Steady-state Probability Vector Research Articles

Markovian phase-type single working vacation queue with breakdown and standby server

We consider a single server queuing model with three service types, each with three phases. In which the customers arrive according to a Poisson process and the service time of each customer follows Hyper-Erlang distribution. We analyzed the model with interruptions, such as breakdown, repair and single working vacation. The standby server works whenever the main server under breakdown. Using the matrix geometric method, the steady-state probability vector of the model is examined and finally, some numerical examples are presented.

Replacement of failed items in a two commodity retrial queueing-inventory system with multi-component demand and vacation interruption

This study investigates a crucial aspect of inventory management, which is the process of replacing failed items. In dynamic commercial environments, it is essential to efficiently and strategically replace failed items to maintain operational efficiency and ensure profitability. We consider a two-commodity retrial queueing-inventory system with vacation interruption. Upon purchasing the first commodity, the second commodity is provided as a complimentary item. In contrast, no item is given as a complimentary for the purchase of the second item. Only the first commodity is stored in a dedicated pooled storage for replacement when it fails. The (s,Q) policy governs replenishing the first commodity while the second is replenished through instantaneous ordering. The model considers the multi-component demand rate for customer arrivals. Server vacations are initiated during customer absence in waiting hall or when the first commodity is unavailable. We formulate a level-dependent quasi-birth-and-death process, and its steady-state probability vector is computed using Neuts and Rao's truncation method. The stability condition for the system is derived, and various system performance measures, including expected total cost, number of replaceable items, and customers in the waiting hall and orbit, are established. The comparative analysis between the system with replacement is done with the regular model without replacement, which revealed the efficiency of replacement. The analysis of multi-component demand towards homogeneous arrival highlights the impact of multi-component demand on boosting customer arrival. Also, parametric sensitivity analysis has been conducted numerically over total cost, mean number of failed items for replacement, and mean number of customers in the waiting hall and orbit.

Analysis of heterogeneous two server queueing system with multiple working vacations and server breakdowns

The steady state analysis and the time-dependent behaviour of a heterogeneous two-server multiple vacation queueing model is examined in this research work. The model also takes into account the possibility of a breakdown happening when server 1 is busy while server 2 is on working vacation and both servers are busy, with the restoration procedure starting right away. The stability condition is established through the use of matrix geometric technique, and the steady-state probability vector of the system?s number of customers has been evaluated as a quasi-birth and death (QBD) process. Some system performance measures are obtained. Numerical analysis for both steady-state and transient analysis are given to our proposed model.

Performance of a dual service station in stochastic inventory system with multi server and optional feedback service

This paper considers a stochastic queueing-inventory system with dual service stations, two groups of heterogeneous multi-servers, two finite waiting halls, and two classes of customers. Station-1 and station-2 provide an inventory sales service and feedback service, respectively. The feedback service option can be given to customers at the service completion epoch. If a customer requires feedback service, moves to an orbit; otherwise, leaves the system permanently. The classical retrial policy is applied to get feedback service. For the replenishment process, the system follows an (S−1,S) base stock ordering policy. This study analyzes the model under four classifications: 1) orbit size is finite and servers have homogeneous service rate; 2) orbit size is finite and servers have heterogeneous service rate; 3) orbit size is infinite and servers have homogeneous service rate; and 4) orbit size is infinite and servers have heterogeneous service rate. The steady-state probability vector for an infinite-size orbit case is computed using the Neuts and Rao truncation method. An expected total cost function is derived with sufficient system indicators for the four classifications. An optimized expected total cost is gained for classifications 2 and 4. The probability that a server is busy, the customer's waiting time, and the customer's loss rate are minimized for classifications 2 and 4.

Modeling of Junior Servers Approaching a Senior Server in the Retrial Queuing-Inventory System

This article deals with the queuing-inventory system, composed of c junior servers, a senior server, two finite waiting halls, and an infinite orbit. On occasion, junior servers encounter challenges during customer service. In these instances, they approach the senior server for guidance in resolving the issue. Suppose the senior server is engaged with another junior server. The approaching junior servers await their turn in a finite waiting area with a capacity of c for consultation. Concerning this, we study the performance of junior servers approaching the senior server in the retrial queuing-inventory model with the two finite waiting halls dedicated to the primary customers and the junior servers for consultation. We formulate a level-dependent QBD process and solve its steady-state probability vector using Neuts and Rao’s truncation method. The stability condition of the system is derived and the R matrix is computed. The optimum total cost has been obtained, and the sensitivity analyses, which include the expected total cost, the waiting time of customers in the waiting hall and orbit, the number of busy servers, and a fraction of the successful retrial rate of the model, are computed numerically.

Controlled Arrivals on the Retrial Queueing–Inventory System with an Essential Interruption and Emergency Vacationing Server

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.

