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Recent technological advances have equipped vehicles with significant computational capacity, enabling Vehicular Fog Computing (VFC) for decentralized processing at the network edge. This study presents a novel queuing theory-based analytical model that addresses the challenges of breakdowns and server vacations in VFC. The model incorporates factors like intermittent connectivity, energy constraints, and vehicle failures. Numerical results show that optimizing repair rates can reduce average delay by 16.67 % , while decreasing vacation time reduces delays by 33.33 % . Conversely, higher breakdown rates increase delays by 50 % , emphasizing the need for effective maintenance strategies. These results offer insights into Quality of Service (QoS) improvements by adjusting decision parameters. The study concludes by highlighting open areas for future research to further enhance VFC performance. [ABSTRACT FROM AUTHOR]

This paper considers an M/M/1 queueing inventory system with (s, S) policy. The server may break down during the working vacation and each customer takes one product away after being served. When there is no product in the system, the system starts a working vacation, and the service continues at a lower rate at that time. The server may breakdown only during a working vacation period. When a vacation breakdown occurs, the system immediately stops service and starts repairing. Firstly, we utilize Markov process theory to construct a three-dimensional Markov chain to analyze the stationary process of the system. Using Gaussian iteration and matrix geometry solution method, the steady-state performance measures about queueing and inventory are obtained. Then, through numerical experiments, we analyze the system parameters’ influence on performance indexes. Finally, we define a cost function and use the genetic algorithm to obtain the optimal inventory level and the lowest cost at a certain set of parameters. [ABSTRACT FROM AUTHOR]

In this article, a working vacation policy-based on bulk arrival feedback retrial queueing system with variable server capacity has been analyzed. The server can serve a minimum of one customer and a maximum of B customers in a batch in accordance with the variable server capacity bulk service rule. As soon as the orbit becomes empty at the time of service completion, the server goes for a working vacation. The server works at a lower speed during a working vacation period. In addition, the steady state probability generating function for system size and orbit size is generated by incorporating the supplementary variables technique (SVT). Further, the conditional decomposition law is shown for this retrial queueing system. Moreover, system performance metrics, and significant special instances are discussed. Finally, the effects of various parameters on the system performance are analyzed numerically. [ABSTRACT FROM AUTHOR]

The cloud storage service, known for its flexible and expandable nature, often has difficulties managing operating costs while ensuring dependable service and quick response times. This investigation presents a novel approach to optimizing cost efficiency in cloud storage systems by applying particle swarm optimization of the Markovian retrial queueing model in a generic setup by incorporating the working vacation and users' discouragement behavior. Some users may opt not to enter the system or join the retry pool to wait for their turn if the server is occupied. After returning from working vacation, if there is one user available for service, the server can interrupt the vacation period. The server is subject to breakdown and can be recovered after getting the repair. In the proposed model, the server is prone to catastrophes and can fail at any time, leading to the entire system breaking down, and no users being able to access it during this period. Chapman–Kolmogorov (CK) steady-state equations associated with the quasi-birth-death (QBD) process are constructed to make a mathematical design. The governing equations framed to derive the queue length distributions and various performance indices are solved using the recursive method and difference equation theory. The fuzzified parameters are used to develop the FM/FM/1/WV model, which is analyzed using a parametric nonlinear programming approach. To determine the optimal design parameters, the cost minimization problem has been done using the quasi-Newton method and particle swarm optimization. This model incorporates features such as server failures, retrials, and catastrophes, thereby reflecting the complex nature of cloud storage operations. A suitable illustration of cloud storage is taken for both classical and fuzzified models to facilitate the numerical results of performance indices and optimal decision descriptors. [ABSTRACT FROM AUTHOR]

In this article, we consider an M/M/C queue in which the arrival rate and service rate depends on the state of the system. In addition, the servers takes working vacation and the system may breakdown. Whenever breakdown takes place, the repair process immediately commences. During the repair period the customers are given service in a reduced service rate. Based on the vacation termination point, two models have been defined. The steady state probability vector of the number of customers in the queue and the stability condition are obtained using Matrix-Geometric method. The stationary waiting time distributions have been obtained. Some illustrative examples are also provided. [ABSTRACT FROM AUTHOR]

