Synchronous Vacation Research Articles

Customer Joining Behavior and Regulation Strategy in Visible Multi-Server Queues with Synchronous Vacations of Partial Servers

Customer Joining Behavior and Regulation Strategy in Visible Multi-Server Queues with Synchronous Vacations of Partial Servers

Modeling and optimization analysis of a retrial system with partial servers synchronous vacation and customer geometric loss

In many actual issues, such as call center and cloud service computing center, retrial systems have been widely applied to derive the performance measures. This study takes up a multi-server retrial system with partial servers synchronous vacation and customer geometric loss. An arriving customer finding no servers available either enters a retrial group according to first-in-first-serve discipline or balks. In the retrial group, only the first customer can retry for his service. A customer from the retrial group finding all servers unavailable decides either balks or reenters the retrial group. Partial servers may be on vacation simultaneously when they become idle at the moment of service completion and system conditions satisfy the administrative policy. This retrial system is analyzed as a quasi-birth-death process, and some general descriptors are developed. Optimization problems, including single-objective and bi-objective, are constructed to help the manager’s decision-making. Numerical results are displayed tabularly and graphically. A potential application in which the proposed model can be applied is provided.

The virtual machines scheduling strategy based on M/M/[formula omitted] queueing model with vacation

To enhance the energy savings of cloud systems and ensure the quality of service, energy saving measures was proposed for virtual machines (VMs) in cloud data centers. Considering the disadvantages of synchronous vacation and asynchronous vacation and the energy consumption caused by frequent VM state switching. In this study, a policy combining synchronous vacation and asynchronous vacation is proposed. The scheduling policy is modeled as a two-dimensional Markov stochastic model with multiple servers. Then, the analytical expressions of the equations are obtained using the quasi-birth-and-death process and the matrix-geometric solution, such as the average delay of tasks, the average number of busy state VMs and the average number of idle state VMs. Eventually, numerical experiments demonstrate that the strategy can decrease the energy consumption of the system effectively.

Performance analysis and evaluation of ternary optical computer based on asynchronous multiple vacations

Ternary optical computer (TOC) has become a research hotspot in the field because of the advantages such as inherent parallelism, numerous trits, low power consumption, extendibility, bitwise allocability and dynamical bitwise reconfigurability. Meanwhile, its performance evaluation attracts more and more attentions from potential users and researchers. To model its computing ecology more accurately, this paper first builds a three-staged TOC service model by introducing asynchronous multi-vacations and tandem queueing, and then proposes a task scheduling algorithm and an optical processor allocation algorithm with asynchronous vacations of some small optical processors after dividing equally the entire optical processor into several small optical processors which can be used independently. At the same time, the analytical model was established to obtain important performance indicators such as response time, the number of tasks and utilization of optical processor, based on M/M/1 and M/M/n queuing system with asynchronous multi-vacations. In addition, relevant numerical simulation experiments are conducted. The results illustrate that the number of small optical processors, vacation rate and the number of small optical processors allowed to be on vacation have important effects on the system performance. Compared with synchronous vacation, asynchronous vacation not only ensures the system to obtain better maintenance but also improves the system performance to some degree.

Task Scheduling Strategy Based on Queueing Model in Cloud Computing and Its Energy Consumption Optimization Analysis

In order to reduce the energy consumption of the cloud computing system while ensuring the response performance of the cloud users, a task scheduling strategy is proposed. In the strategy, the local processor of the mobile device continues to work, the virtual machines in a physical machine sleep synchronously, and the virtual machines in different physical machine sleep asynchronously. For the heterogeneous physical machines in cloud computing, a queueing model with a synchronous multiple vacation is established. By using the quasi-birth-death process and the matrix geometric solution, the steady-state distribution of the queueing model is given, and the expressions of the average response time of the tasks and the average power of the system are derived. The numerical results show that there is a power-performance trade-off for cloud computing when setting the assigning probability of the tasks to the local processor. Through the improved whale optimization algorithm, the task scheduling strategy is optimized with the minimum system cost.

