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In this modern era, the security and safety of financial and personal digital data are crucial considerations. Blockchain technology offers a secure framework for the transmission of data between devices, without the interference of malicious actors. With the growing demand for this technology, it has become essential to evaluate the impact of its processes on system characteristics to forecast transactions behavior and ensure necessary quality indicators, as well as the stability of system components while the blockchain technology is in operation. Queueing models are a highly effective mathematical tool for analyzing and optimizing the performance of blockchain systems. In this study, a more generic queueing model is utilized to evaluate a blockchain application. The transactions arrive in batches according to a Markovian arrival process, and the service times for the block-generation (the first stage) and blockchain-building (the second stage) follow phase-type distributions. Using matrix-analytic method, we study the model in steady-state. The stationary distribution and the stability condition of the system have been derived. Further, by determining key system performance measures, illustrative numerical examples covering a wide range of batch size and service time distributions have been discussed. In conclusion, our hope is that the methodology and findings presented in this paper will pave the way for the queueing analysis of more extensive and diverse blockchain systems in practice. Furthermore, we believe that it will inspire a series of promising future quantitative and qualitative studies aimed at the development of blockchain technologies. [ABSTRACT FROM AUTHOR]

Stochastic population models with mild catastrophes have gained much attention in recent years due to their wide application in a variety of areas including computer-communications systems. This article considers a population model in which both the arrival process of individuals and catastrophes occur as per the Markovian arrival process (MAP) and are independent of each other. The killing mechanism takes place according to the binomial distribution. The steady-state analysis of the model is carried out using the vector generating function approach and the population size distributions at an arbitrary, post-catastrophe, and pre-arrival epochs are presented in terms of the roots of the characteristic equation. Several performance measures of the system are studied in detail, and the impact of critical parameters is duly investigated. [ABSTRACT FROM AUTHOR]

We investigate a continuous review inventory queuing system in the present study that has two heterogeneous servers: Server-2, which is reliable, and Server-1, which is unreliable. An exponentially distributed random time is used to describe the repair process when server-1 has an interruption. On the other hand, server-2 is completely dependable, but it goes on vacation when the system is empty. These two goods can be reordered under ordering regulations. To ensure customer satisfaction, an emergency replenishment of one item with no lead time occurs when the on-hand inventory level falls to zero. We use the matrix analytic approach for the QBD process under a steady-state probability vector. We also take into account the overall cost and the busy time. Furthermore, numerical data shows the benefits of the suggested approach in a range of random circumstances. [ABSTRACT FROM AUTHOR]

The Markovian arrival process (MAP) is a versatile counting process with dependent and non‐identically distributed inter‐arrival times following the phase‐type distribution. In this article, we study the classical δ$$ \delta $$‐shock model and a mixed δ$$ \delta $$‐shock model by assuming the MAP of shocks. We derive explicit expressions for the reliability and the mean lifetime of the system. Further, we study an optimal replacement policy based on the MAP. We illustrate the developed results through several numerical examples. [ABSTRACT FROM AUTHOR]

We consider a single server queueing system with a finite capacity buffer in which the server is subject to breakdowns and repairs. At the time of a breakdown, the customers, if any, present in the system will either decide to stay or abandon according to a probabilistic mechanism. This decision is made by every customer present in the system independent of the others. We assume that the arrivals of customers occur according to a Markovian arrival process (MAP). The shocks that create breakdowns of the servers are also assumed to follow a MAP independent of the other MAP. The service and the repair times of the server are assumed to follow (possibly) different phase type (PH) distributions. An arriving customer finding the buffer full will be lost. Further, an arriving customer finding the server under repair may decide to balk or join the queue at that instant. This queueing system, which generalizes some of the well-known queueing models in the literature, is studied as a GI/ M/ 1- type with a finite state space. The model is analyzed in steady-state and illustrative numerical examples including an optimization problem of interest are presented. [ABSTRACT FROM AUTHOR]