An M/M/1 retrial queue with working vacation and interruption in Bernoulli schedule under Ncontrol pattern is investigated in this article. To describe the system, we employ a QBD analogy. The model’s stability condition is deduced. The stationary probability distribution is generated by utilizing the matrix-analytic technique. The performance measures and special cases are designed. The model’s firmness is demonstrated numerically. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the single server queueing model where arrival of units is based on Markovian arrival process and service provided by server is in phase type distribution (MAP/PH/1) with working vacation and working breakdown. The study of working vacation and working breakdown is very important and applicable in real life. The absence of a server during vacation or server breakdown due to any reason may cause service interruption in queueing systems. We consider the repair process in two phases to recover the server from breakdown to working state with improved service rate phase-wise. Arrival process is based on Neuts' versatile point process that follows MAP and service times follow phase type distributions. The arrival rate, the service rate, working vacation rate and working breakdown are mutually independent. Matrix geometric solution method is used to obtain stationary probability vectors. Long run probabilities for all the states are derived in the results. Also with the different combination of arrival and service process the flow of average expected length with several parameters have been presented graphically. [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]

A single server queueing system with working vacation operating in a multi-level environment controlled by a random switch is considered. The environment has N levels and the random switch puts the server in one of the N levels to perform a service. The server resides in a level and reports to the random switch immediately after completing the service of the last customer in the queue. The random switch initiates an assignment process immediately at the arrival of a customer. The assignment process takes a random time and immediately at the end of the assignment interval, the server is put to operate in any one of the N levels with a positive probability. In each level r of the environment, the queueing system behaves like an M(λr,1)/M(μr,1)/1 queue and switches to a working vacation at a random time and serves with a lesser rate μr,2. During the working vacation, all newly arriving customers are lost. When the server is with the control switch, customers are permitted to join in the queue. For this model, time-dependent state probabilities are explicitly found and the corresponding steady-state probabilities are deduced. Some key performance measures are also obtained. A numerical study is also made. [ABSTRACT FROM AUTHOR]

We study the M/M/1 working vacation queueing system with N-policy and different arrival rates in this paper. The arrival rate of customers in a different period is different. Using the method of matrix geometry solution, We have given the steady-state performance indicators of this system. In addition, we also get the conditional stochastic decomposition structures of the queue length as well as the waiting time. The last part of the article show the numerical analyses and the trend of each performance indicators. [ABSTRACT FROM AUTHOR]

A Markovian retrial queue with reneging and working vacation under N-control pattern is investigated in this article. To describe the system, we employ a QBD analogy. The model's stability condition is deduced. The stationary probability distribution is generated by utilizing the matrix-analytic technique. The conditional stochastic decomposition of the line length in the orbit is calculated. The performance measures and special cases are designed. The model's firmness is demonstrated numerically. [ABSTRACT FROM AUTHOR]

In this article, an M/M/1 queueing model with bi-level network service provider, balking process and bi-level vacation policy that comprises of working vacation and complete vacation after fixed service, is developed. Matrix form expressions have been derived for the distributions of the queued customers with some performance metrics with the help of matrix geometric method. The maximum entropy principle is also used to derive the distributions of the steady state probabilities of queue size. The cost function has been formed to optimize the decision variables of the system. We perform the cost optimization by employing the steepest descent search method. Numerical illustrations along with the sensitivity analysis have been drawn to validate the model. Finally, the conclusions of the investigation done are drawn by mentioning the novel features and future scope. [ABSTRACT FROM AUTHOR]

