Bernoulli Schedule Research Articles

A Queueing-Inventory System with Modified Delayed Vacation under Bernoulli Schedule

A Queueing-Inventory System with Modified Delayed Vacation under Bernoulli Schedule

An Optimization of Home Delivery Services in a Stochastic Modeling with Self and Compulsory Vacation Interruption

This study presents and discusses the home delivery services in stochastic queuing-inventory modeling (SQIM). This system consists of two servers: one server manages the inventory sales processes, and the other server provides home delivery services at the doorstep of customers. Based on the Bernoulli schedule, a customer served by the first server may opt for a home delivery service. If any customer chooses the home delivery option, he hands over the purchased item for home delivery and leaves the system immediately. Otherwise, he carries the purchased item and leaves the system. When the delivery server returns to the system after the last home delivery service and finds that there are no items available for delivery, he goes on vacation. Such a vacation of a delivery server is to be interrupted compulsorily or voluntarily, according to the prefixed threshold level. The replenishment process is executed due to the (s,Q) reordering policy. The unique solution of the stationary probability vector to the finite generator matrix is found using recursive substitution and the normalizing condition. The necessary and sufficient system performance measures and the expected total cost of the system are computed. The optimal expected total cost is obtained numerically for all the parameters and shown graphically. The influence of parameters on the expected number of items that need to be delivered, the probability that the delivery server is busy, and the expected rate at which the delivery server’s self and compulsory vacation interruptions are also discussed.

Uncertainty Quantification and Global Sensitivity Analysis of an M/G/1 Retrial Queue with Bernoulli Schedule

Uncertainty Quantification and Global Sensitivity Analysis of an M/G/1 Retrial Queue with Bernoulli Schedule

Tail Asymptotics for a Retrial Queue with Bernoulli Schedule

In this paper, we study the asymptotic behaviour of the tail probability of the number of customers in the steady-state M/G/1 retrial queue with Bernoulli schedule, under the assumption that the service time distribution has a regularly varying tail. Detailed tail asymptotic properties are obtained for the conditional probability of the number of customers in the (priority) queue and orbit, respectively, in terms of the recently proposed exhaustive stochastic decomposition approach. Numerical examples are presented to show the impacts of system parameters on the tail asymptotic probabilities.

A MARKOVIAN TWO COMMODITY QUEUEING–INVENTORY SYSTEM WITH COMPLIMENT ITEM AND CLASSICAL RETRIAL FACILITY

This paper explores the two-commodity (TC) inventory system in which commodities are classified as major and complementary items. The system allows a customer who has purchased a free product to conduct Bernoulli trials at will. Under the Bernoulli schedule, any entering customer will quickly enter an orbit of infinite capability during the stock-out time of the major item. The arrival of a retrial customer in the system follows a classical retrial policy. These two products' re-ordering process occurs under the \((s, Q)\) and instantaneous ordering policies for the major and complimentary items, respectively. A comprehensive analysis of the retrial queue, including the system's stability and the steady-state distribution of the retrial queue with the stock levels of two commodities, is carried out. The various system operations are measured under the stability condition. Finally, numerical evidence has shown the benefits of the proposed model under different random situations.

HETEROGENEOUS TWO SERVER QUEUE WITH BREAKDOWN AND WITH VARIANT REPAIR POLICY

In this paper, we consider a system with two heterogeneous servers Markovian queue. In which the system breakdown occurs when the system is in busy mode. Immediately the system undergoes repair. After completion of the repair, the system either undergoes optional repair mode or becomes busy mode based on a Bernoulli schedule. It is assumed that the number of repairs follows the Poisson process and the repair periods follow an exponential distribution. The model has been solved in steady-state using the matrix analytic method. Some performance measures and numerical results are obtained.

Analysis of an unreliable N – policy queue with set-up period under Bernoulli schedule and re-service approach

This research article investigates the stationary N-policy with a set-up time of a single server bulk arrival queueing model rendering two-heterogeneous optional repeated service (THORS) with the factor of unreliability. The server studied here is allowed to take a single vacation after each busy period following Bernoulli Schedule (BS). An extensive analysis of the proposed model is carried out using the Supplementary Variable Technique (SVT) under some suitable transformations. The joint distribution of the server’s state and queue length are obtained under both elapsed and remaining times along with their double transforms. Probability generating function (PGF) of the queue size distribution together with the mean system size of the model are determined for any arbitrary time point and service completion epoch, besides the busy period distribution and various pivotal system characteristics. A suitable linear cost structure of the underlying model is developed, and with the help of a difference operator, a locally optimal N-policy at a lower cost is obtained. Finally, some numerical experiments are carried out in support of the theory.