The purpose of this study is to present a subgeometric convergence formula for the stationary distribution of the finite-level M/G/1-type Markov chain when taking its infinite-level limit, where the upper boundary level goes to infinity. This study is carried out using the fundamental deviation matrix , which is a block-decomposition-friendly solution to the Poisson equation of the deviation matrix. The fundamental deviation matrix provides a difference formula for the respective stationary distributions of the finite-level chain and the corresponding infinite-level chain. The difference formula plays a crucial role in the derivation of the main result of this paper, and the main result is used, for example, to derive an asymptotic formula for the loss probability in the MAP/GI/1/ N queue. [ABSTRACT FROM AUTHOR]

This work focuses over the performability analysis of a multi-server retrial queueing model with phase-type inter-retrial times in cellular networks. It is considered that the pattern of the new call arrival and handoff call arrival follows marked Markovian arrival process. The service times of both types of calls are phase-type (PH) distributed with different parameters, and inter-failure & inter-repair times of channels are exponentially distributed. For the prioritization of handoff calls, G channels are kept in reserve for handoff calls. When all the available channels, say S , are busy at the arrival epoch of a handoff call, the handoff call will be dropped. Whereas a new call will be blocked and will have an option to join the orbit of infinite capacity or leave the system without getting the connection, if at least S − G channels are busy. A new call in the orbit, termed as retrial call, retries to get the connection after a random interval which follows PH distribution. This model is analyzed as a level-dependent-quasi-birth-death process by applying an efficient method. Through numerical illustrations, the behavior of performance measures depending on the various relevant intensities is discussed. [ABSTRACT FROM AUTHOR]

This paper presents a study of fork–join systems. The fork–join system breaks down each customer into numerous tasks and processes them on separate servers. Once all tasks are finished, the customer is considered completed. This design enables the efficient handling of customers. The customers enter the system in a MAP flow. This helps create a more realistic and flexible representation of how customers arrive. It is important for modeling various real-life scenarios. Customers are divided into K ≥ 2 tasks and assigned to different subsystems. The number of tasks matches the number of subsystems. Each subsystem has a server that processes tasks, and a buffer that temporarily stores tasks waiting to be processed. The service time of a task by the k-th server follows a PH (phase-type) distribution with an irreducible representation ( β k , S k ), 1 ≤ k ≤ K . An analytical solution was derived for the case of K = 2 when the input MAP flow and service time follow a PH distribution. We have efficient algorithms to calculate the stationary distribution and performance characteristics of the fork–join system for this case. In general cases, this paper suggests using a combination of Monte Carlo and machine learning methods to study the performance of fork–join systems. In this paper, we present the results of our numerical experiments. [ABSTRACT FROM AUTHOR]

The retrial queueing inventory system with working vacation, flush out, balking, breakdown, and repair, as well as a constant retrial rate and orbital client collision are all examined in this study. We made the assumption that customers arrive through a Markovian arrival process and that they would get phase-type services from the server. The inventory is replenished using a (s, S) and (s, Q) strategy, and it is expected that the replenishment time will follow an exponential distribution. If there are zero inventory items, no customers in the orbit, or both, the server will go into working vacation mode. When a customer retries an orbit while the server is serving arriving customers, the orbital customer may collide with the arriving customer during that retry, in which case both of them will be shifted back into orbit; otherwise, the orbital customer may avoid colliding with the arriving customer and may rejoin the orbit for another retry. The number of customers in the orbit and the inventory level may be found in the steady state. A cost analysis is produced along with the establishment of various important performance measures. Moreover, some numerical examples are provided to clarify our mathematical notion. [ABSTRACT FROM AUTHOR]

Considering a single server with two queues that is prone to unreliability. The server offers a kitting process and performs necessary checks and rectifications when required. The arrival of items follows a Markovian arrival process, while the service is distributed based on a phase type distribution. The incoming products may exhibit issues such as poor quality or defects. If either of the queues is empty, the server is unable to provide the requested service and remains inactive. Furthermore, if all queues are empty, the server goes into a vacation mode. Breakdowns, repairs, instances of customers leaving without service (reneging), and vacation periods are all modeled using an exponential distribution. To gain insights into the performance of the queueing model, various performance metrics are analyzed and represented through 2D and 3D graphs. [ABSTRACT FROM AUTHOR]

