Multiple Adaptive Vacations Research Articles

A study on <i>M</i><SUP align="right"><i>X</i></SUP>/<i>G</i>(<i>a</i>, <i>b</i>)/<i>1</i> queue with server breakdown without interruption and controllable arrivals during multiple adaptive vacations

A single server model, after completion of a bulk service, if there is no breakdown with probability (1 − ψ) and queue length (queue) ≥ 'a', then the bulk service continues, otherwise, the server performs closedown work is analysed. At the end of bulk service, if there is a breakdown occurs with probability (ψ), then the server performs renovation process. After that, if the queue is ≥ 'a', then he performs bulk service otherwise the server perform closedown work follows a vacation. After that, if the queue is less than 'a', then he takes at most 'M' vacations successively. After 'M' vacations, if the queue is still less than 'a', then he remains in the system. However, the customers enter the service station with probability 'p' (0 ≤ p ≤ 1) during multiple adaptive vacations. The probability generating function (PGF) of queue size and performance measures are obtained and cost model is developed.

A study on <i>M</i><SUP align="right"><i>X</i></SUP>/<i>G</i>(<i>a</i>, <i>b</i>)/<i>1</i> queue with server breakdown without interruption and controllable arrivals during multiple adaptive vacations

A single server model, after completion of a bulk service, if there is no breakdown with probability (1 − ψ) and queue length (queue) ≥ 'a', then the bulk service continues, otherwise, the server performs closedown work is analysed. At the end of bulk service, if there is a breakdown occurs with probability (ψ), then the server performs renovation process. After that, if the queue is ≥ 'a', then he performs bulk service otherwise the server perform closedown work follows a vacation. After that, if the queue is less than 'a', then he takes at most 'M' vacations successively. After 'M' vacations, if the queue is still less than 'a', then he remains in the system. However, the customers enter the service station with probability 'p' (0 ≤ p ≤ 1) during multiple adaptive vacations. The probability generating function (PGF) of queue size and performance measures are obtained and cost model is developed.

Performance Analysis and Cost Optimization of Non-Markovian Bulk Queue with 'p'- Entering Discipline during Multiple Adaptive Vacations

The steady state analysis of M^x/G/1 queue with Multiple Adaptive Vacation (MAV), where the customers enter during the server vacations with probability 'p' (0 ≤ p ≤ 1) is considered. The server provides service to all the entering customers with service follows general distribution. Upon the system being found to be empty, the server immediately takes a vacation of random period. When the server returns to system after a vacation, if the queue length is still empty, he avails another vacation and so on until he completes M number of vacations in successions and the server remains idle and starts service when the server finds a customer in the queue. The probability generating function at an arbitrary time epoch is obtained using Supplementary variable technique. Basic system characteristics such as expected value of the length of the queue, busy period, idle time and expected value of the waiting time are obtained. Cost model is presented and algorithm for finding optimum cost is also presented for the proposed model. Numerical illustration is also provided.

A Novel Model of Stock Data Mining with M/G/1 Queue for Evaluation of Stock Crash

Data mining is the process of searching the information from a large amount of data. In order to evaluate the stock crash this paper proposes general decrementing service M/G/1 queue system with multiple adaptive vacations to find information related to stock crash in data about Shanghai Composite Index. We use the probability generating function (P.G.F.) of stationary queue length and LST of waiting time, and their stochastic decomposition to calculate Existing money flow. Existing Money flow calculation model is improved based on the stationary queue length and LST of waiting time. We program to achieve the stock of existing money flow algorithm, and get the number of existing money flow. The improved algorithm can early warn the stock market crash. The empirical result shows that: There will be a rise in price before the Stock Market Crash, and the stock of existing money inflow begin to decrease. The stock market crash fell for at least six months. The stock market crash fell by at least fifty-five percent. Most of the stock market crash fell by over seventy-percent. The stock market crash down time is inversely proportional to the magnitude of the decline. If the down time is short, the magnitude of the decline is large. If the down time is long, the magnitude of the decline is small. The stock market crash is great harm to investors.

Reliability Analysis of M/G/1 Repairable Queueing System with Multiple Adaptive Vacations and p-Entering Discipline

This paper considers the reliability of an M/G/1 queue with multiple adaptive vacations in which the arriving customers enter the system with probability p (0＜p≤1) during vacations. Through appropriate assumptions, the model is studied by the total probability decomposition technique and the tool of Laplace transform. Some reliability indices are studied. Moreover, we give some numerical examples to observe the effect of various parameters on the reliability indices.