In this paper, we consider two queuing models. Model 1 considers a single-server working vacation queuing system with interdependent arrival and service processes. The arrival and service processes evolve by transitions on the product space of two Markovian chains. The transitions in the two Markov chains in the product space are governed by a semi-Markov rule, with sojourn times in states governed by the exponential distribution. In contrast, in the second model, we consider independent arrival and service processes following phase-type distributions with representation (α , T) of order m and (β , S) of order n, respectively. The service time during normal working is the above indicated phase-type distribution whereas that during working vacation is a phase-type distribution with representation (β , θ S) , 0 < θ < 1 . The duration of the latter is exponentially distributed. The latter model is already present in the literature and will be briefly described. The main objective is to make a theoretical comparison between the two. Numerical illustrations for the first model are provided. [ABSTRACT FROM AUTHOR]

Any arriving customer who arrives and finds that the server is free, enters the service station and the remaining customers connect into the orbit. When the normal busy server is running, the system may at any time become defective due to a disaster. All users are forced to quit the system due to a disaster, which also brings about the failure of the main server. When a primary server breaks, it is shipped out for repair, and the repair process starts instantly. The server stops running as soon as the orbit is empty at a typical service finish instant. During the working breakdown or working vacation, the replacement server offers arriving customers a lower level of service. The arriving customer receives service instantly if the server is idle. If not, he will choose whether to leave the system without service or returning to receive service. Using the supplementary variable technique, we calculate the steady state PGF for system and orbit sizes. We generate performance measures and particular cases. With the use of specific numerical examples, we analyse the model. [ABSTRACT FROM AUTHOR]

Any arriving customer who arrives and finds that the server is free, enters the service station and the remaining customers connect into the orbit. When the normal busy server is running, the system may at any time become defective due to a disaster. All users are forced to quit the system due to a disaster, which also brings about the failure of the main server. When a primary server breaks, it is shipped out for repair, and the repair process starts instantly. The server stops running as soon as the orbit is empty at a typical service finish instant. During the working breakdown or working vacation, the replacement server offers arriving customers a lower level of service. The arriving customer receives service instantly if the server is idle. If not, he will choose whether to leave the system without service or returning to receive service. Using the supplementary variable technique, we calculate the steady state PGF for system and orbit sizes. We generate performance measures and particular cases. With the use of specific numerical examples, we analyse the model. [ABSTRACT FROM AUTHOR]

An M/G/1 feedback retrial queue with working vacation and a waiting server is taken into consideration in this study. Both retrial times and service times are assumed to follow general distribution and the waiting server follows an exponential distribution. During the working vacation period customers are served at a lesser rate of service. Before going for a vacation the server waits for some arbitrary amount of time and so is called a waiting server. We obtain the probability generating function (PGF) for the number of customers and the mean number of customers in the invisible waiting area by utilizing the supplementary variable technique. We compute the mean waiting time. Out of interest a few special cases are conferred. Numerical outcomes are exhibited. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

• We discuss reliability analysis of a fault tolerant system with imperfect coverage. • We explore system reliability, MTTF and other queuing measures. • The cost-effective ratio is evaluated to upgrade and improve availability. • We obtain optimal control parameters via a direct search approach and PSO. Due to widespread usage in many real time systems, reliability modeling and cost optimization of fault tolerance system have drawn attention of the practitioners. The fault tolerance in these systems can be provided by the support of maintenance and redundant components that help in smooth operation of the system in spite of failure of some active components. This investigation deals with the performance modeling of a fault-tolerant system consisting of a finite number of active (online) and standby components. During the switching from active to standby, the recovery procedure is performed, which may be imperfect. In case of imperfect recovery, the system reboot takes place. The maintenance of all the components is managed by a repairman (server) which is subject to failure. When the server is interrupted for rendering the service, functioning does not get stopped due to the system switch-over from perfect working to working breakdown mode. The system works even when the server is on working vacation and performs repair jobs of the failed components. The machine repair model based on Markovian process is developed to derive the transient probabilities and other performance indices of the fault tolerant system using Laplace transforms and matrix analytical method. Using the direct search strategy and particle swarm optimization, the cost-benefit analysis is done. The optimal design of the control parameters for the fault-tolerant system are presented by framing a cost-effective ratio function. The model is examined computationally by performing the numerical simulation and cost optimization. [ABSTRACT FROM AUTHOR]