The two service stations have been connected in series in our tandem queueing inventory model, replicating the real-life environment that provides customer service by providing seperated service without congestion. Customers arrive at Station 1, which has infinite servers (self-service), according to the Markovian arrival process, where they get an information about the inventory. After completion of service in Station 1, the customers can choose whether to move to the buffer in front of Station 2, where a single server gives service for buying inventory, or can exit the system with the predetermined probability. Once the customer who demands an inventory enters the buffer, they are instantly attached to an inventory from the unattached inventory of maximum capacity S, and the inventory replenishment follows $ (s, Q) $ ordering policy. In Station 2, the server provides two services that the customers can choose with a predefined probability. The server in Station 2 may break down at any time during service. When the server experiences the empty buffer, the server automatically moves to vacation until it finds the customer in the system. Exponentially distributed service, lead, breakdown, and repair times have been considered. The joint probability distribution of the number of customers in Station 1, the available inventory level, and the number of customers in the Station 2 buffer has been found. Numerical calculations have been done to show the sensitive analysis of all parameters.

We consider a single server queueing model in which the customers arrive according to a Markovian Arrival Process (MAP). We modeled this queueing system with many kinds of service where the customers can choose any kind from the server. Each kind of service follows Phase-type (PH) distribution. We analyzed the model with breakdown, repair, single working vacation and standby server. A standby server works only when the main server is breakdown while serving the customer during a busy or vacation period. Using the matrix-geometric approach, we examined the model. A few real-world examples have been given, and several graphical and numerical examples have also been provided.

In this work, a single server implements a two-commodity inventory queueing system. We assume that both commodities have a finite capacity. Customers arrive by a Markovian Arrival Process, there is a need for a single item, and either or both types of commodities are required, and this requirement is modeled using certain probabilities. The lead times are exponentially distributed, and the service times have a PH distribution. We use matrix analytical techniques to investigate the queueing inventory system and adopt an (s, S)-type replenishment policy that is dependent on the type of commodity. In the steady state, the joint and individual probability distribution of the Esystem, inventory level, and server status is obtained. A few significant performance measures are attained. Our mathematical concept is then illustrated with a few numerical examples. [ABSTRACT FROM AUTHOR]

A single server retrial queueing model with non-preemptive priority was examined in this research. The arrival of priority consumers follow a marked Markovian arrival pattern, and both high priority and low priority service times are according to phase type distribution. Matrix analytic method are used to examine the steady state analysis of this model. Various system performance measures, cost analysis and busy period analysis also examined in this model. In additionally, by using some system performance measures we provide the numerical illustration with numerically and graphically. [ABSTRACT FROM AUTHOR]

Queueing models in which the services are provided in groups (or blocks or batches) have found to be very useful in real-world applications and such queues been extensively analysed in the literature. In this paper we see one such group service queueing model with balking, reneging and immediate feedback. The arrival processes is a Markovian arrival, where, the arriving customer may balk the system while the server is idle and the pool is empty. Customers are provided service in groups of varying size from 1 to the fixed constant, say, N. The service time of a batch follows the phase type distribution corresponding to the each size of the group. A group's service time is taken as the highest of the service times of each customers who make up the group. The group of customers who are dissatisfied with the service then that group will get the service immediately. Here, the feedback of a group is defined as the average of the feedback of each customers who make up the group. During the admission period the customers may renege. We calculated the steady state probabilities by using the matrix geometric method, then, by using it we computed few performance measures. We have studied the busy period and the distribution of waiting time is derived. Results are illustrated with some graphical representations. [ABSTRACT FROM AUTHOR]

