Matrix-geometric Method Research Articles

Cost Analysis of Fuzzy Discrete (s, S) Queueing Inventory System with Positive Lead Time and Optimization using Genetic Algorithm

In this article, a queueing inventory model with discrete time (DQIM) FGEOM/FGEOM/1 with (s, S) replenishment policy incorporating fuzzy numbers as input parameters is considered. The system has a fuzzy pentagonal number arrival rate according to a Bernoulli process and a fuzzy pentagonal number service rate that follows a geometric distribution. Here, S represents the highest level of stock where the process of replenishment is stopped, and s represents the lowest level of stock at which replenishment is started again. Using matrix geometric method, the steady-state solution is obtained followed by derivation of various fuzzy performance measures. Further, the total cost function is defined as a two-variable function of the minimum and maximum stock level. Genetic algorithm is employed to optimize the total cost. Various examples are presented to highlight the dependence of cost on input parameters. The use of PFN in DQIS and genetic algorithm in the optimization of DQIS is introduced in this paper for the first time. JEL Codes: C44, C61, C62, D11, D12, L89 Received: 17/07/2024. Accepted: 29/09/2024. Published: 04/10/24.

The analysis of P2P networks with malicious peers and repairable breakdown based on Geo/Geo/1+1 queue

The incredible growth of Peer-to-Peer (P2P) networks has brought with it some complex challenges, such as trust issues and high bandwidth consumption. To address these challenges, this paper analyzes the “free-riding” behavior, system energy consumption, and the benefits of requesting and service peers in the network. A Geo/Geo/1+1 queuing model is built with malicious peers which includes several strategies such as repairable breakdown, synchronized multiple working vacations, differentiated service, and waiting threshold. The matrix-geometric solution method is used to obtain steady-state distribution and performance measures. By conducting numerical experiments and analyzing the impact of each parameter, it is possible to optimize the system's performance and reduce energy consumption. With careful adjustments to parameter values, significant cost savings of requesting peers and energy conservation can be achieved. The resulting analysis provides a comprehensive understanding of the behavior of P2P networks, and the strategies proposed in the study can be used to optimize the performance of P2P networks.

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.

Chat service in a multichannel system under competition where customers are boundedly rational

Many service systems in various type of industry provide services via chat technologies. This type of service system is characterized by an interesting combination of properties. First, the customer is actively involved in service execution, since the service is comprised of responses of the server and the customer to each other, until the customer's request is satisfied. In addition, in such a system, the service can be provided by a single server in a parallel manner to a number of customers. Motivated by these and others interesting properties of the chat service, we formulate and analyze the stochastic queueing system as a two-dimensional quasi-birth-and-death process and derive its steady-state probabilities using matrix geometric methods. By means of economic analysis, we provide a scheme for deriving the optimal capacity of parallel chats and the optimal response effort that should be made by the server, considering that higher effort lengthens the response time but increases the probability of successful service completion. We next investigate a queueing game that considers a multi-channel service system consisting of a chat service and a traditional multi-server call center, and we consider strategic customers under both a centralized and a decentralized scenario. We show that competition between service channels may lead to lower service utilization even though customers enjoy higher expected utility. Finally, customer bounded rationality is considered and it is shown that customers who are less rational may enjoy higher utility. Sensitivity analyses are conducted for all scenarios.

Energy-efficient production control of a make-to-stock system with buffer- and time-based policies

Increasing energy efficiency in manufacturing has significant environmental and cost benefits. Turning on or off a machine dynamically while considering the production rate requirements can offer substantial energy savings. In this work, we examine the optimal policies to control production and turn on and off a machine that operates in working, idle, off, and warmup modes for the case where demand inter-arrival, production, and warmup times have phase-type distributions. The optimal control problem that minimises the expected costs associated with the energy usage in different energy modes and the inventory and backlog costs is solved using a linear program associated with the underlying Markov Decision Process. We also present a matrix-geometric method to evaluate the steady-state performance of the system under a given threshold control policy. We show that when the inter-arrival time distribution is not exponential, the optimal control policy depends on both the current phase of the inter-arrival time and inventory position. The phase-dependent policy implemented by estimating the current phase based on the time elapsed since the last arrival yields a buffer- and time-based policy to control the energy mode and production. We show that policies that only use the inventory position information can be effective if the control parameters are chosen appropriately. However, the control policies that use both the inventory and time information further improve the performance.