Analysis of customer’s impatience in queues with Bernoulli schedule server working vacations and vacation interruption

Analysis of customer’s impatience in queues with Bernoulli schedule server working vacations and vacation interruption

Cost Optimization of Queueing System with Working Vacation, Setup, Feedback, Reneging, and Retention of Reneged Customers

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.

Analysis and optimisation of a M/M/1/WV queue with Bernoulli schedule vacation interruption and customer’s impatience

Abstract In this investigation, we establish a steady-state solution of an infinite-space single-server Markovian queueing system with working vacation (WV), Bernoulli schedule vacation interruption, and impatient customers. Once the system becomes empty, the server leaves the system and takes a vacation with probability p or a working vacation with probability 1 − p, where 0 ≤ p ≤ 1. The working vacation period is interrupted if the system is non empty at a service completion epoch and the server resumes its regular service period with probability 1 − q or carries on with the working vacation with probability q. During vacation and working vacation periods, the customers may be impatient and leave the system. We use a probability generating function technique to obtain the expected number of customers and other system characteristics. Stochastic decomposition of the queueing model is given. Then, a cost function is constructed by considering different cost elements of the system states, in order to determine the optimal values of the service rate during regular busy period, simultaneously, to minimize the total expected cost per unit time by using a quadratic fit search method (QFSM). Further, by taking illustration, numerical experiment is performed to validate the analytical results and to examine the impact of different parameters on the system characteristics.

Steady State Analysis of Impulse Customers and Cancellation Policy in Queueing-Inventory System

This article discusses the queueing-inventory model with a cancellation policy and two classes of customers. The two classes of customers are named ordinary and impulse customers. A customer who does not plan to buy the product when entering the system is called an impulse customer. Suppose the customer enters into the system to buy the product with a plan is called ordinary customer. The system consists of a pool of finite waiting areas of size N and maximum S items in the inventory. The ordinary customer can move to the pooled place if they find that the inventory is empty under the Bernoulli schedule. In such a situation, impulse customers are not allowed to enter into the pooled place. Additionally, the pooled customers buy the product whenever they find positive inventory. If the inventory level falls to s, the replenishment of Q items is to be replaced immediately under the (s, Q) ordering principle. Both arrival streams occur according to the independent Markovian arrival process (MAP), and lead time follows an exponential distribution. In addition, the system allows the cancellation of the purchased item only when there exist fewer than S items in the inventory. Here, the time between two successive cancellations of the purchased item is assumed to be exponentially distributed. The Gaver algorithm is used to obtain the stationary probability vector of the system in the steady-state. Further, the necessary numerical interpretations are investigated to enhance the proposed model.

Cost optimization of single server retrial queueing model with Bernoulli schedule working vacation, vacation interruption and balking

In present paper an M/M/1 retrial queueing model with working vacation interruption using Bernoulli schedule and Balking is analyzed under classical retrial policy. When sever is occupied in regular busy period customer enters in system with probability b and during working vacation of server, customer joins the orbit with probability v. Whenever no customers are in the system after vacation completion then either server returns to normal free state with probability q or goes for multiple working vacation with probability 1-q. Server provides the service at lower rate during working vacation than normal busy period. In this paper steady state probabilities have been obtained using probability generating function technique. Some important performance measures of this model are also evaluated numerically and some results are shown graphically using MATLAB software.

Impact of Bernoulli and non-Bernoulli working vacation schedule on single server Markovian queue with disaster

Impact of Bernoulli and non-Bernoulli working vacation schedule on single server Markovian queue with disaster

Cost optimization of single server retrial queueing model with Bernoulli schedule working vacation, vacation interruption and balking

Cost optimization of single server retrial queueing model with Bernoulli schedule working vacation, vacation interruption and balking

Bernoulli Vacation Scheduling (BVS) Model for Energy efficient intrusion detection in WSN

Cutting-edge wireless sensor networks, security is the most important challenge that has to be considered. Most of the applications of WSN are vulnerable to attacks, as they are deployed in any kind of environment. So, the intrusion detection system is the second line of defense for WSN. The intruder should be detected in WSN more efficiently. We introduce the Bernoulli scheduling rule such that several trials are performed to get the detection of an intruder to be successful once. Since the sensor nodes are always active for detecting an intruder, the energy consumption becomes high. To reduce the consumption of energy, sleep/wakeup, also called small interval vacations are used for the sensors which are almost considered to be like the mechanism of fluid queues. In our proposed work, the vacations are introduced so that the energy used is minimized whenever the sensor is not required to be active. Thus, in our proposed work, since WSN is vulnerable to many harmful attacks, the Bernoulli scheduling rule is applied and the sensor nodes have a vacation state whenever the sensor need not be active, to save the energy consumed. The probability of detecting an intruder is calculated and performance is compared. We have applied a fluid queuing model for the vacations of small intervals in the sensor nodes to minimize the energy consumption and the obtained result is compared with the energy consumed when the vacations are not applied. Thus, the results are simulated for both energy consumed and the number of success rates of detection of an intruder, most importantly the comparison is analyzed with the past work of wsn.