In this study, a production inventory model with phase type service times where customers join the system occur according to a Markovian arrival process is discussed. When the inventory level is positive, if an arriving customer finds the server idle gets into service immediately. Served customer leaves the system and the on-hand inventory is decreased one unit of item at service completion epoch. Otherwise, the customer enters into a waiting space (queue) of infinite capacity and waits for get served. The production facility produces items according to an (𝑠,𝑆) policy. The production is switched on when the inventory level depletes to 𝑠 and the production remains on until the inventory level reaches to the maximum level 𝑆. Once the inventory level becomes 𝑆, the production process is switched off. Applying the matrix-geometric method, we carry out the steady-state analysis of the production inventory model and perform a few illustrative numerical examples includes the effect of parameters on the system performance measures and an optimization study for the inventory policy

In recent times, we have encountered new situations that have imposed restrictions on our ability to visit public places. These changes have affected various aspects of our lives, including limited access to supermarkets, vegetable shops, and other essential establishments. As a response to these circumstances, we have developed a continuous review retrial queueing–inventory system featuring a single server and controlled customer arrivals. In our system, customers arriving to procure a single item follow a Markovian Arrival Process, while the service time for each customer is modeled by an exponential distribution. Inventories are replenished according to the (s , Q) reordering policy with exponentially distributed lead times. The system controls arrival in the waiting space with setup time. The customers who arrive at a not allowed situation decide to enter an orbit of infinite size with predefined probability. Orbiting customers make retrials to claim a place in the waiting space, and their inter-retrial times are exponentially distributed. The server may experience essential interruption (emergency situation) which arrives according to Poisson process. Then, the server goes for an emergency vacation of a random time which is exponentially distributed. In the steady-state case, the joint probability of the number of customers in orbit and the inventory level has been found, and the Matrix Geometric Method has been used to find the steady-state probability vector. In numerical calculations, the convexity of the system and the impact of F-policy and emergency vacation in the system are discussed. [ABSTRACT FROM AUTHOR]

Two types of services are explored in this paper: regular server and main server, both of which provide both regular and optional services. Customers arrive using the Markovian Arrival Process (MAP), and service time is allocated based on phase type. The regular server uses the main server as a resource. Customers' service at the primary server is disrupted as a result. When the queue size is empty, the main server can take several vacations. This system has been represented as a QBD Process that investigates steady state with the use of matrix analytic techniques, employing finite-dimensional block matrices. Our model's waiting time distribution has been examined in more detail during the busy times. The system's key parameters are assessed, and a few graphs and numerical representations are constructed. [ABSTRACT FROM AUTHOR]

In this paper, we study a continuous-time single server queueing system with an infinite system of capacity, a two-stage vacation policy with imperfect service, setup, breakdown, delay time, phase-type of repair and customer reneging. The Markovian Arrival Process is used for the arrival of a customer and the phase-type distribution is used when offering service. This encompasses the policy of two vacations: a single working vacation and multiple vacations. Using the Matrix-Analytic Method to approach the system generates an invariant probability vector for this model. Henceforth, the busy period, waiting time distribution and cost analysis are the additional findings. The indicators are secured as a result of this performance. The outcomes result of numerical order can be graphically interpreted in the form of 2D and 3D. [ABSTRACT FROM AUTHOR]

In this paper, a classical queueing system with two types of heterogeneous servers has been considered. The Markovian Arrival Process (MAP) is used for the customer arrival, while phase type distribution (PH) is applicable for the offering of service to customers as well as the repair time of servers. Optional service are provided by the servers to the unsatisfied customers. The server-2 may get breakdown during the busy period of any type of service. Though the server-2 got breakdown, server-2 has a capacity to provide the service at a slower rate to the current customer who is receiving service when the moment of server-2 struck with breakdown. In the period of vacation/closedown of server-1 and the server-2 is in working breakdown or under repair process, the arrival of customers may balk the system due to the impatient. Stability conditions has derived for our system and the stationary probability vector was evaluated by using the matrix analytical method. This model also examined at the analysis of busy period, waiting time distribution and system performance measures. The numerical illustrations are provided with the aid of two dimensional and three dimensional graphs. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a production inventory system with service times. Customers arrive in the system according to a Markovian arrival process. The service times follow a phase-type distribution. We assume that there is an infinite waiting space for customers. Arriving customers demand only one unit of item from the inventory. The production facility produces items according to an (s, S)-policy. Once the inventory level becomes the maximum level S, the production facility goes on a vacation of random duration. When the production facility returns from the vacation, if the inventory level depletes to the fixed level s, it is immediately switched on and starts production until the inventory level becomes S. Otherwise, if the inventory level is greater than s on return from the vacation, it takes another vacation. The vacation times are exponentially distributed. The production inventory system in the steady-state is analyzed by using the matrix-geometric method. A numerical study is performed on the system performance measures. Besides, an optimization study is discussed for the inventory policy. [ABSTRACT FROM AUTHOR]

