Supplementary Variable Method Research Articles

Analysis of M<SUP align="right">[X]</SUP>/G<SUB align="right">k/1 retrial queue with Bernoulli-2 vacation, search for customers and hot stand-by

A batch arrival retrial queueing model with k optional stages of service, extended Bernoulli vacation and stand-by is studied. After completion of the ith stage of service, the customer may have the option to choose (i + 1)th stage of service with probability θi or may leave the system with probability qi = 1 − θi, I = 1, 2, …, k − 1 and qk = 1. After service completion of each customer, the server may take a vacation with probability v1 and extend a vacation with probability v2 or rejoin the system after the first vacation with probability 1 − v2. Busy server may get to breakdown and the stand-by server provides service only during the repair times. At the completion of service, vacation or repair, the server searches for the customer in the orbit (if any) with probability α or remains idle with probability 1 − α. By using the supplementary variable method, steady state probability generating function for system size, some system performance measures like Lq, Ls, Wq and Ws are discussed. Simulation results are given using MATLAB.

Stochastic analysis of a preemptive retrial queue with orbital search and multiple vacations

This paper deals with a preemptive priority M/G/1 retrial queue with orbital search and exhaustive multiple vacations. By using embedded Markov chain technique and the supplementary variable method, we discuss the necessary and sufficient condition for the system to be stable and the joint queue length distribution in steady state as well as some important performance measures and the Laplace–Stieltjes transform of the busy period. Also, we establish a special case and the stochastic decomposition laws for this preemptive retrial queueing system. Finally, some numerical examples and cost optimization analysis are presented.

An unreliable discrete-time retrial queue with probabilistic preemptive priority, balking customers and replacements of repair times

This paper deals with a discrete-time $Geo/G/1$ retrial queueing system with probabilistic preemptive priority and balking customers, in which the server is subject to starting failures and replacements in the repair times may occur with some probability. If the server is found busy at an arrival epoch, the newly arriving customer either interrupts the customer in service to begin its own service with probability $p$ or enters the orbit with probability $1-p$. When an arriving customer (external or repeated) finds the server free, he must turn on the server. If the server is activated successfully, the customer receives service immediately. Otherwise, the server undergoes a repair process. If an external arrival finds that the server is under repair, he decides either to join the orbit with probability $q$ or leaves the system completely (balking) with probability $1-q$. Applying the supplementary variable method and the generating function technique, we analyze the Markov chain underlying the considered queueing model and derive the stationary distributions under different system states, the generating functions for the number of customers in the orbit and in the system, as well as some crucial performance measures in steady state. Especially, some corresponding results under special cases are directly obtained by setting appropriate parameter values. Further, some numerical examples are provided to examine the effect of various system parameters on queueing characteristics. Finally, an operating cost function is formulated to discuss numerically a cost optimization problem.

Queuing system in E- Commerce

This paper explores an investigation on Queuing framework in E-trade. Online business (electronic trade) is the acquiring and selling of product and adventures, or the transmitting of benefits or data, over an electronic system, basically the web. It is the trading of things or administrations using PC systems like Internet or online casual networks. Here the Business is led utilizing PCs, phones, fax machines, standardized tag per users, Visas, ATM or other electronic apparatuses without the trading of paper-based records or physically moving to a strip mall. It incorporates exercises, for instance, acquirement, request passage, trade handling, online installment, verification, stock control, request satisfaction, shipment, and client support. In the Queuing arrangement of E trade, the clients touches base in Batch with Reneging happens at the set-up time organize and Optional second stage where breakdown happens just as fix procedure will be done. Moreover, we accept that the clients may renege amid the Initial stage because of eagerness (Slow server) or Non-Satisfaction in acquiring the items from the specific site. At the season of Repair process, Strictly Non-acceptability of the clients is executed. The model is well analyzed and solved by supplementary variable method .The system performance measures are derived. Numerical analysis and graphical representation of the model clearly justifies the model to a fullest satisfaction.

