Matrix-geometric Approach Research Articles

Performance analysis and ANFIS computing of a Markovian queuing model with intermittently accessible server under a hybrid vacation policy

In this study, we investigate a heterogeneous queueing model with intermittent server availability, server catastrophes, and a hybrid vacation policy. Our focus is on a specific scenario: server 1 is always available, while server 2 may experience breakdowns or vacations, making it intermittently accessible. Using the matrix-geometric approach (MGA), we derive matrix-based expressions for the stationary probability distribution of the number of customers in the system and various system performance measures. Additionally, we evaluate the cost function per unit of time to determine optimal values for the system’s decision variables. Furthermore, we employ an adaptive neural fuzzy inference system (ANFIS) based on soft computing technology to compare and analyze the numerical results obtained. Through this comprehensive analysis, our study contributes to the understanding and optimization of this complex queueing system, attracting the attention of researchers in the field and offering practical insights for real-world applications.

Performance approximations of Markov Modulated Poisson Process arrival queue with Markovian Service using matrix geometric approach

Queue analysis of correlated work of the arrival and the service process are not available much in literature. This research paper applies matrix geometric approach to study the MMPP/MSP/1 queuing model under the quasi-birth death(QBD) process. Customers admitted in the system are having two different intensity of arrival followed by Markov Modulated Poisson Process(MMPP). MMPP is a versatile class of MAP and it is necessary to adapt the first characterization of the queuing model. In service mechanism Markov Service Process(MSP) is adopted. As like MMPP, MSP is also a versatile and can capture the correlation of the modulated arrivals. The methodology presented in this research work calculates the metric performances not depending on the series of infinite state of the system. The assumed model is formulated as QBD process to obtain the steady state probability of the system. The complexity model formulated involves structures matrices with appropriate dimension. Neut?s pioneered the matrix geometric approach to such complex models which involves the rate computation of rate matrix R. The rate matrix R is approximated to derive the performance metrics of the system and then compute the cost for the underlying Markov chain. To attain this approximation the sensitivity analysis is done with numerical results.

Analysis of single server repairable queueing system with many kinds of service, single working vacation and standby server

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.

HETEROGENEOUS SERVER RETRIAL QUEUEING MODEL WITH FEEDBACK AND WORKING VACATION USING ARTIFICIAL BEE COLONY OPTIMIZATION ALGORITHM

This research delves into the dynamics of a retrial queueing system featuring heterogeneous servers with intermittent availability, incorporating feedback and working vacation mechanisms. Employing a matrix geometric approach, this study establishes the steady-state probability distribution for the queue size in this complex heterogeneous service model. Additionally, a range of system performance metrics is developed, alongside the formulation of a cost function to evaluate decision variable optimization within the service system. The Artificial Bee Colony (ABC) optimization algorithm is harnessed to determine service rates that minimize the overall cost. This work includes numerical examples and sensitivity analyses to validate the model's effectiveness. Also, a comparison between the numerical findings and the neuro-fuzzy results has been examined by the adaptive neuro fuzzy interface system (ANFIS).

Two commodity queueing inventory system with random common lifetime, two demand classes and pool of customers

This paper analyzes a two-commodity perishable queueing inventory system with random common lifetimes commodity-wise, positive service time, and individual ordering policy. We have considered two types of customers: priority customers and non-priority customers. There is a buffer of finite capacity for the priority customers and a pool with infinite capacity made available for the non-priority customers. The arrival of these customers is occurred by two independent Poisson processes. The lifetimes of each of the commodities and service times follow independent exponential distributions. The matrix geometric approach is used to analyze the model's steady-state behavior and we computed various system performance measures. This article examines the effects of the common lifetime parameters on the measures of performance, such as average cycle time, the average number of reorders due to the realization of the common lifetime of commodities, etc. Also, a sensitivity analysis of the system parameters on the performance measures is carried out. Further, a cost function is constructed based on the performance measures, and its convexity is examined numerically.

A multi-server retrial queueing-inventory system with asynchronous multiple vacations

This study deals with asynchronous server vacation and customer retrial facilities in a multi-server queueing-inventory system. Customers arrive according to a Poisson process. The system comprises identical servers, a finite-size waiting area, and a storage area. The service time is distributed exponentially. If each server finds insufficient customers and items in the system after the busy period, they start a vacation. Once the server’s vacation is over and it recognizes there is no chance of getting busy, it goes into an idle state; otherwise, it will take another vacation. Each server’s vacation period occurs independently of the other servers. The system accepts a control policy for inventory replenishment. For the steady-state analysis, Neuts and Rao’s matrix geometric approximation approach is used owing to the structure of an infinitesimal generator matrix. The necessary stability condition is computed. After calculating the sufficient system performance measures, an expected total cost of the system will be constructed and numerically incorporated with the parameters. Additional numerical analyses are conducted to examine customers’ waiting time in the queue and orbit and the expected customer loss rate.