This paper considers the customers' equilibrium and socially optimal joining–balking behavior in a single-server Markovian queue with a single working vacation and Bernoulli interruptions. The model is motivated by practical service systems where the service rate can be adjusted according to whether or not the system is empty. Specifically, we focus on a single-server queue in which the server's service rate is reduced from a regular to a lower one when the system becomes empty. This lower rate period is called a working vacation for the server which may represent that part of the service facility is under a maintenance process or works on other non-queueing job, or simply for saving the energy (for a machine server case). In this paper, we assume that the working vacation period is terminated after a random period or with probability p after serving a customer in a non-empty system. Such a system is called a queue with single working vacation and Bernoulli interruptions. Customers are strategic and can make choice of joining or balking based on different levels of system information. We consider four scenarios: fully observable, almost observable, almost unobservable, and fully unobservable queue cases. Under a reward-cost structure, we analyze the customer's equilibrium and social-optimal strategies. In addition, the effects of system parameters on optimal strategies are illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

In this manuscript, a queueing system with two optional vacation policies, power-saving mode under reneging and retention of reneged customers in both vacations is analyzed. If the server is free, it chooses either of the vacations, classical vacation or working vacation. During vacations, the customers may get impatient due to delays and may leave the system, but they are retained in the system with some convincing mechanisms. On vacation completion, if the system is empty, the server is turned off to facilitate better utilization of the resources. Some of the operating system characteristics are derived using the probability generating functions technique. The numerical results are graphically represented by using MATLAB software. [ABSTRACT FROM AUTHOR]

In this manuscript, Markovian queueing system with working vacation, Bernoulli schedule interruption, setup time under feedback, reneging of impatient customers, and retention of reneged customers are analyzed. The unsatisfied customers on service completion may either leave the system with probabilities  or may rejoin the queue with complementary probabilities during working vacations and regular service, respectively. The waiting customers in the queue may lose patience due to vacations and decide to leave without getting the service with probability q. They may be retained in the system via some convincing mechanisms with probability (1-q). The mean system length, probability of server in various states, mean sojourn time are obtained using the probability generating function method. The MATLAB software is used for representing the observed behavior graphically. [ABSTRACT FROM AUTHOR]

In this present paper, an M/M/1 retrial queueing model with a waiting server subject to breakdown and repair under working vacation, vacation interruption is considered. Customers are served at a slow rate during the working vacation period, and the server may undergo breakdowns from a normal busy state. The customer has to wait in orbit for the service until the server gets repaired. Steady-state solutions are obtained using the probability generating function technique. Probabilities of different server states and some other performance measures of the system are developed. The variation in mean orbit size, availability of the server, and server state probabilities are plotted for different values of breakdown parameter and repair rate with the help of MATLAB software. Finally, cost optimization of the system is also discussed, and the optimal value of the slow service rate for the model is obtained. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, an M/G/1 G-queue with server breakdown, working vacations and Bernoulli vacation interruption is considered. In the normal busy period, the arrival of a negative customer not only takes away the positive customer being in service, but also causes the server break down. During the repair time, the system is out of service until the repair is completed. And during the working vacation period, the vacation will be interrupted with probability p (0 ≤ p ≤ 1) or continues with probability p = 1 - p, if there are customers in the system at a service completion instant. By applying the matrix-analytic method and the supplementary variable technique, the probability generating functions of the queue length and the server state are obtained. Finally, the sensitivity analysis and cost analysis of the model are presented. [ABSTRACT FROM AUTHOR]