Queuing System Modeling for Supermarkets

Supermarkets may not have the IT needs that service-oriented businesses do, but that does not mean they cannot use computers in their day-to-day operations. Many supermarkets are adopting complex computer systems that allow them to regulate many of their practices, providing business leaders with more information to make decisions with and making some processes automatic, saving supermarkets both time and money. For the complete satisfaction of the customers, in system modeling, various queuing system is followed by the super markets. In all those queuing system, various queuing issue have to be dealt with. Among those, on of the Queuing problem is inspected and it is considered as a Non Markovian Queuing issues. For the above portrayed Queuing problem, by the usage of supplementary variable method, performance measures like number of customers in the system, average length of the queue, waiting time of the customers in the system as well as in the queue are derived. In addition, idle time of the server and the time spent by the server for each service are determined. The model is well justified by means of a Numerical portrayal and a grow graphical examination is done toward the end to favour the model

Performance and sensitivity analysis of an M/G/1 queue with retrial customers due to server vacation

This paper treats an M/G/1 queue with retrial customers due to server vacation, which can be used to model a hospital service system. When an arriving customer finds the server on vacation at his arrival epoch, he either enters the retrial group with probability p or leaves the system with probability 1-p. Otherwise, the new arriving customer begins his service immediately if the server is idle upon his arrival or joins the original queue with infinite capacity under FCFS if the sever is busy at his arrival epoch. First, we develop the steady-state analysis by the supplementary variable method and the generating function approach, and give some important performance measures. Moreover, we obtain the Laplace-Stieltjes transform of the sojourn time in the system of an arbitrary customer and prove that the Little’s law still holds in our model. Finally, sensitivity analysis and optimal joining policy are given for illustrative purpose.

The MAP/G/1 G‐Queue with Unreliable Server and Multiple Vacations

In this paper, we consider a MAP/G/1 G‐queues with unreliable server and multiple vacations . The arrival of a negative customer not only removes the customer being in service, but also makes the server under repair. The server leaves for a vacation as soon as the system empties and is allowed to take repeated (multiple) vacations. By using the supplementary variables method and the censoring technique, we obtain the queue length distributions. We derive the mean of the busy period based on the renewal theory. Furthermore, we analyze some main reliability indexes and investigate some important special cases.

A computational method for finding the availability of opportunistically maintained multi-state systems with non-exponential distributions

Availability is one of the most important performance measures of a repairable system. Among various mathematical methods, the method of supplementary variables is an effective way of modeling the steady-state availability of systems governed by non-exponential distributions. However, when all the underlying probability distributions are non-exponential (e.g., Weibull), the corresponding state equations are difficult to solve. To overcome this challenge, a new method is proposed in this article to determine the steady-state availability of a multi-state repairable system, where all the state sojourn times, as well as the maintenance times, are generally distributed. As an indispensable step, the well-posedness and stability of the system’s state equations are illustrated and proved using C0 operator semigroup theory. Afterwards, based on the generalized Integral Mean Value Theorem, the expression for system steady-state availability is derived as a function of state probabilities. Then, the original problem is transformed into a system of linear equations that can be easily solved. A simulation study and an instance studied in the literature are used to demonstrate the applications of the proposed method in practice. These numerical examples illustrate that the proposed method provides a new computational tool for effectively evaluating the availability of a repairable system without relying on simulation.

Analysis of a Single Server Retrial Queue with Server Vacation and Two Waiting Buffers Based on ATM Networks

This paper presents a discrete-time Geo/G/1 retrial queue with two waiting buffers to model an ATM network, in which the server begins a single vacation in cases where the system is empty at the instant of a service completion. New arriving customer who finds the server being on vacation can decide to either enter the retrial buffer with some probability p or leave the system with complementary probability 1−p. But the new arriving customer can begin its service immediately if he finds the server idle and join the original buffer if he finds the server busy. We first carry out an extensive analysis of the model by using the supplementary variable method and the generating function approach, and give some performance measures, such as server’s state probabilities and mean queue lengths in the original buffer, retrial buffer, and in the system. Secondly, we give the generating function of the sojourn time of a customer in the system and prove that Little’s law still holds in our model. Sensitivity analysis and cost optimization are finally given for illustrative purposes.

Analysis of Markov-modulated fluid polling systems with gated discipline

<p style='text-indent:20px;'>In this paper we present two different analytical descriptions of the fluid polling model with Markov modulated load and gated discipline. The fluid arrival to the stations is modulated by a common continuous-time Markov chain (the special case when the modulating Markov chains are independent is also included). The fluid is removed at the stations during the service period by a station dependent constant rate. <p style='text-indent:20px;'>The first analytical description is based on the relationships of steady-state fluid levels at embedded server arrival and departure epochs. We derive the steady-state vector Laplace transform of the fluid levels at the stations at arbitrary epoch and its moments. The second analytical description applies the method of supplementary variables and results in differential equations, from which the joint density function of the fluid levels can be obtained. <p style='text-indent:20px;'>We also propose computational methods for both analytical descriptions and provide numerical examples to illustrate the numeric computations.