Building performance models for software services in DevOps is costly and error-prone. Accurate service demand distribution estimation is critical to precisely modeling queueing behaviors and performance prediction. However, current estimation methods focus on capturing the mean service demand, disregarding higher-order moments of the distribution that still can largely affect prediction accuracy. To address this limitation, we propose to estimate higher moments of the service demand distribution for a microservice from monitoring traces. We first generate a closed queueing model to abstract software performance and use it to model the departure process of requests completed by the software service as a Markovian arrival process (MAP). This allows formulating the estimation of service demand into an optimization problem, which aims to find the first multiple moments of the service demand distribution that maximize the likelihood of the MAP using generated the measured inter-departure times. We then estimate the service demand distribution for different classes of service with a maximum likelihood algorithm and novel heuristics to mitigate the computational cost of the optimization process for scalability. We apply our method to real traces from a microservice-based application and demonstrate that its estimations lead to greater prediction accuracy than exponential distributions assumed in traditional service demand estimation approaches for software services. [ABSTRACT FROM AUTHOR]

We consider a single server phase type queueing model with server vacation, repair, breakdown, degrading service, starting failure and closedown. When the arrival rate of the customer follows the Markovian Arrival Process (MAP) and the service rate of the server follows the phase-type distribution. If no one is in the system when the server is back from the vacation, then the server will wait until the customer arrives. If the customer arrives at the moment with no starting failure, then he provides service, otherwise the server immediately goes to the repair process. Here, the service rate declining until degradation fixed. After completion of K services the degradation is addressed. During the period of service, the server may get a breakdown at any moment, and then the server immediately goes for a repair process. After completing the service, he switches to the close-down process, and then he goes on vacation. Using the Matrix-Analytic method, The stationary probability vector representing the total number of customers in the system is examined. The analysis of the busy period, the mean waiting time, and cost analysis are discussed. A few significant performance measures are attained. Finally, some numerical examples are given. [ABSTRACT FROM AUTHOR]

This paper presents a study of fork–join systems. The fork–join system breaks down each customer into numerous tasks and processes them on separate servers. Once all tasks are finished, the customer is considered completed. This design enables the efficient handling of customers. The customers enter the system in a MAP flow. This helps create a more realistic and flexible representation of how customers arrive. It is important for modeling various real-life scenarios. Customers are divided into K≥2 tasks and assigned to different subsystems. The number of tasks matches the number of subsystems. Each subsystem has a server that processes tasks, and a buffer that temporarily stores tasks waiting to be processed. The service time of a task by the k-th server follows a PH (phase-type) distribution with an irreducible representation (βk, Sk), 1≤k≤K. An analytical solution was derived for the case of K=2 when the input MAP flow and service time follow a PH distribution. We have efficient algorithms to calculate the stationary distribution and performance characteristics of the fork–join system for this case. In general cases, this paper suggests using a combination of Monte Carlo and machine learning methods to study the performance of fork–join systems. In this paper, we present the results of our numerical experiments.