Energy saving strategy and Nash equilibrium of hybrid P2P networks

This paper proposes a penalty strategy with differentiated service rate based on the free riding phenomenon in P2P networks, and establishes an M/M/c+d queueing model. Based on this model, a sleep/wakeup mechanism is introduced for the peers at the service end, and a single asynchronous vacation strategy is adopted to reduce the energy consumption of the system. In addition, the energy consumption of peers in each state is quantified, and the relationship between the energy consumption and parameters of the system is analyzed. In order to avoid excessive requests for unnecessary services from requesting nodes and increasing energy consumption of the system, this paper analyzes the Nash equilibrium between the arrival rate and the net profit of a single node, and then studies the optimization of social profit. The stationary distribution of queueing model is obtained by the method of matrix geometric solution, the performance indicators of the system are constructed, and the system performance is analyzed by numerical experiments. Experimental results show that the model developed in this paper has a significant penalty effect on free riding behavior, and that the single asynchronous vacation strategy not only saves more than 10% of the total energy consumption compared with the single synchronous vacation strategy, but also makes the hybrid P2P networks more flexible and efficient.

Analysis and Performance Evaluation of Markovian Feedback Multi-Server Queueing Model with Vacation and Impatience

This paper deals with a finite capacity multi-server Markovian queueing model with Bernoulli feedback, synchronous multiple vacation policy and customers’ impatience (balking and reneging). By employing certain customer retention mechanism, impatient customers can be retained in the system. Applications of the suggested queueing model can be found in a wide variety of practical systems including modern information and communication technology (ICT) networks, call centers, and manufacturing systems. Using the recursive method, the steady state probabilities of the model are obtained. Various performance measures are derived. Then, some important particular cases are provided. Finally, different numerical examples are presented to demonstrate how the different parameters of the model influence the behavior of the stationary characteristics of the system.

Strategic joining in an M/M/K queue with asynchronous and synchronous multiple vacations

We study customers’ equilibrium joining strategies in an M/M/K queue with asynchronous and synchronous multiple vacations. Arriving customers face four different information levels, that is, fully observable, almost observable, almost unobservable and fully unobservable cases, and they decide whether to join or balk the system based on their service utility. In this study, we analyse customers’ equilibrium strategies in terms of the social welfare. It is found that the equilibrium social welfare under an asynchronous vacation policy is higher than that under a synchronous vacation policy when the traffic density is low, the opposite relation exists when the traffic density is high. Furthermore, when the traffic density is high enough, the difference between the two vacation policies in terms of social welfare can be negligible. Finally, if all customers follow the equilibrium strategies, compared to the situation with no information provided for customers, the social planner would prefer to reveal the queue length. Such a finding has the managerial implication for waiting line managers who want to maximise the social welfare of customers.

An Energy Efficient Task Scheduling Strategy in a Cloud Computing System and its Performance Evaluation using a Two-Dimensional Continuous Time Markov Chain Model

With ongoing energy shortages and rises in greenhouse emissions worldwide, increasing academic attention is being turned towards ways to improve the efficiency and sustainability of cloud computing. In this paper, we present a performance analysis and a system optimization of a cloud computing system with an energy efficient task scheduling strategy directed towards satisfying the service level agreement of cloud users while at the same time improving the energy efficiency in cloud computing system. In this paper, we propose a novel energy-aware task scheduling strategy based on a sleep-delay timer and a waking-up threshold. To capture the stochastic behavior of tasks with the proposed strategy, we establish a synchronous vacation queueing system combining vacation-delay and N-policy. Taking into account the total number of tasks and the state of the physical machine (PM), we construct a two-dimensional continuous-time Markov chain (CTMC), and produce an infinitesimal generator. Moreover, by using the geometric-matrix solution method, we analyze the queueing model in the steady state, and then, we derive the system performance measures in terms of the average sojourn time and the energy conservation level. Furthermore, we conduct system experiments to investigate the proposed strategy and validate the system model according to performance measures. Statistical results show that there is a compromise between the different performance measures when setting strategy parameters. By combining different performance measures, we develop a cost function for the system optimization. Finally, by dynamically adjusting the crossover probability and the mutation probability, and initializing the individuals with chaotic equations, we present an improved genetic algorithm to jointly optimize the sleep parameter, the sleep-delay parameter and the waking-up threshold.

A Virtual Machine Scheduling Strategy with a Speed Switch and a Multi-Sleep Mode in Cloud Data Centers

With the rapid growth of energy costs and the constant promotion of environmental standards, energy consumption has become a significant expenditure for the operating and maintaining of a cloud data center. To improve the energy efficiency of cloud data centers, in this paper, we propose a Virtual Machine (VM) scheduling strategy with a speed switch and a multi-sleep mode. In accordance with the current traffic loads, a proportion of VMs operate at a low speed or a high speed, while the remaining VMs either sleep or operate at a high speed. Commensurate with our proposal, we develop a continuous-time queueing model with an adaptive service rate and a partial synchronous vacation. We construct a two dimensional Markov chain based on the total number of requests in the system and the state of all the VMs. Using a matrix geometric solution, we mathematically estimate the energy saving level and the response performance of the system. Numerical experiments with analysis and simulation show that our proposed VM scheduling strategy can effectively reduce the energy consumption without significant degradation in response performance. Additionally, we establish a system utility function to trade off the different performance measures. In order to determine the optimal sleep parameter and the maximum system utility function, we develop an improved Firefly intelligent searching Algorithm.