Analysis of a discrete-time repairable queue with disasters and working breakdowns

In this paper, we analyse a discrete-time queue with a primary server of high service capacity and a substitute server of low service capacity. Disasters that only arrive during the busy periods of the primary server remove all customers from the system and make the primary server breakdown. When the primary server fails and is being repaired, the substitute server handles arriving customers. Applying the embedded Markov chain technique and the supplementary variable method, we determine the distribution of the system length at departure epochs and the joint distribution of the queue length and server’s state at an arbitrary instant. Then we derive the sojourn time distribution. We also provide the probability generating function of the time between failures. Some numerical examples are delivered to give an insight into the impact of system parameters on performance measures and a cost function.

Reliability analysis for a cold standby system under stepwise Poisson shocks

ABSTRACT In this paper, a two-dissimilar-component cold standby repairable system maintained by a repairman is investigated. The system fails due to either intrinsic factors or external shocks. Assume that shocks arrive according to a stepwise Poisson process, the intensity of which varies with respect to the number of failures of the working component. Repair after failure is imperfect. Using the supplementary variable method and the vector Markov process theory, we obtain some important reliability index for the system. By the renewal reward theorem, an optimal replacement policy is derived by minimising the average cost rate of the system. Numerical examples are presented to validate the theoretical results.

Efficient computational analysis of non-exhaustive service vacation queues: BMAP/R/1/N(∞) under gated-limited discipline

• A finite-buffer vacation queue with batch Markovian arrival process has been analyzed. • The server is subjected to serve under gated-limited service discipline. • Proposed analysis is based on the successive substitution and the supplementary variable method. • The results have been matched with the corresponding infinite-buffer queue. • Numerical results are presented for different service- and vacation-time distributions. This paper analyzes the finite-buffer single server queue with vacation(s). It is assumed that the arrivals follow a batch Markovian arrival process ( BMAP ) and the server serves customers according to a non-exhaustive type gated-limited service discipline. It has been also considered that the service and vacation distributions possess rational Laplace-Stieltjes transformation (LST) as these types of distributions may approximate many other distributions appeared in queueing literature. Among several batch acceptance/rejection strategies, the partial batch acceptance strategy is discussed in this paper. The service limit L (1 ≤ L ≤ N ) is considered to be fixed, where N is the buffer-capacity excluding the one in service. It is assumed that in each busy period the server continues to serve until either L customers out of those that were waiting at the start of the busy period are served or the queue empties, whichever occurs first. The queue-length distribution at vacation termination/service completion epochs is determined by solving a set of linear simultaneous equations . The successive substitution method is used in the steady-state equations embedded at vacation termination/service completion epochs. The distribution of the queue-length at an arbitrary epoch has been obtained using the supplementary variable technique. The queue-length distributions at pre-arrival and post-departure epoch are also obtained. The results of the corresponding infinite-buffer queueing model have been analyzed briefly and matched with the previous model. Net profit function per unit of time is derived and an optimal service limit and buffer-capacity are obtained from a maximal expected profit. Some numerical results are presented in tabular and graphical forms.

Transient study of a queueing system with one unreliable server, batch arrivals, two types of verification, loss and vacation

This paper studies a transient regime for an unreliable queue with repairs, two delays, geometric loss and Bernoulli schedule server vacations. The server takes always a vacation when during the service, no customer arrives and the queue is empty. The customer whose service is interrupted stays in the service, waiting for delay, repair and second delay (verification), after that, he completes his service. Using a supplementary variable method, transient solutions, performance and reliability measures and stochastic decomposition of the system are determined. Numerical results are sketched out to illustrate the effect of the system parameters on the main performances.

Availability analysis of a general time distribution system with the consideration of maintenance and spares

A one component system with the mean time to failure following general time distributions is used as the subject and which is also affected by corrective maintenance (CM), preventive maintenance (PM), stock level, degrading and failed components repair turnaround. The opportunity of operating CM and PM is governed by the state of the system and also the stocks on hand. By defining the system's state space and the sojourn time of each state, the state equation is derived with the supplementary variable method. Since in the state equation all the time distributions involved are not exponential, the regular method is difficult to resolve. A new method is proposed to cope with this problem. Two important parameters are introduced into the equation and the method proposed in the proposition is perfectly general. By this method the original problem can be transformed in a form as a regular Markov problem. Then the solving process is given and a vector Cauchy sequence of iteration variables is constructed to obtain the final solution. In each calculation the solution of the system's steady-state is derived by the minimum norm theory. Finally, a number of numerical examples are illustrated and some interesting results are derived.