Machine repair system with threshold recovery policy, unreliable servers and phase repairs

ABSTRACT This study aims to discuss queueing modeling of machine repair problems under threshold recovery policy (Q), server unreliability and k-type phase repairs. When one or more machines in the system fails/fail, the service facility (or all R -servers) is activated and repairs the failed machine according to the first come, first served (FCFS) discipline. The service facility may experience a partial or complete breakdown while providing service and needs to be repaired in required phases under the threshold recovery policy (Q). The matrix geometric approach is applied to solve the system governing equations, and a sensitivity analysis is presented to understand the effect of various parameters on system performance metrics. The greedy selection method and the golden section method are used to compute optimal rates of R ∗ , μ ∗ of service facility in order to minimize the cost function of the system.

Cost Optimization of an Unreliable server queue with two stage service process under hybrid vacation policy

In this investigation, an unreliable server Markovian queueing model is developed for a service system by considering two stage service process and hybrid vacation policy. By including the features of combination of working vacation (WV) and complete vacation (CV), the steady state probability distribution of the queue size of two stage service model via matrix geometric approach has been established. The cost function has been formulated to evaluate the optimal values of the decision variables of the service system. Particle swarm optimization (PSO) and Artificial bee colony (ABC) optimization algorithms are employed to compute the optimal service rates at optimum cost. To validate the model, numerical illustrations along with sensitivity analysis have been provided.

Analysis of Stochastic State-Dependent Arrivals in a Queueing-Inventory System with Multiple Server Vacation and Retrial Facility

This article analyses a four-dimensional stochastic queueing-inventory system with multiple server vacations and a state-dependent arrival process. The server can start multiple vacations at a random time only when there is no customer in the waiting hall and the inventory level is zero. The arrival flow of customers in the system is state-dependent. Whenever the arriving customer finds that the waiting hall is full, they enter into the infinite orbit and they retry to enter the waiting hall. If there is at least one space in the waiting hall, the orbital customer enters the waiting hall. When the server is on vacation, the primary (retrial) customer enters the system with a rate of λ1(θ1). If the server is not on vacation, the primary (retrial) arrival occurs with a rate of λ2(θ2). Each arrival rate follows an independent Poisson distribution. The service is provided to customers one by one in a positive time with the rate of μ, which follows exponential distribution. When the inventory level drops to a fixed s, reorder of Q items is triggered immediately under (s,Q) ordering policy. The stability of the system has been analysed, and using the Neuts matrix geometric approach, the stationary probability vectors have been obtained. Moreover, various system performance measures are derived. The expected total cost analysis explores and verifies the characteristics of the assumed parameters of this model. The average waiting time of a customer in the waiting hall and orbit are investigated using all the parameters. The monotonicity of the parameters is verified with its characteristics by the numerical simulation. The discussion about the fraction time server being on vacation suggests that as the server’s vacation duration reduces, its fraction time also reduces. The mean number of customers in the waiting hall and orbit is reduced whenever the average service time per customer and average replenishment time are reduced.

Analysis of Stochastic M/M/c/N Inventory System with Queue-Dependent Server Activation, Multi-Threshold Stages and Optional Retrial Facility

The purpose of this article is to examine the server activation policy (SAP) in a multi-server queuing-inventory system (MQIS). The queue has a total of c number of multi-threshold stages as well as c-homogeneous servers. The activation of each server begins one by one if there is an adequate queue length and inventory in the system; otherwise, they remain idle. The server deactivation process continues until the queue length exceeds the manageable level (predetermined stages) or there is insufficient stock. In addition, when we assume the length of the two successive threshold levels is one, the server activation policy model becomes a regular multi-server model. The Neuts matrix geometric approach is used to discuss the stability condition, stationary probability vector. The Laplace–Stieltjes transform (LST) is used to analyse the waiting time distributions of the queue and orbital customers. Additionally, significant system performance metrics and sensitivity analysis are used to investigate the effects of various parameters and cost values. In the comparative result between the server activation model (SAM) and without the server activation model (WSAM) on the expected total cost, we obtain the minimised cost in the SAM. Moreover, the results are obtained by assuming that the length of the intervals between the two successive threshold levels is to be taken into account as the non-uniform length. The expected inventory level, reorder rate, and waiting time of a customer in the waiting hall and orbit were explored numerically by the parameter analysis.