In this article, we consider a batch arrival MX/M/1 queue with two-stage vacations policy that comprises of single working vacation and multiple vacations, denoted by MX/M/1/SWV + MV. Using the matrix analytic method, we derive the probability generating function (PGF) of the stationary system size and investigate the stochastic decomposition structure of stationary system size. Further, we obtain the Laplace–Stieltjes transform (LST) of stationary sojourn time of a customer by the first passage time analysis. At last, we illustrate the effects of various parameters on the performance measures numerically and graphically by some numerical examples. [ABSTRACT FROM AUTHOR]

This paper presents a steady-state analysis of an M/M/1 retrial queue with working vacations, in which the server is subject to starting failures. The proposed queueing model is described in terms of the quasi-birth-death (QBD) process. We first derive the system stability condition. We then use the matrix-geometric method to compute the stationary probability distribution of the orbit size. Some performance measures for the system are developed. We construct a cost model, and our objective is to determine the optimal service rates during normal and vacation periods that minimize the expected cost per unit time. The canonical particle swarm optimization (CPSO) algorithm is employed to deal with the cost optimization problem. Numerical results are provided to illustrate the effects of system parameters on the performance measures and the optimal service rates. These results depict the system behaviour and show how the CPSO algorithm can be used to find numerical solutions for optimal service rates. [ABSTRACT FROM AUTHOR]

In this paper, an M/M/1 queue with working vacation and vacation interruption is investigated. The server is supposed to interrupt the vacation and return back to the normal working period, if there are at least N customers waiting in the system at a service completion instant during the working vacation period. Otherwise, the server continues the vacation until the system is nonempty after a vacation ends or there are at least N customers after a service ends. In terms of the quasi birth and death process and matrix-geometric solution method, we obtain the distributions for the stationary queue length. Moreover, we demonstrate stochastic decomposition structures of the queue length and waiting time, and obtain the distributions of the additional queue length and additional delay for the case N = 2. Finally, numerical examples are presented. [ABSTRACT FROM AUTHOR]

In this paper, an M/M/1 retrial queue with working vacation, orbit search and balking is considered. Using the matrix-analytic method, we obtain the necessary and sufficient condition for system to be stable. We also derive the stationary probability distribution and some performance measures. Then we give the conditional stochastic decomposition for the queue length in the orbit when the server is busy. Finally, we show the effect of the model parameters on the system's characteristics by some numerical examples. [ABSTRACT FROM AUTHOR]

This study deals with the performance modeling of finite Markov M/M/1/L/WV model for the fault-tolerant machining system (FTMS). The concepts of redundancy along with the provision of dissimilar warm standbys are taken into account in order to maintain the pre-required high reliability of the system. The repairman is allowed to take a vacation in case of no workload of broken down machines. The failed machines are also repaired with slower rate by the repairman during working vacation period. The analytical method, namely matrix method, is implemented for evaluating the transient queue size distribution and closed form expressions of the performance metrics of multi-component FTMS. Moreover, cost function is constructed, which can be further used to control the system cost and other factors. Numerical simulation is also carried out to exhibit the effect of parameters on various system indices. [ABSTRACT FROM AUTHOR]

Highlights • Consider a standby system with working vacations and retrial of failed components. • We compute the steady-state availability using the matrix-analytic method. • We develop the reliability function and mean-time-to-failure. • Numerical examples are used to conduct sensitivity analysis. Abstract In this paper, we consider a repairable system consisting of M primary components, S spare components, and a repairman. In cases where none of the components in the system is failed, the repairman leaves the system for multiple vacations. During a vacation period, the repairman lowers the repair rate rather than halting repairs together. The system does not include a waiting space. If a failed component finds the repairman free upon arrival, then it immediately occupies the repairman and is being repaired. If a failed component does not find a free repairman upon arrival, then it leaves the service area to join the retrial group (orbit) to try again for a repair. For this system, the matrix-analytic method is used to compute the steady-state availability. We develop the reliability function and mean-time-to-failure (MTTF) based on the Laplace transform technique. Numerical examples are given to assess the effects of system parameters on the system reliability, MTTF, and steady-state availability. [ABSTRACT FROM AUTHOR]