Cost optimization analysis for an $$M^{X}/M/c$$ M X / M / c vacation queueing system with waiting servers and impatient customers

This paper deals with the study of an $$M^{X}/M/c$$ Bernoulli feedback queueing system with waiting servers and two different policies of synchronous vacations (single and multiple vacation policies). During vacation period, the customers may leave the system (reneging), and using certain customer retention mechanism, the reneged customers may be retained in the system. The probability generating function (PGF) has been used to obtain the steady state probabilities of the model. Various performances measures of the system are derived. Then, a cost model is developed. Further, a cost optimization problem is considered using quadratic fit search method. Finally, a variety of numerical illustrations are discussed for the applicability of the model.

Performance Analysis of Cloud Systems with Load Dependent Virtual Machine Activation and Sleep Modes

Due to the advancements in virtualization technology, it is now an up and coming field and has become a more appealing area of internet technology. Since there is a rapid growth for the demand of computational power increases by scientific, business, and web-applications, it leads to the creation of large-scale data centers. These data centers consume enormous amounts of electrical power. In this article, the authors study energy saving methods by consolidation and by switching off those virtual machines which are not in use. According to this policy, c virtual machines continue serving the customer until the number of idle server attains the threshold level d; then d idle servers take synchronous vacation simultaneously, otherwise these servers would begin serving the customers. Numerical results are provided to demonstrate the applicability of the proposed model for data center management in particular, to quantify the tradeoff theoretically between the conflicting aims of energy efficiency and QoS.

Computational analysis of multi-server discrete-time queueing system with balking, reneging and synchronous vacations

This paper proposes a discrete-time multi-server queue with multiple synchronous vacations under balking and reneging. Arriving customers decide whether to join the system or balk on the basis of some state-dependent joining/balking probabilities, and renege according to a geometric distribution when servers are busy. The servers take a vacation together if there are no customers in the system at a service completion instant. When the servers are on vacation, an arriving customer activates an impatience timer which is geometrically distributed. The inter-arrival times, service times and vacation times are assumed to be independent and geometrically distributed. We obtain closed-form expressions and develop a computational algorithm for calculating the steady-state probabilities. Specifically, we establish the application of the proposed framework in analyzing a multi-server queueing system with synchronous vacation under balking and reneging. Applications of such models can be found in a wide variety of real-time systems including call centers, computer and communication systems, cloud computing, quality control and maintenance in industrial establishments. We develop a cost model to determine the optimal service rate. Various performance measures and numerical examples are sketched out to demonstrate the impact of the proposed method. Some special cases of the model have also been discussed. Finally, we show that in the limiting case the results converge to the corresponding continuous-time counterparts.

Algorithmic computation of steady-state probabilities in an almost observable GI/M/c queue with or without vacations under state dependent balking and reneging

We consider an almost observable GI/M/c/N queue with customer impatience with or without multiple synchronous vacations under state-dependent balking. Upon arriving, a customer joins or refuses to join the system based on certain state dependent joining/balking probabilities. Once an arriving customer joins the system and finds all the servers busy, it initiates an exponentially distributed impatience timer with random duration T. An equilibrium balking strategy is discussed for constant balking as a special case of state-dependent balking. In the constant balking case, the waiting time of a customer in a queueing system has been associated with a linear cost-reward structure for estimating the net benefit if a customer chooses to participate in the system. A steady-state system of equations is obtained using the supplementary variable technique. After that, a recursive algorithm is proposed to obtain the stationary system-length distribution at pre-arrival and arbitrary epochs using those steady-state equations, from which the mean system sojourn time, the average reneging rate and the average balking rate are derived. Finally, we produce numerical results relating to system performance and its net benefit when investigated for different model parameters. The proposed model has applications in the modeling of balking and impatient behavior of incoming calls in a call center, multi-core computing, multi-path routing in delay sensitive communications networks.