In this study, a production inventory model with phase type service times where customers join the system occur according to a Markovian arrival process is discussed. When the inventory level is positive, if an arriving customer finds the server idle gets into service immediately. Served customer leaves the system and the on-hand inventory is decreased one unit of item at service completion epoch. Otherwise, the customer enters into a waiting space (queue) of infinite capacity and waits for get served. The production facility produces items according to an (s, S) policy. The production is switched on when the inventory level depletes to s and the production remains on until the inventory level reaches to the maximum level S. Once the inventory level becomes S, the production process is switched off. Applying the matrix-geometric method, we carry out the steady-state analysis of the production inventory model and perform a few illustrative numerical examples includes the effect of parameters on the system performance measures and an optimization study for the inventory policy. [ABSTRACT FROM AUTHOR]

This paper investigates a population model which grows as per the Markovian arrival process and is influenced by binomial catastrophes that occur according to renewal process. That is, when a catastrophe attacks, an individual (element) of the population survives with probability p or dies with probability 1 - p , independent of anything else. Using the supplementary variable technique, the steady-state vector generating function (VGF) of the population size distribution at post-catastrophe epoch is obtained in terms of the infinite product of matrices. Further, the VGF of the population size distribution at arbitrary and pre-catastrophe epochs are also deduced. To make the model valuable for practitioners, a step-wise computing process for evaluation of the distribution of population size at various epochs is given. A recursive formula to compute factorial moments of the population size is also presented. Finally, some numerical results are included to illustrate the impact of parameters on the behavior of the model. [ABSTRACT FROM AUTHOR]

This study investigates a multi-server queueing-inventory system in a random environment. The system has an unlimited waiting space and an inventory capacity of $S$ units. The customer arrives at the system according to the Markovian arrival process. Whenever all the servers are busy, an arriving customer either goes to an unlimited waiting space or leaves the system. And if there is a free server with positive inventory, then the arriving customer gets service immediately. Inventories are filled under the instantaneous replenishment policy. We consider that the service time follows a phase-type distribution, and we assess the joint probability distribution of the inventory level and the number of customers in the steady-state case. On top of that, we extract the sojourn time distributions of arbitrary customers using the Laplace-Stieltjes transform. Finally, a few numerical examples are provided to illustrate our mathematical model.

We consider a tandem queue consisting of two stations. We demonstrate that customer numbers and waiting times at both stations are independent of each other in the steady-state. As an optimal design, in the second station, We consider that there is a single channel that has the total service capacity of two parallel channels instead of two identical channels. Then, we indicate that the optimal system works faster than the tandem queue system with parallel channels at the second station. Also, when the total service capacity is limited, the mean waiting time in the system has been minimized in the optimal system obtained the mean waiting time lower limit. We compared it to those found through a numerical example. [ABSTRACT FROM AUTHOR]

The functioning of complex systems relies on subsystems (modules) that in turn are composed of multiple units. In this paper, we focus on modular systems that might fail due to wear on their units or environmental conditions (shocks). The lifetimes of the units follow a phase-type distribution, while shocks follow a Markovian Arrival Process. The use of Matrix-Analytic methods and a bottom-up approach for constructing the system generator is proposed. The use of modular structures, as well as its implementation by the Modular Matrix-Analytic (MoMA) algorithm, make our methodology flexible in adapting to physical changes in the system, e.g., incorporation of new modules into the current model. After the model for the system is built, the modules are seen as a 'black box', i.e., only the contribution of the module as a whole to system performance is considered. However, if required, our method is able to keep track of the events within the module, making it possible to identify the state of individual units. Compact expressions for different reliability measures are obtained with the proposed description, optimal maintenance strategies based on critical operative states are suggested, and a numerical application based on a k-out-of-n structure is developed. [ABSTRACT FROM AUTHOR]

Energy consumption is becoming a significant part of overall operational cost in cloud data centers. For the purpose of satisfying the Service Level Agreement (SLA) of cloud users while enhancing the energy efficiency in cloud computing systems, in this paper we propose an energy-saving mechanism with a sleep mode. Taking into consideration the traffic's correlation and the stochastical behavior of data arrival requests in a random cloud environment with the proposed energy-saving mechanism, we model the system as a MAP/M/ N / N + K queue with a synchronous multi-vacation. Then, we present a theoretical basis for analyzing and evaluating the system performance by taking a state transition rate matrix in the steady state. Next, we investigate the change trends for the energy saving rate of the system and the average latency of tasks by carrying out numerical experiments. Moreover, we give a [ABSTRACT FROM AUTHOR]