Analysis of Stock-Dependent Arrival Process in a Retrial Stochastic Inventory System with Server Vacation

The present study deals with the stock-dependent Markovian demand of a retrial queueing system with a single server and multiple server vacation. The items are restocked under a continuous review (s,Q) ordering policy. When there is no item in the system, the server goes on vacation. Further, any arrival demand permits entry into an infinite orbit whenever the server is on vacation. In the Matrix geometric approach with the Neuts-Rao truncation technique, the steady-state joint distribution of the number of customers in orbit, the server status, and the inventory level is obtained. Under the steady-state conditions, some significant system performance measures, including the long-run total cost rate, are derived, and the Laplace-Stieltjes transform is also used to investigate the waiting time distribution. According to various considerations of uncontrollable parameters and costs, the merits of the proposed model, especially the important characteristics of the system with stock dependency over non-stock dependency, are explored. Ultimately, the important facts and ideas behind this model are given in conclusion.

Optimization of retrial queue with unreliable servers subject to imperfect coverage and reboot delay

ABSTRACT A retrial queue with constant retrial policy, unreliable servers and the failed servers may not be detected is investigated in this paper. Upon arrival, a customer either is attended to one of the available servers or enters a retrial orbit and retries in a random time as all servers are not available. The unreliable server can be breakdown when he is working. The breakdown server may not be detected due to some fault matters. Under such a situation, the repair facility will issue a reboot operation to locate the non-detected breakdown server. Once the breakdown server is detected and located, he will be sent for repair. After utilizing the general theory of matrix-geometric approach to determine the stationary distribution, some performance measures are proposed. A single-objective model aiming at minimizing the average cost is constructed. We apply particle swarm optimization algorithm and genetic algorithm to find the optimum combination of parameters. We also formulate a bi-objective model aiming at minimizing the expect cost and the mean waiting time of customers in orbit. Two multi-objective algorithms are developed to solve it.

M/M/1 Queueing Model with Working Vacation and Two Type of Server Breakdown

In this paper the steady state probability distribution of the number of customers in single server Markovian queue is obtained by using matrix geometric approach with working vacation where server may breakdown in working vacation state as well as in busy state. The arrival of the customer is depends upon the server’s state and arrival follows FCFS discipline. The server is made available for rendering alternate service to customers. The inter-arrival time of the customers, service time, vacation duration, and lifetime and repair time of the server follows an exponential distribution. Numerical illustrations are made to examine the validity of analytical results. Sensitivity analysis is also made of in order to discover the outcome of various parameters on the system performance physical appearance.

Analysis of Interconnected Arrivals on Queueing-Inventory System with Two Multi-Server Service Channels and One Retrial Facility

Present-day queuing inventory systems (QIS) do not utilize two multi-server service channels. We proposed two multi-server service channels referred to as T1S (Type 1 n-identical multi-server) and T2S (Type 2 m-identical multi-server). It includes an optional interconnected service connection between T1S and T2S, which has a finite queue of size N. An arriving customer either uses the inventory (basic service or main service) for their demand, whom we call T1, or simply uses the service only, whom we call T2. Customer T1 will utilize the server T1S, while customer T2 will utilize the server T2S, and T1 can also get the second optional service after completing their main service. If there is a free server with a positive inventory, there is a chance that T1 customers may go to an infinite orbit whenever they find that either all the servers are busy or no sufficient stock. The orbital customer can request for T1S service under the classical retrial policy. Q(=S−s) items are replaced into the inventory whenever it falls into the reorder level s such that the inequality always holds n<s. We use the standard (s,Q) ordering policy to replace items into the inventory. By varying S and s, we investigate to find the optimal cost value using stationary probability vector ϕ. We used the Neuts Matrix geometric approach to derive the stability condition and steady-state analysis with R-matrix to find ϕ. Then, we perform the waiting time analysis for both T1 and T2 customers using Laplace transform technique. Further, we computed the necessary system characteristics and presented sufficient numerical results.

Computation of Invariant Measures and Stationary Expectations for Markov Chains with Block-Band Transition Matrix

This paper deals with the computation of invariant measures and stationary expectations for discrete-time Markov chains governed by a block-structured one-step transition probability matrix. The method generalizes in some respect Neuts’ matrix-geometric approach to vector-state Markov chains. The method reveals a strong relationship between Markov chains and matrix continued fractions which can provide valuable information for mastering the growing complexity of real-world applications of large-scale grid systems and multidimensional level-dependent Markov models. The results obtained are extended to continuous-time Markov chains.