On a retrial queue with abandoned customers and multi-optional vacations

This paper treats an M/G/1 retrial queue with abandoned customers and multi-optional vacations. If the server is found busy upon a new arrival, the customer being served could be abandoned by the server with some probability. Once the system is empty after a service completion, the server takes the first essential vacation (FEV) immediately. At the end of the FEV, if the system is still empty, the server will take another optional vacation depending on the server's choice; whereas, if the server finds a nonempty system, it will wait for a new arrival or a repeated customer from the orbit. The sufficient and necessary condition for the system to be stable is provided by using the embedded Markov chain technique. Moreover, the steady-state analysis is carried out via the supplementary variable method. The Laplace-Stieltjes transform of the sojourn time distribution is also provided. Finally, numerical examples with cost optimization analysis are given.

Performance analysis of a discrete-time &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;$ Geo/G/1$&lt;/tex-math&gt;&lt;/inline-formula&gt; retrial queue with non-preemptive priority, working vacations and vacation interruption

This paper is concerned with a discrete-time $ Geo/G/1$ retrial queueing system with non-preemptive priority, working vacations and vacation interruption where the service times and retrial times are arbitrarily distributed. If an arriving customer finds the server free, his service commences immediately. Otherwise, he either joins the priority queue with probability $ α$, or leaves the service area and enters the retrial group (orbit) with probability $ \mathit{\bar{\alpha }}\left( = 1-\alpha \right)$. Customers in the priority queue have non-preemptive priority over those in the orbit. Whenever the system becomes empty, the server takes working vacation during which the server can serve customers at a lower service rate. If there are customers in the system at the epoch of a service completion, the server resumes the normal working level whether the working vacation ends or not (i.e., working vacation interruption occurs). Otherwise, the server proceeds with the vacation. Employing supplementary variable method and generating function technique, we analyze the underlying Markov chain of the considered queueing model, and obtain the stationary distribution of the Markov chain, the generating functions for the number of customers in the priority queue, in the orbit and in the system, as well as some crucial performance measures in steady state. Furthermore, the relation between our discrete-time queue and its continuous-time counterpart is investigated. Finally, some numerical examples are provided to explore the effect of various system parameters on the queueing characteristics.

Batch Renewal Arrival Process Subject to Geometric Catastrophes

We consider a stochastic model where a population grows in batches according to renewal arrival process. The population is prone to be affected by catastrophes which occur according to Poisson process. The catastrophe starts the destruction of the population sequentially, with one individual at a time, with probability p. This process comes to an end when the first individual survives or when the entire population is eliminated. Using supplementary variable and difference equation method we obtain explicit expressions of population size distribution in steady-state at pre-arrival and arbitrary epochs, in terms of roots of the associated characteristic equation. Besides, we prove that the distribution at pre-arrival epoch is asymptotically geometric. Based on our theoretical work we present few numerical results to demonstrate the efficiency of our methodology. We also investigate the impact of different parameters on the behavior of the model through some numerical examples.

Performance prediction of interactive telemedicine

Abstract Telemedicine is a remote medical practice, which utilizes information and communication technology to provide healthcare and exchange of health information for distance places. Diagnosis and prescription for rural and remote area patients are the main advantages of this system. The major categories of telemedicine are store-and-forward, remote patient monitoring and interactive telemedicine. Interactive telemedicine is a real-time interaction between patients and doctors. This method can be conducted in a nursing home or at the patient's house via a video-conferencing system. Moreover, this is more convenient for patients, than to obtain health care with a caretaker in person. However, Internet-based networks are congested with many users. We considered interactive telemedicine as a queueing system and analyzed it by using the method of supplementary variables. We obtained the performance measures through a probability generating function. The numerical results of the queueing system may be useful to provide service satisfactorily and update the utility function of interactive telemedicine.

Cost optimisation analysis of retrial queue with K optional phases of service under multiple working vacations and random breakdowns

In this paper, we consider a cost optimisation analysis of M/G/1 retrial queue with k optional phases of service under multiple working vacations and vacation interruption, where each phase consists of an optional re-service. An arriving customer may balk the system at some particular times. When the orbit becomes empty at regular service completion instant, the server goes for a working vacation. During a working vacation period, the server works in lower service rate. The normal busy server may get to breakdown and the service channel will fail for a short interval of time. We analyse the steady state probability generating function for the system size by using the supplementary variable method. Some important system performance measures are obtained. Finally, some numerical examples and cost optimisation analysis are presented.