A matrix-analytic method for the steady state analysis of a Markovian queueing system with scrap items

This paper studies a mathematical analysis of scrap item-based single server stochastic queueing, which consists of storage space for the scrap items collected from the customers. The primary customers arrive in the finite waiting hall to sell scrap or recyclable products. The server collects the customer’s scrap items and stores them in the storage space. If the waiting hall is full whenever the primary customers arrive, they can join an infinite orbit or permanently leave the system under the Bernoulli trial. Whenever the orbital customers find at least one free space in the waiting hall, they can enter it. The server begins vacation once the storage reaches its maximum capacity or no more customers exist in the waiting hall. The server switches to regular mode whenever at least one free space exists in the storage, and one customer is in the waiting hall. Otherwise, he is not working in vacation mode. Customers in the waiting hall may lose their patience after waiting for a long time and may either enter orbit or leave the system permanently. Whenever the storage level (scrap item level) reaches a certain limit, the server places the procurement order for sending the items to the recycling unit or re-manufacturer. The system’s steady-state probability vector and stability condition have been derived by truncating the orbit at a point. Various system performances, such as expected total cost, profit cost, and waiting time for customers in the waiting hall, are defined, and a numerical study is carried out to analyze the performance of the proposed model. The numerical findings indicate that disabling working in vacation mode increases waiting time and total cost while working in vacation mode results in minimized waiting time and total cost.

Performance analysis of a queueing system based on working vacation with repairable fault in the P2P network

In recent years, peer-to-peer (P2P) as a means of content distribution and sharing service are receiving extensive attention. In this paper, the hybrid P2P network structure model is combined with the knowledge of queueing theory. In order to address the problem generated by unnecessary online behavior of nodes, the working sleep mechanism for the serving nodes is introduced, and the synchronous multiple working vacation strategy is also adopted to decrease the system energy consumption. Then a queueing model with working vacation and repairable fault is built, which used to simulate the resource transmission process of P2P network and to study the performance of P2P sharing system. The steady-state probability vector is derived by employing quasi-birth-and-death process and matrix-geometric solution method. And the expressions for system performance indicators, such as the average delay, are obtained by employing Gauss–Seidel iteration method. Through numerical experiments, the performance indicators of the P2P resource sharing system are studied, which provides a theoretical basis to decrease online energy consumption of nodes in the hybrid P2P network. And through employing Nash equilibrium and social optimal strategy, the value of the social maximum benefit under the social optimization is obtained.

Markovian queueing model with single working vacation and catastrophic

A markovian queueing system with a single working vacation and different service rates have been considered. Service times during the vacation period, service times during the service period as well as vacation times are all exponentially distributed. If the queue length is increasing, the server converts the service parameter from µv to µb, and a normal working time begins. According to the Poisson process, a catastrophe event with parameter α occurs only if there is a customer in the system. This type of queue model has been analysed using Matrix Geometric Method (MGM) to find steady state probability vectors. Using it some performance measure is also determined.

Stochastic Decomposition of Geo/Geo/1 Production Inventory System

This paper considers an (s, S) inventory system with production in which demand occurs according to a Bernoulli process and service time follows a geometric distribution. The maximum inventory that can be accommodated in the system is S. When the items in the on-hand inventory are reduced to a pre-assigned level of s due to service, production is started. The production time follows a geometric distribution. The production process is stopped as soon as the inventory level reaches the maximum. Customer who arrives during the inventory level is zero is assumed to be lost. Using the stochastic decomposed solution obtained for the steady-state probability vector, we analyzed the inventory cycle. A suitable cost function is defined using the performance measures. Numerical experiments are also incorporated to highlight the minimum value of the cost function against the parameter values.

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.

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.

Analysis of an M/PH/1 Retrial Queueing-Inventory System with Level Dependent Retrial Rate

We analyze a queueing-inventory system which can model airline and railway reservation systems. An arriving customer to an idle server joins for service immediately with exactly one item from inventory at the moment of service completion if there are some on-hand inventory, or else he accesses to a buffer of varying size (the buffer capacity varies and equals to the number of the items in the inventory with maximum size S). When the buffer overflows, the customer joins an orbit of infinite capacity with probability p or is lost forever with probability 1−p. Arrivals form a Poisson process, and service time has phase type distribution. The time between any two successive retrials of the orbiting customer is exponentially distributed with parameter depending on the number of customers in the orbit. In addition, the items have a common life time with exponentially distributed. Cancellation of orders is possible before their expiry and intercancellation times are assumed to be exponentially distributed. The stability condition and steady-state probability vector have been studied by Neuts–Rao truncation method using the theory of Level Dependent Quasi-Birth-Death (LDQBD) processes. Several stationary performance measures are also computed. Furthermore, we provide numerical illustration of the system performance with variation in values of underlying parameters and analyze an optimization problem.