Data-driven control of a production system by using marking-dependent threshold policy

As increasingly more shop-floor data becomes available, the performance of a production system can be improved by developing effective data-driven control methods that utilize this information. We focus on the following research questions: how can the decision to produce or not to produce at any time be given depending on the real-time information about a production system?; how can the collected data be used directly in optimizing the policy parameters?; and what is the effect of using different information sources on the performance of the system? In order to answer these questions, a production/inventory system that consists of a production stage that produces to stock to meet random demand is considered. The system is not fully observable but partial production and demand information, referred to as markings is available. We propose using the marking-dependent threshold policy to decide whether to produce or not based on the observed markings in addition to the inventory and production status at any given time. An analytical method that uses a matrix geometric approach is developed to analyze a production system controlled with the marking-dependent threshold policy when the production, demand, and information arrivals are modeled as Marked Markovian Arrival Processes. A mixed integer programming formulation is presented to determine the optimal thresholds. Then a mathematical programming formulation that uses the real-time shop floor data for joint simulation and optimization (JSO) of the system is presented. Using numerical experiments, we compare the performance of the JSO approach to the analytical solutions. We show that using the marking-dependent control policy where the policy parameters are determined from the data works effectively as a data-driven control method for manufacturing.

Resource allocation strategies for protein purification operations

We analyze resource allocation challenges in protein purification operations where differences in scientist capabilities can lead to significantly different outcomes. We use queuing models to capture the underlying dynamics and quantify the performance of different strategies based on solutions obtained using the matrix-geometric approach. We show that certain partial flexibility structures coupled with appropriate priority rules can yield very efficient system performance. We also define a new server utilization metric that can be very effective in rank ordering strategies. Through numerical studies, we provide useful rules for the biomanufacturers to achieve higher profits and shorter lead times.

Performance analysis of a GI/D- $$MSP/1/\infty$$ M S P / 1 / ∞ queueing system under different service phase initiations

The main objective of this paper is to derive some analytic results concerning GI/D- $$MSP/1/\infty$$ queueing model in which customers arrive according to a discrete-time renewal process. We assume that the service process is correlated and its configuration is designed through discrete-time Markovian service process under different service phase initiations to start a busy period. The system-length distributions at various time epochs have been obtained by employing the matrix-geometric approach. An explicit closed form expression for the waiting-time distribution in the queue measured in slots has also been derived. Computational experiences with a variety of numerical results are discussed to show the effect of system parameters on the performance measures. This queueing model has potential applications in the domain of computer networks, telecommunication systems, and wireless local communication networks for evaluating the various system performances. An application of this type of queueing model in the domain of network coding in wireless communications is also discussed. This provides relevant information to the system analysts, researchers, and industry people who are interested to investigate congestion problems using queueing theory.

An M/M/1 Queue with Working Vacation and Vacation Interruption Under Bernoulli Schedule

This work deals with M/M/1 queue with Vacation and Vacation Interruption Under Bernoulli schedule. When there are no customers in the system, the server takes a classical vacation with probability p or a working vacation with probability 1-p, where . At the instants of service completion during the working vacation, either the server is supposed to interrupt the vacation and returns back to the non-vacation period with probability 1-q or the sever will carry on with the vacation with probability q. When the system is non empty after the end of vacation period, a new non vacation period begins. A matrix geometric approach is employed to obtain the stationary distribution for the mean queue length and the mean waiting time and their stochastic decomposition structures. Numerous graphical demonstrations are presented to show the effects of the system parameters on the performance measures.

Analysis of a queueing system in random environment with an unreliable server and geometric abandonments

This paper studies a single server queueing model in a multi-phase random environment with server breakdowns and geometric abandonments, where server breakdowns only occur while the server is in operation. At a server breakdown instant (i.e., an abandonment opportunity epoch), all present customers adopt the so-called geometric abandonments, that is, the customers decide sequentially whether they will leave the system or not. In the meantime, the server abandons the service and a repair process starts immediately. After the server is repaired, the server resumes its service, and the system enters into the operative phaseiwith probabilityqi,i= 1, 2, …,d. Using probability generating functions and matrix geometric approach, we obtain the steady state distribution and various performance measures. In addition, some numerical examples are presented to show the impact of parameters on the performance measures.