Performance analysis and ANFIS computing of an unreliable Markovian feedback queueing model under a hybrid vacation policy

This study covers the stationary analysis of an unreliable Markovian queueing system with feedback based on a hybrid vacation policy. Customers arrive at the station at random using a Poisson process. Service time is distributed exponentially. The necessary and sufficient requirements for system stability have been demonstrated. Many performance metrics of this queueing system, such as the stationary queue length distribution, have been obtained from the steady-state probability distribution of the queueing model and the steady-state queue length using the matrix geometric method. To establish the model’s performance indicators, a few formulas have also been developed. The Adaptive Neuro-Fuzzy Inference System (ANFIS) was used to develop rules from the input datasets and generate different performance indices. Finally, the impact of system parameters and cost analysis was studied using numerical examples.

Queueing-Inventory Systems with Catastrophes under Various Replenishment Policies

We discuss two queueing-inventory systems with catastrophes in the warehouse. Catastrophes occur according to the Poisson process and instantly destroy all items in the inventory. The arrivals of the consumer customers follow a Markovian arrival process and they can be queued in an infinite buffer. The service time of a consumer customer follows a phase-type distribution. The system receives negative customers which have Poisson flows and as soon as a negative customer comes into the system, he causes a consumer customer to leave the system, if any. One of two inventory policies is used in the systems: either (s,S) or (s,Q). If the inventory level is zero when a consumer customer arrives, then this customer is either lost (lost sale) or joins the queue (backorder sale). The system is formulated by a four-dimensional continuous-time Markov chain. Ergodicity condition for both systems is established and steady-state distribution is obtained using the matrix-geometric method. By numerical studies, the influence of the distributions of the arrival process and the service time and the system parameters on performance measures are deeply analyzed. Finally, an optimization study is presented in which the criterion is the minimization of expected total costs and the controlled parameter is warehouse capacity.

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.

Performance analysis of a multi server retrial queueing system with unreliable server, discouragement and vacation model

In this paper, we investigate a Markovian multi server retrial queueing system with unreliable servers, discouraged customers and vacation. The model is explored with a view to reduce the impact of unreliable servers on the system performance. This is attained by providing regular maintenance to the servers through synchronised vacation. Further, the customer facing service interruption on account of server breakdown is directed to the orbit from where the customer can retry seeking service. In this study, the stationary system size distribution is framed by quasi-birth–death (QBD) process. The rate matrix and steady state probabilities are established using matrix geometric method (MGM). The performance measures are derived by using matrix form expressions. The effects induced by the system parameters on the performance metrics are numerically and graphically analysed. Through numerical investigation, we provide insightful information that enables to manage the system to give enhanced system performance.

Analysis of a Heterogeneous Queuing Model with Intermittently Obtainable Servers under a Hybrid Vacation Schedule

This paper investigates the concept of a Markovian queueing model with heterogeneous, intermittently available servers with feedback under a hybrid vacation policy. Both the asymmetric transition representation and the hybrid vacation policy are addressed in this article. The necessary and sufficient conditions for system stability are presented. In addition, the steady-state probability distribution of the queueing model was derived by employing the matrix geometric method. Furthermore, a few formulae were constructed to determine the model’s performance indicators. Finally, the influence of system parameters was also investigated using some numerical examples.

Analysis of a production inventory system with MAP arrivals, phase-type services and vacation to production facility

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.

MAP/PH/1 Üretim Envanter Modeli

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.

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.

Queueing and Reliability Analysis of Unreliable Multi-server Retrial queue with Bernoulli feedback

The study consists of an unreliable multiserver retrial queue with feedback where clients may resist to join the orbit and the service provider is erratic. In this model we have considered c service providers at a time and if all the c servers are busy, the incoming clients wait in the retrial orbit. The model is unfolded using Matrix Geometric Method (MGM) to obtain the rate matrix and stationary probabilities. The system with retrials is modelled and the performance measures and reliability indices are procured. These derived quotients are then visualised and validated with the help of tables and graphs. Further, the cost analysis of the model is carried out and the optimal cost for the system is obtained using Particle Swarm Optimization (PSO) algorithm for matrix method. The optimal repair and retrial rates are obtained using PSO technique which helps the computer and manufacturing industry to work towards better decision policies.

Optimization of M/M/2 Queueing Model with Working Vacations

The paper deals with an M/M/2 Queueing Model with working vacations and reneging of customers due to impatience. The matrix geometric method is used to find the distribution of the number of customers in the system. A cost function is constructed to obtain the optimal value of the service rate to optimize (minimize) the cost function using the Quadratic Fit Search Method (QFSM). Further, the effects on the system's performance measures using numerical analysis and graphical representation are studied.