Optimal and Sensitivity Analysis of Vacation Queueing System with F-Policy and Vacation Interruption

In this article, the investigation on a randomized arrival control policy for the prospective customers in the finite capacity queueing system with working vacation and vacation interruption is done. The impatience behavior of the customers is also considered in modeling and assumptions to the studied problem to make it more realistic. In the investigated queueing model, at the epoch when the number of customers in the system reaches system’s capacity, newly arriving customers are not allowed to join the system for service and referred as lost customers. As the length of the queue decreases to a pre-specified threshold value F, the server commences a start-up for allowing to join the customers according to an exponential distribution and starts allowing newly arriving customers to join the system for service. The steady-state probability distribution and vector representation of various system performance measures are derived using matrix-analytic approach. The cost optimization problem is also formulated, and the particle swarm optimization algorithm is implemented to determine the optimal decision parameters to achieve the minimal expected cost. Finally, some numerical results in tables and graphs are provided for the illustrative and comparative purpose which help the system analyst in decision making from the performance and economic perspective.

A $ {BMAP/BMSP/1} $ queue with Markov dependent arrival and Markov dependent service batches

Batch arrival and batch service queueing systems are of importance in the context of telecommunication networks. None of the work reported so far consider the dependence of consecutive arrival and service batches. Batch Markovian Arrival Process( \begin{document}$ BMAP $\end{document} ) and Batch Markovian Service Process ( \begin{document}$ BMSP $\end{document} ) take care of the dependence between successive inter-arrival and service times, respectively. However in real life situations dependence between consecutive arrival and service batch sizes also play an important role. This is to regulate the workload of the server in the context of service and to restrict the arrival batch size when the flow is from the same source. In this paper we study a queueing system with Markov dependent arrival and service batch sizes. The arrival and service batch sizes are assumed to be finite. Further, successive inter-arrival and service time durations are also assumed to be correlated. Specifically, we consider a \begin{document}$ BMAP/BMSP/1 $\end{document} queue with Markov dependent arrival and Markov dependent service batch sizes. The stability of the system is investigated. The steady state probability vectors of the system state and some important performance measures are computed. The Laplace-Stieltjes transform of waiting time and idle time of the server are obtained. Some numerical examples are provided.

An M/M/1 Queueing-Inventory System with Working Vacations, Vacation Interruptions and Lost Sales

We consider a single server queueing-inventory system under (s,Q) replenishment strategy with working vacations, vacation interruptions and lost sales. The server provides service at a lower rate during working vacations than while in normal mode of service. If a working vacation realizes while providing service in that mode, then the server continues in the present status until the current service is completed. Upon that it switches to the normal mode of service, provided there is at least one customer waiting. If no customer is waiting at this point of time, it goes for vacation. We also assume that if there are customers in the system at a service completion epoch during a working vacation, the server will come back to the normal working mode, otherwise the server will stay in the working vacation mode. With the system having infinite capacity, the stability condition for the system is obtained, followed by computation of the steady-state probability vector and discussion. Various performance measures are evaluated. In addition, the busy period analysis is provided and the stationary waiting time distribution in the queue is derived. Numerical illustrations are provided to illustrate the system performance, and an optimization problem is also discussed.

Machine interference problem involving unsuccessful switchover for a group of repairable servers with vacations

The purpose of this paper is to explore the multiple-server machine interference problem with standby unsuccessful switchover and Bernoulli vacation schedule. Failure times of operating and standby machines are assumed to have exponential distributions and repair times of the failed machines and vacation times of servers are also assumed to have exponential distributions. After the completion of service, the server either goes for a vacation or may continue serving for the next machine. The vacation policy we considered is a single vacation policy. In practice, the switchover may experience a significant failure. The matrix analytical method and recursive method are employed to obtain the steady-state probability vectors, and closed-form expressions of some important system characteristics are obtained. The problem of cost optimization dealt with a number of numerical examples is provided by the Quasi-Newton method, the pattern search method, and the Nelder-Mead simplex direct search method. Expressions of various system characteristics are derived. Sensitivity analysis is performed numerically for system parameters. This paper presents the first time that machine interference problem with unsuccessful switchover for a group of repairable servers with vacations has been obtained, which is quite useful for the decision makers.

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.

Availability Analysis of a Repairable Series-Parallel System with Redundant Dependency

This paper investigates the steady-state availability of a repairable series-parallel system with redundant dependency. The different types of components and repairmen are taken into account, the failure rate of the operating component varies as the number of other failed components and the repair rate of the failed component is constant in each parallel redundant subsystem. To quantify the redundant dependency, a modified failure dependence function is introduced to determine the failure rate of the components in each subsystem. Markov theory and matrix analysis method are used to get the steady-state probability vector of each subsystem and the steady-state availability of the entire system. A numerical example is presented to illustrate the obtained results and to analyze the effect of redundant dependency class on the system availability.

A Queueing Inventory System with Server Working Breakdown

This paper considers a single server inventory queueing system with two types of server breakdowns, say Type 1(T1) breakdown and Type 2 (T2) breakdowns, and working breakdown. T1 breakdown occurs in regular service period whereas T2 breakdown occurs in the duration of working breakdown. The inter arrival time between any two customers and the times of occurrences of both types of breakdowns are all independent exponential distributions. The commencement of repairing process of T1 breakdown follows Bernoulli’s whereas the commencement of repairing process of T2 breakdown is instantaneous. By using matrix method, we obtain the steady state probability vector of the finite capacity queueing inventory system. Finally, the numerical examination of model sensitiveness is performed.