity clusters in central Ionian Islands (Greece)

The spatiotemporal evolution of seismicity of an accurately relocated earthquake catalog in the central Ionian Islands, Greece, is explored, by the systematic identification of the earthquake clusters over the period September 2016–December 2019. Manual waveform phase picking was the input for the relocation leading to 13,632 earthquakes that compose a data set with a completeness magnitude as low as Mc = 1.5. The clustering procedure engages a temporal stochastic point process, the Markovian Arrival Process (MAP), for an initial separation of the background seismicity from potential seismic excitations, using the changes in the seismicity rate and a density-based clustering algorithm, DBSCAN, for the detection of elevated spatial density areas. A high concentration of temporally persistent clusters is identified along the western coastline of Lefkada Island, in the parallel step-over faults between the two major fault branches of the Kefalonia Transform Fault Zone and in the Myrtos Gulf area in Kefalonia, which are spatially correlated to the positive static stress changes induced by the coseismic slip of the Mw6.5 November 17, 2015 Lefkada main shock. The method presented remarkable potential in the identification of clusters in 3-D space along with the seismicity migration during the temporal evolution of seismic sequences. Multiple secondary faults of the Kefalonia segment can be revealed and the dominant triggering mechanisms can be illuminated contributing towards the understanding of the regional faulting properties and mechanics, an outcome that cannot be achieved solely by studying the large earthquakes, due to their rarity. The temporal and spatial clustering properties of microseismicity shed light on the preparation phase of the large earthquakes as being part of this process, thus contributing to the seismic hazard assessment.

A postponed inventory system with modified M vacation policy

In this article, we analyse a postponed inventory system with a single server under modified M vacation policy, where the server can take atmost M inactive mode. We assume that the demand process follows a Markovian arrival process and (s, S) ordering policy with exponential lead time. During the inactive mode, the server can be idle or go on vacation, which occurs due to the depletion of inventory. In every inactive mode, server avails the inactive idle period first followed by a vacation period. Inactive idle period and vacation period follow independent phase type distribution. The demand that arrives during the server inactive mode enters the pool of infinite size. The server selects a demand one by one on FCFS rule from the pool, as long as the inventory level is greater than the reorder point and inter selection time follows exponential distribution. A quasi birth and death process is formulated to analyse the system and solved by using the matrix-geometric method. We explicit some system performance measures on the steady state and some illustrative examples are discussed numerically.

Моделирование крупнейшей в мире железнодорожной сортировочной станции с использованием теории массового обслуживания

The problem of scenario forecasting of the operation of railway stations in the conditions of changing traffic flow is considered. An approach to modeling based on the use of the queuing theory apparatus is proposed, which allows taking into account the presence of significant random impacts on the system operation. The largest railway marshalling yard in the world - Bailey Yard (Nebraska, USA) was chosen as the object. The article presents a mathematical model of its operation in the form of a queuing network, in which, unlike queuing systems, branching and circular movement of applications are allowed. This allows you to display various routes of the movement of cars in the structure of the station. To describe several incoming transport flows, a BMAP stream (Branch Markovian Arrival Process) is used, so it is possible to take into account the parameters of each of them. Analytical results for such mathematical objects are not known, so the model is studied numerically. The amount of traffic flow varies in a series of experiments. Based on the results obtained, the current and maximum capacity of the station, «bottlenecks» in its structure are determined and recommendations are developed to improve the efficiency of its operation.

Analysis of MAP/PH/1 queueing model with immediate feedback, starting failures, single vacation, standby server, delayed repair, breakdown and impatient customers

We have analysed the queueing model with a single server in which customers arrive according to the Markovian arrival process, the service process according to phase type distributions and the standby server who is serving at a lower rate also follows the phase type distribution. The total number of customers are present in the system under the steady-state probability vector has been probed by using matrix-analytic method (MAM). We have examined the stability condition, the analysis of the busy period and derived some important performance measures of our model. Numerical results and graphical representation are discussed for the proposed model.

Modelling and analysis of D-BMAP/D-MSP/1 queue using RG-factorization

ABSTRACT The current investigation of our paper is concentrated on an infinite-buffer single-server queueing model with discrete-time batch Markovian arrival process and Markovian service process. The steady-state behaviour of this discrete-time queueing system at various significant time epochs are presented. We first analyze the system-length distribution at outside observer’s epoch using the -factorization technique combined with the spectral expansion method to compute the stochastic matrix . For the purpose of comparative study, a comprehensive analysis of the system-length distribution at outside observer’s epoch is also carried out using the roots method. Next, we obtain the explicit closed-form expressions of the system-length distributions at pre-arrival, intermediate, post-departure and random epochs by developing the relations among them in equilibrium state. The analysis of waiting-time distribution in the queue measured in slots of an arbitrary customer in an arriving batch is also discussed. Computational experiences with adequate numerical results are conferred in the structure of tables and graphs. Finally, we suggest that our queueing model seems to be fitted for outpatient service of any specialized hospital in a smart city.

Analysis of an aggregate loss model in a Markov renewal regime

In this article we consider an aggregate loss model with dependent losses. The loss occurrence process is governed by a two-state Markovian arrival process (MAP2), a Markov renewal process that allows for (1) correlated inter-loss times, (2) non-exponentially distributed inter-loss times and, (3) overdisperse loss counts. Some quantities of interest to measure persistence in the loss occurrence process are obtained. Given a real OpRisk database, the aggregate loss model is estimated by fitting separately the inter-loss times and severities. The MAP2 is estimated via direct maximization of the likelihood function, and severities are modeled by the heavy-tailed, double-Pareto Lognormal distribution. In comparison with the fit provided by the Poisson process, the results point out that taking into account the dependence and overdispersion in the inter-loss times distribution leads to higher capital charges.