A matrix geometric solution of a multi-server queue with waiting servers and customers’ impatience under variant working vacation and vacation interruption

This paper deals with a M/M/c queueing system with waiting servers, balking, reneging, and K-variant working vacations subjected to Bernoulli schedule vacation interruption. Whenever the system is emptied, the servers wait for a while before synchronously going on vacation during which services are offered with a lower rate. We obtain the steady-state probabilities of the system using the matrix-geometric method. In addition, we derive important performance measures of the queueing model. Moreover, we construct a cost model and apply a direct search method to get the optimum service rates during both working vacation and regular working periods at lowest cost. Finally, numerical results are provided.

MAP/PH/1 queue with discarding customers having imperfect service

In this paper, we consider two queueing models. Model I is on a single-server queueing system in which the arrival process follows MAP with representation D = (D0,D1) of order m and service time follows phase-type distribution (β, S) of order n. When a customer enters into service, a generalized Erlang clock is started simultaneously. The clock has k stages. The pth stage parameter is θp for 1 ≤ p ≤ k. If a customer completes the service in between the realizations of stages k1 and k2 (1 < k1 < k2 < k) of the clock, it is a perfect one. On the other hand, if the service gets completed either before the kth1 stage realization or after the kth2 stage realization, it is discarded because of imperfection. We analyse this model using the matrix-geometric method. We obtain the expected service time and expected waiting time of a tagged customer. Additional performance measures are also computed. We construct a revenue function and numerically analyse it. In Model II, a single server queueing system in which all assumptions are the same as in Model I except the assumption on service time, is considered. Up to stage k1 service time follows phase-type distribution (α′ , T′) of order n1 and beyond stage k1, the service time follows phase type distribution (β′ , S′) of order n2. We compare the values of the revenue function of the two models

Production and energy mode control of a production-inventory system

Energy efficiency in manufacturing can be improved by controlling energy modes and production dynamically. We examine a production-inventory system that can operate in Working, Idle, and Off energy modes with mode-dependent energy costs. There can be a warm-up delay to switch between one mode to another. With random inter-arrival, production and warm-up times, we formulate the problem of determining in which mode the production resource should operate at a given time depending on the state of the system as a stochastic control problem under the long-run average profit criterion considering the sales revenue together with energy, inventory holding and backlog costs. The optimal solution of the problem for the exponential inter-arrival, production and warm-up times is determined by solving the Markov Decision Process with a linear programming approach. The structure of the optimal policy for the exponential case uses two thresholds to switch between the Working and Idle or Working and Off modes. We use the two-threshold policy as an approximate policy to control a system with correlated inter-event times with general distributions. This system is modelled as a Quasi Birth and Death Process and analyzed by using a matrix-geometric method. Our numerical experiments show that the joint production and energy control policy performs better compared to the pure production and energy control policies depending on the system parameters. In summary, we propose a joint energy and production control policy that improves energy efficiency by controlling the energy modes depending on the state of the system.

An efficient optimization procedure for location-inventory problems with (S-1, S) policy and retrial demands

In this research, we used an easy and efficient computation procedure for deriving performance measures of (S-1, S) inventory problems with M/M/c replenishment under a retrial demand scenario. We prove some properties of the studied problem and also under back-ordered demands for comparison. We apply our method to a capacity design problem and a location-inventory design problem. For both problems, we used our method combined with direct search and genetic algorithm (GA) for optimization and compare with matrix geometric method (MGM) based evaluation and other optimization methods, including enumeration and FICO Xpress Mosel. Numerical examples show our method is about ten times more efficient than MGM-based evaluation and outperforms the other optimization methods. To find if the solution can be improved, we compare GA with other meta-heuristics, including simulated annealing, particle swarm optimization, and ant colony optimization. Finally, a sensitivity study for the location-inventory design problem is investigated.

Markovian queueing model with catastrophe, unreliable and backup server

In this article, we considered a finite size Markovian queue with single server. When the server breaks down, in order to facilitate the customer, backup server is provided. When system happened to undergo catastrophe, the customers are being removed and by restoration time the system get back to its normal state. Here, we have analysed the number of times the system reached its capacity. By utilising matrix geometric method, model has been solved and measures of effectiveness are done. Also numerical examples and graphical representation are given.