An M/G/1 retrial queue with single working vacation under Bernoulli schedule

In this paper, an M/G/1 retrial queue with general retrial times and single working vacation is considered. We assume that the customers who find the server busy are queued in the orbit in accordance with a first-come-first-served (FCFS) discipline and only the customer at the head of the queue is allowed access to the server. During the normal period, if the orbit queue is not empty at a service completion instant, the server begins a working vacation with specified probabilityq(0 ≤ q ≤ 1), and with probability 1 − q, he waits for serving the next customer. During the working vacation period, customers can be served at a lower service rate. We first present the necessary and sufficient condition for the system to be stable. Using the supplementary variable method, we deal with the generating functions of the server state and the number of customers in the orbit. Various interesting performance measures are also derived. Finally, some numerical examples and cost optimization analysis are presented.

Analysis of MAP/PH(1), PH(2)/2 Queue with Bernoulli Schedule Vacation, Bernoulli Feedback and Renege of Customers

A classical queueing model with two servers in which the inter arrival times follow Markovian arrival process, the service times are phase type distributed and the remaining random variables are exponentially distributed is studied in this paper. The resulting QBD process is investigated in the steady state by employing matrix-analytic method. We have also done the busy period analysis of our model and discussed about the waiting time distribution for our system. Some of the performance measures are provided. Finally, a few numerical and graphical examples are given.

Performance analysis of the service periods of IEEE 802.11ad MAC

AbstractToday, we witness an explosion in wireless networks that are accessed by high‐performance mobile devices running a wide variety of demanding applications. The IEEE 802.11ad is a standard targeted at meeting these design requirements. Analytical modeling and performance evaluation is a crucial step needed for the evolution of any standard. In this paper, we model the service periods (SPs) of the IEEE 802.11ad hybrid medium access control as a special case of periodically gated queueing system (PGQS) with Bernoulli scheduling. We first state and prove some basic theorems related to the probability generating functions of the PGQS. Using these results, we derive the average delay and the delay variations (jitter) of packets belonging to SP of IEEE 802.11ad. We show that as the arrival rates increases, the delay increases linearly until the critical arrival rate, and beyond that, it becomes unbounded. Further, a practical scheme for the admission control mechanism of 802.11ad is also presented. To validate the model, analytical results are compared with simulation results.

Analysis of a bulk queue with unreliable server, immediate feedback, N-policy, Bernoulli schedule multiple vacation and stand-by server

Abstract We have studied the behaviour of a non-Markovian bulk service queueing model with unreliable server, stand-by server, loss and feedback, N-policy and Bernoulli schedule multiple vacation. The stand-by server is employed only during the main server’s repair period. At each service completion the main server may have an option to go for a short vacation. After each short vacation completion or service completion, if the system size is less than ‘a’, the server takes a sequence of vacation until the queue size reaches at least ‘N’. The concept of loss and feedback is also incorporated in this model. The PGF of queue size and some important performance measures are derived. An extensive numerical result is illustrated.

Analysis of an M^([X])/G(a,b)/1 Unreliable G-queue with Loss, Instantaneous Bernoulli Feedback, Vacation and Two Delays of Verification

This paper deals with a batch arrival that customers arrive to the system according to a compound Poisson process. The customer’s behavior is incorporated according to which loss with a certain probability and the server begins to provide a service only when a queue size minimum say ‘a’ and maximum service capacity is ‘b’. Once the server completes the service, the unsatisfied customers may get the same service under Bernoulli schedule is termed as instantaneous Bernoulli feedback. The occurrence of negative customer cause the server to fail and removes a group of customers or an amount of work if present upon its arrival. As soon as the failure instant, the service channel send to the two delays of verification, the first verification delay starts before the repair process and the second verification delay begins after the repair process. We use the generating function method to derive the stationary queue size distribution. Some important performance measures such as different states of the system and the expected length of the queue explicitly. Some important special cases and numerical examples are determined.