A single-channel finite-buffer queueing model with a general independent input stream of customers, exponential processing times and a working vacation policy is considered. Every time , when the server becomes idle, an exponentially distributed single working vacation period is being initialized, during which the processing is provided with another (slower) rate. After the completion of the vacation period, the service is being continued normally, with the original speed. Using the idea of an embedded Markov chain, the systems of Volterra-type integral equations for the time-dependent queue-size distributions, conditioned by the initial buffer state and related to each other, are built for models beginning the operation in normal and working vacation modes, separately. The solutions of the corresponding systems written for the Laplace transforms are obtained in compact forms using the linear algebraic approach. The numerical illustrative examples are attached as well. [ABSTRACT FROM AUTHOR]

We consider anM/M/c queue with multiple working vacation and impatient customers. The server serves the customers at a lower rate rather than completely halts the service during this working vacation period. The impatience of the customer's arises when they arrive during the working vacation period, where the service rate of the customer's is lower than the normal busy period. The queue is analyzed for multiple working vacation policies. The policy of a MWV demands the server to keep taking vacation until it finds at least a single customer waiting in the system at an instant vacation completion. On returning of the server from his vacation along with finding at least one customer in the system, the server changes its service rate, thereby giving rise to a non-vacation period; otherwise the server immediately goes for another WV. We formulate the probability generating function for the number of customers present when the server is both in a service period as well as in a working vacation period. We further derive a closed-form solution for various performance measures such as the mean queue length and the mean waiting time. The stochastic decomposition properties are verified for the model. [ABSTRACT FROM AUTHOR]

In this paper, we investigate an M/M/1 queue with a two-stage vacation policy which comprises of single working vacation and single vacation. Using the matrix-analytic method, we obtain the distribution of stationary system size, and then the decomposition structures of the stationary system size and the sojourn time are demonstrated. Furthermore, we study the waiting time by first-passage time analysis. Meanwhile, the busy-cycle analysis is provided by the limiting theorem of alternative renewal process. Finally, several numerical examples are presented in the paper. [ABSTRACT FROM PUBLISHER]

This investigation is concerned with the performance modeling of machining system operating under the admission control F-policy and server working vacation policy. The repair of failed machines is provided by an unreliable server, who also renders the service with the slower rate rather than completely terminating the service during the vacation period. The failed machines are allowed to enter the system till the system capacity ( K) is full; then after failed machines are not allowed to join the system until the system size again decreases to the prespecified threshold level ' F'. At that instant, the server takes start-up time in order to start allowing the failed machines to enter into the system for the repair job. Numerical method based on successive over-relaxation is applied to obtain the steady-state probabilities and various performance indices including the cost function. The numerical simulation is performed to explore the sensitivity of the system indices with respect to various parameters. Quasi-Newton method and direct search method are used to determine the optimal service rate and threshold parameter. [ABSTRACT FROM AUTHOR]

We consider an M=G=1 retrial queue with general retrial times and single working vacation. During the working vacation period, customers can be served at a lower rate. Both service times in a vacation period and in a service period are generally distributed random variables. Using supplementary variable method we obtain the probability generating function for the number of customers and the average number of customers in the orbit. Furthermore, we carry out the waiting time distribution and some special cases of interest are discussed. Finally, some numerical results are presented. [ABSTRACT FROM AUTHOR]

We consider an M/M/1 queue with two vacation policies which comprise single working vacation and multiple vacations, denoted by M/M/1/SMV+MV. Using two methods (calledR-matrix method andG-matrix method), we obtain the stationary distribution of queue length (including the customer being in service) and make further analysis on the stationary numbers of customers in the working vacation and vacation period, respectively. The stochastic decomposition results of stationary queue length and the sojourn time of a customer are also derived. Meanwhile, we show that a simple and direct method of decomposition developed in Liuet al.[Stochastic decompositions in the M/M/1 queue with working vacations, Oper. Res. Lett. 35 (2007), pp. 595–600] is also applicable to our model. Furthermore, busy period is analysed by the limiting theorem of alternative renewal process. Finally, some boundary properties and numerical analysis on performance measures are presented. [ABSTRACT FROM PUBLISHER]