Development of computational algorithm for multiserver queue with renewal input and synchronous vacation

In this paper, we develop a new computational algorithm for calculating the Markov chain stationary distribution based on a Taylor series approach, where the Taylor series coefficients are expressed in closed-form in terms of the fundamental matrix of the underlying Markov chain. Additionally, we provide an approximate expression for the remainder term of the Taylor series that can be computed in an efficient manner. Specifically, we demonstrate the application of the proposed framework in analyzing a multi-server queueing system with synchronous vacation. The only required assumption of the proposed framework is that the entries of the transition matrix are differentiable functions with respect to a control parameter. Numerical examples are sketched out to illustrate the accuracy of the proposed method.

Transient Analysis of ak-out-of-n:GSystem withN-Policy, Repairmen’s Multiple Vacations, and Redundant Dependency

This paper analyzes ak-out-of-n:Grepairable system withN-policy, repairmen’s multiple synchronous vacations, and redundant dependency. When there is no failed component in the system, the repairmen leave for a vacation, the duration of which follows a phase type distribution. Upon returning from vacation, they should take another vacation if there are less thanNfailed components waiting in the system. This pattern continues until at leastNfailed components are waiting. Moreover, the redundant dependency which is a special kind of failure dependency is taken into account in the multicomponent system. Under such assumptions, the availability, the rate of occurrence of failures, and the reliability of the system are derived in transient regime by applying the quasi-birth-and-death process. Furthermore, the Runge-Kutta method is carried out to numerically discuss the time-dependent behavior of the system reliability measures. Finally, a special case of the system is presented to show the validity of our model.

Analysis of an M/M/cQueueing System with Impatient Customers and Synchronous Vacations

We consider an M/M/cqueueing system with impatient customers and a synchronous vacation policy, where customer impatience is due to the servers’ vacation. Whenever a system becomes empty, all the servers take a vacation. If the system is still empty, when the vacation ends, all the servers take another vacation; otherwise, they return to serve the queue. We develop the balance equations for the steady-state probabilities and solve the equations by using the probability generating function method. We obtain explicit expressions of some important performance measures by means of the two indexes. Based on these, we obtain some results about limiting behavior for some performance measures. We derive closed-form expressions of some important performance measures for two special cases. Finally, some numerical results are also presented.

Multi-server machine repair problems under a (V, R) synchronous single vacation policy

This paper considers a machine repair problem with M operating machines and S standbys, in which R repairmen are responsible for supervising these machines and operate a (V,R) vacation policy. With such policy, if the number of the failed machines is reduced to R−V (R>V) (there exists V idle repairmen) at a service completion, these V idle servers will together take a synchronous vacation (or leave for other secondary job). Upon returning from the vacation, they do not take a vacation again and remain idle until the first arriving failed machine arrives. The steady-state probabilities are solved in terms of matrix forms and the system performance measures are obtained. Algorithmic procedures are provided to deal with the optimization problem of discrete/continuous decision variables while maintaining a minimum specified level of system availability.

Analysis of the Machine Repair Models With Multi-Threshold Synchronous Vacations

Abstract In this paper, we present an analysis of a machine repair problem with M operating machines and S standbys, in which c repairmen are responsible for supervising these machines and follow a multi-threshold, synchronous vacation policy. With such a policy, at a repair completion instant, if the number of the failed machines in the system is less than a preset threshold value, partial idle servers together take a single vacation (or leave for a random amount of time doing other secondary job). At the end of a vacation, they return to repair the failed machines (or waits for the arrival). The steady-state probabilities are solved by using the matrix-analytic method, and formulae of system performance measures are thereby obtained. A cost model is then developed to formulate an optimization problem to find the minimum cost. The direct search and quasi-Newton methods are implemented to determine the optimal number of servers, the optimal threshold policy and the optimal repair rate.

A Note on a Multi-server Queue with Vacations of Multiple Groups of Servers

We study a Markovian queueing system with unreliable servers and two possible groups of servers taking two different types of vacations. Once the number of idle of servers reaches one of the two critical numbers in the system, a group of servers will take a single synchronous vacation (or leave for doing a secondary job). After taking the vacation, the servers return to attend the queue (or stay idle for the next arriving customer). Therefore, there can be a maximum of two groups of servers taking two different types of vacations at a time (doing some secondary jobs). The servers are subject to failure when attending the queue. The stability condition for the system and the stationary performance measures are developed by using the matrix-analytical method. A cost model is constructed to determine the optimal vacation policy and the optimal service rate of each server. Some numerical results are presented to illustrate the optimization procedures.