Reliability Analysis of an Extended Shock Model and Its Optimization Application in a Production Line

Firstly, a two-unit cold standby shock model with multiple adaptive vacations is introduced, in which the startup and replacement of repair facility are also considered. Secondly, using supplementary variable method and Laplace transform, some important reliability indices are derived, such as availability, failure frequency, mean vacation period, mean renewal cycle, mean startup period, and replacement frequency. Finally, a production line controlled by two cold-standby computers is modeled to present numerical illustration and its optimal part-time job policy at a maximum profit.

Stationary distributions and optimal control of queues with batch Markovian arrival process under multiple adaptive vacations

We consider an infinite-buffer single server queue with batch Markovian arrival process (BMAP) and exhaustive service discipline under multiple adaptive vacation policy. That is, the server serves until system emptied and after that server takes a random maximum number H different vacations until either he finds at least one customer in queue or the server have exhaustively taken all the vacations. The maximum number H of vacations taken by the server is a discrete random variable. We obtain queue-length distributions at various epochs such as, service completion/vacation termination, pre-arrival, arbitrary, post-departure and pre-service. The proposed analysis is based on the use of matrix-analytic procedure to obtain queue-length distribution at a post-departure epoch. Later we use supplementary variable method and simple algebraic manipulations to obtain the queue-length distribution at other epochs using queue-length distribution at post-departure epoch. Some important performance measures, like mean queue lengths and mean waiting times have been obtained. Several other vacation queueing models can be obtained as a special case of our model, e.g., single-, multiple-vacation model and queues with exceptional first vacation time. Finally, the total expected cost function per unit time is considered to determine a locally optimal multiple adaptive vacation policy at a minimum cost.

General decrementing service M/G/1 queue with multiple adaptive vacations

In this paper, we introduce the multiple adaptive vacation policy and the general decrementing service rule based on the classical M/G/1 queueing systems, and obtain the P.G.F. (Probability Generating Function) of stationary queue length by using the embedded Markov chain method and regeneration cycle approach. Then, the LST (Laplace Stieltjes Transform) of stationary waiting time is also derived according to the independence between the waiting time and arrival process. At last some special cases are given to show the general properties of the new model, and some numerical results are shown to compare the mean queue length and waiting time of special cases.

Pure Decrementing Service M/G/1 Queue with Multiple Adaptive Vacations

In this paper, an M/G/1 queue model with the multiple adaptive vacations and pure decrementing service policy is studied based on the classical M/G/1 queueing model. The PGF of stationary queue length is derived by using an embedded Markov chain method. The LST of stationary waiting time is also given according to the independence between waiting time and arrival process. The average probabilities of various system states are obtained. Some special cases are given to show the general properties of the new model.

The discrete-time Geom/G/1 queue with multiple adaptive vacations and server Setup/Closedown times

In this paper, we discuss the discrete-time Geom/G/1 queue with multiple adaptive vacations and server setup/closedown times. We consider two types of closedown time policies which differ from each other in the completion instant of the closedown times and contrast the effects of them on the queueing system. We derive the probability generating functions (PGFs) and stochastic decomposition results of the steady-state queue length and waiting time distributions through the method of embedded Markov chain in both cases. Many discussed models related to Geom/G/1 queue are special cases of our model.

Steady state analysis on the batch arrival Geo/G/1 queue with multiple adaptive vacations

This paper concentrates on the steady state analysis on the discrete-time batch arrival Geo/G/1 queue with vacations. In this model, after serving all customers in the system, the server will take a random maximum number of vacations before returning to the service mode. The distributions of busy period, vacation period and vacation cycle have been given. Meanwhile, the PGFs for the off-line period and on-line period are expressed explicitly. The PGFs are also derived for the steady-state queue length, the unfinished work as well as the waiting time of an arbitrary customer in the FCFS and LCFS systems. According to the relationship of the discrete-time and continuous-time systems, we also have obtained the mean queue length, the mean unfinished work and the mean waiting time in continuous-time system. Several vacation models are special cases of the vacation model presented in this study. Moreover, we give some numerical examples to observe the effect of the vacation policy on the queue model.

Discrete Time Geo/G/1 Queue with Multiple Adaptive Vacations

This paper treats the discrete time Geometric/G/1 system with vacations. In this system, after serving all customers in the system, the server will take a random maximum number of vacations before returning to the service mode. The stochastic decomposition property of steady-state queue length and waiting time has been proven. The busy period, vacation mode period, and service mode period distributions are also derived. Several common vacation policies are special cases of the vacation policy presented in this study.