This paper deals with the steady state behavior of an M/G/1 multiple working vacation queue with server breakdown. The server works with different service times rather than completely stopping service during a vacation. Both service times in a vacation period and in a regular service period are assumed to be generally distributed random variables. The system may breakdown at random and repair time is arbitrary. Further, just after completion of a customer's service the server may take a multiple working vacation. Supplementary variable technique is employed tofind the probability generating function for the number of customers in the system. The mean number of customers in the system is calculated. Some particular cases of interest are discussed. Numerical results are also presented. [ABSTRACT FROM AUTHOR]

The power management issue has always been a critical concern in cloud computing for supporting rapid growth of data centers. In this paper, our strategy is to implement working vacation (WV) to lower and eliminate unnecessary power consumed by idle servers. Two green systems are first proposed where one implements a single WV and the other implements multiple WVs in an operational cycle. The effect of various service rates and WV lengths on system delay and operating state probabilities is compared and studied. A cost function is developed by taking response time, system holding cost and power consumption cost into consideration. Control procedures in both green systems are mapped into Petri net-based models which contribute to designing a multiple decision process and describing system behaviors. The issue of determining the optimal service rate and WV length to obtain the cost optimality within response time guarantee is studied. The proposed Green control ( $$\mu $$ , $$\Theta )$$ policy combined with a heuristic algorithm allows cloud providers to solve constrained optimization problems. Simulation results show that significant cost savings and response time improvement can be validated as compared to a typical system. [ABSTRACT FROM AUTHOR]

Abstract: We first analyze a multi-server queue with working vacations. Using a quasi birth-and-death (QBD) process and a matrix-geometric solution method, the steady state distribution of the queue length is derived. Furthermore, a fluid flow model driven by this working vacation queue is discussed, we obtain the sets of differential equations satisfied by the stationary joint distribution of the buffer content, by which we gain the simple structure of the Laplace transform (LT) of the stationary distribution of the buffer content. Finally, we give the probability of empty buffer content and the mean of the buffer content based on the relationship between the LT and the Laplace–Stieltjes transform (LST) of the stationary distribution. [Copyright &y& Elsevier]

Abstract: In this paper we study a MAP/PH/1 queueing model in which the server is subject to taking vacations and offering services at a lower rate during those times. The service is returned to normal rate whenever the vacation gets over or when the queue length hits a specific threshold value. This model is analyzed in steady state using matrix analytic methods. An illustrative numerical example is discussed. [Copyright &y& Elsevier]

Abstract: This paper treats an M/G/1 queue with single working vacation and vacation interruption under Bernoulli schedule. Whenever the system becomes empty at a service completion instant, the server goes for a single working vacation. In the working vacation, a customer is served at a lower speed, and if there are customers in the queue at the instant of a service completion, the server is resumed to a regular busy period with probability p (i.e., the vacation is interrupted) or continues the vacation with probability . Using the matrix analytic method, we obtain the distribution for the stationary queue length at departure epochs. The joint distribution for the stationary queue length and service status at the arbitrary epoch is also obtained by using supplementary variable technique. We also develop a variety of stationary performance measures for this system and give a conditional stochastic decomposition result. Finally, several numerical examples are presented. [Copyright &y& Elsevier]

Abstract: Consider a GI/M/1 queue with start-up period and single working vacation. When the system is in a closed state, an arriving customer leading to a start-up period, after the start-up period, the system becomes a normal service state. And during the working vacation period, if there are customers at a service completion instant, the vacation can be interrupted and the server will come back to the normal working level with probability p (0⩽ p ⩽1) or continue the vacation with probability 1− p. Meanwhile, if there is no customer when a vacation ends, the system is closed. Using the matrix-analytic method, we obtain the steady-state distributions for the queue length at both arrival epochs and arbitrary epochs, the waiting time and sojourn time. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a finite buffer size discrete-time multiple working vacation queue with changeover time. Employing the supplementary variable and embedded Markov chain techniques, we derive the steady state system length distributions at different time epochs. Based on the various system length distributions, the blocking probability, probability mass function of sojourn time and other performance measures along with some numerical examples have been discussed. Then, we use the parabolic method to search the optimum value of the service rate in working vacation period under a given cost structure. [ABSTRACT FROM AUTHOR]