Compound Binomial Model with Batch Markovian Arrival Process

A compound binomial model with batch Markovian arrival process was studied, and the specific definitions are introduced. We discussed the problem of ruin probabilities. Specially, the recursion formulas of the conditional finite-time ruin probability are obtained and the numerical algorithm of the conditional finite-time nonruin probability is proposed. We also discuss research on the compound binomial model with batch Markovian arrival process and threshold dividend. Recursion formulas of the Gerber–Shiu function and the first discounted dividend value are provided, and the expressions of the total discounted dividend value are obtained and proved. At the last part, some numerical illustrations were presented.

Adaptive and Robust Routing With Lyapunov-Based Deep RL in MEC Networks Enabled by Blockchains

The most recent development of the Internet of Things brings massive timely sensitive and bursty data flows. Also, joint optimization on storage, computation, and communication is in need for multiaccess edge computing frameworks. The adaptive network control has been explored using deep reinforcement learning (RL), but it is not sufficient for bursty network traffic flows, especially when the network traffic pattern may change over time. We formulate the routing control in an environment with time-variant link delays as a Lyapunov optimization problem. We identify that there is a tradeoff between optimization performance and modeling accuracy when the propagation delays are included. We propose a novel deep RL (DRL)-based adaptive network routing method to tackle the issues mentioned above. A Lyapunov optimization technique is used to reduce the upper bound of the Lyapunov drift, improving queuing stability in networked systems. By modeling the network traffic pattern using the Markovian arrival process, we show that network routing problems can be modeled as Markov decision processes and value-iteration-based RL methods can be used to solve them. We design a blockchain-based protocol using proof of elapsed time consensus mechanism to ensure a trustworthy network statistics information exchange for the routing framework. Experiment results show that the proposed method can learn a routing policy and adapt to the changing environment. The proposed method outperforms the baseline backpressure method in multiple settings and converges faster than existing methods. Moreover, the DRL module can effectively learn a better estimation of the long-term Lyapunov drift and penalty functions, providing superior results in terms of the backlog size, end-to-end latency, age of information, and throughput. Furthermore, the blockchain-based network statistics exchange can provide the routing framework against malicious nodes. In addition, the proposed model performs well under various topologies, and thus can be used in general cases.

Analysis of a Queueing Model with Batch Markovian Arrival Process and General Distribution for Group Clearance.

In this paper we consider a single server queueing model with under general bulk service rule with infinite upper bound on the batch size which we call group clearance. The arrivals occur according to a batch Markovian point process and the services are generally distributed. The customers arriving after the service initiation cannot enter the ongoing service. The service time is independent on the batch size. First, we employ the classical embedded Markov renewal process approach to study the model. Secondly, under the assumption that the services are of phase type, we study the model as a continuous-time Markov chain whose generator has a very special structure. Using matrix-analytic methods we study the model in steady-state and discuss some special cases of the model as well as representative numerical examples covering a wide range of service time distributions such as constant, uniform, Weibull, and phase type.

A MAP/PH/1 Perishable Inventory System with Dependent Retrial Loss

This paper considers a continuous review perishable inventory system with an infinite orbit. The replenishment of items in the inventory is governed by (s, Q) policy. The Markovian Arrival Process characterizes the arrival pattern of primary customers and the service time follows phase-type distribution. Inventoried items are subject to decay with some preassigned linear rate. The items are purchased from an outside source and lead time is exponentially distributed. An unsatisfied primary customer may enter an orbit or exit the system with a predetermined probability. The retrial customers, those who entered into the orbit, try to access the idle server with a linear rate. An unsatisfied retrial customer may enter or exit the orbit with prefixed probability based on the inventory level. Stationary probability vectors are obtained using the Matrix Analytic method. Numerical optimization of the profit function is carried out for different parameter values.

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 because they can maintain smaller intervehicle gaps and travel together in larger platoons than human-driven vehicles (HVs). Various policies have been proposed to regulate AV travel on highways, yet no in-depth comparison of these policies exists. To address this shortcoming, we develop a queueing model for a multilane highway and analyze two policies: 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. We connect the service rate to intervehicle gaps (governed by a Markovian arrival process) and congestion, and measure the performance using mean travel time and throughput. Our analysis shows 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 only in terms of throughput. 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 and show that for highly congested highways, a moderate number of AVs can make a substantial improvement (e.g., 22% AVs can improve throughput by 30%), and when all vehicles are AVs, throughput can be increased by over 400%. This paper was accepted by Jayashankar Swaminathan, operations management.

Analysis of $$MAP, PH_{2}^{OA}/PH_{1}^{I}, PH_{2}^{O}/1$$ retrial queue with vacation, feedback, two-way communication and impatient customers

This article concentrates on the steady-state analysis of a constant retrial queueing system with impatient customers, vacation, feedback, and two types of arrivals, namely the incoming calls which are made by the customers and the outgoing calls which are made by the server during the idle period. The incoming calls arrive at the system by following the Markovian Arrival Process(MAP) and service times of incoming/outgoing calls follow phase-type (PH) distribution, and the rest of the random variables are exponentially distributed. We have framed our model for analyzing some of the basic situations/problems in telecommunication systems. With the support of matrix analytic method, the invariant analysis of our system has been carried out. We have also discussed the busy period and have performed the cost analysis for our model. At last, we have validated our model through numerical and graphical exemplifications.