Abstract: This paper analyzes the -policy M/M/1/K queueing system with working vacation and an exponential startup time. The -policy deals with the issue of controlling arrivals to a queueing system, and the server requires a startup time before allowing customers to enter the system. For the queueing systems with working vacation, the server can still provide service to customers rather than completely stop the service during a vacation period. The matrix-analytic method is applied to develop the steady-state probabilities, and then obtain several system characteristics. We construct the expected cost function and formulate an optimization problem to find the minimum cost. The direct search method and Quasi-Newton method are implemented to determine the optimal system capacity , the optimal threshold and the optimal service rates at the minimum cost. A sensitivity analysis is conducted to investigate the effect of changes in the system parameters on the expected cost function. Finally, numerical examples are provided for illustration purpose. [Copyright &y& Elsevier]

Abstract: We consider a discrete-time single-server queueing model where arrivals are governed by a discrete Markovian arrival process (DMAP), which captures both burstiness and correlation in the interarrival times, and the service times and the vacation duration times are assumed to have a general phase-type distributions. The vacation policy is that of a working vacation policy where the server serves the customers at a lower rate during the vacation period as compared to the rate during the normal busy period. Various performance measures of this queueing system like the stationary queue length distribution, waiting time distribution and the distribution of regular busy period are derived. Through numerical experiments, certain insights are presented based on a comparison of the considered model with an equivalent model with independent arrivals, and the effect of the parameters on the performance measures of this model are analyzed. [Copyright &y& Elsevier]

Abstract: This paper deals with a single server working vacation queueing model with multiple types of server breakdowns. In a working vacations queueing model, the server works at a different rate instead of being completely idle during the vacation period; the arrival rate varies according to the server’s status. It is assumed that the server is subject to interruption due to multiple types of breakdowns and is sent immediately for repair. Each type of breakdown requires a finite random number of stages of repair. The life time of the server and the repair time of each phase are assumed to be exponentially distributed. We propose a matrix–geometric approach for computing the stationary queue length distribution. Various performance indices namely the expected length of busy period, the expected length of working vacation period, the mean waiting time and average delay, etc. are established. In order to validate the analytical approach, by taking illustration, we compute numerical results. The sensitivity analysis is also performed to explore the effect of different parameters. [Copyright &y& Elsevier]

Abstract: This paper studies the M/M/1 machine repair problem with working vacation in which the server works with different repair rates rather than completely terminating the repair during a vacation period. We assume that the server begins the working vacation when the system is empty. The failure times, repair times, and vacation times are all assumed to be exponentially distributed. We use the MAPLE software to compute steady-state probabilities and several system performance measures. A cost model is derived to determine the optimal values of the number of operating machines and two different repair rates simultaneously, and maintain the system availability at a certain level. We use the direct search method and Newton’s method for unconstrained optimization to repeatedly find the global minimum value until the system availability constraint is satisfied. Some numerical examples are provided to illustrate Newton’s method. [Copyright &y& Elsevier]

Abstract: In this paper, we study a discrete time Geom/Geom/1 queue with multiple working vacations. Using the quasi birth and death chain and matrix-geometric solution method, we give distributions for the number of customers in system and the waiting time of a customer and their stochastic decomposition structures, and obtain distributions of the additional number of customers and additional delay. Furthermore, we derive the formulae of expected regular busy period and expected busy cycle. Finally, by numerical examples, we analyze the effect of the parameters on the expected queue length and sojourn time. [Copyright &y& Elsevier]