Supplementary Variable Technique Research Articles

Batch service queue with change over times and Bernoulli schedule vacation interruption

In this paper, we model a discrete-time renewal input queue with change over times and Bernoulli schedule vacation interruption under batch service (a, c, b) policy. The service, working vacation and change over times are geometrically distributed. At the instants of a service completion, the vacation is interrupted and the server is resumed to a regular busy period with probability 1 − q if there are c or more customers in the system, or continues the vacation with probability q. Employing the supplementary variable and recursive techniques, we have derived the steady state queue length distributions at various epochs. Some performance measures and a cost model have been presented and an optimum service rate has been obtained using geneticalgorithm. Numerical results showing the effect of the parameters of the modelon the key performance measures are presented.

An MX/G/1 queue with randomized working vacations and at most J vacations

This paper treats a bulk arrival queue with randomized working vacation policy. Whenever the system becomes empty, the server takes a vacation. During the vacation period, customers are to be served at a lower rate. Once the vacation ends, the server will return to the normal working state and begin to serve the customers in the system if any. Otherwise, the server either remains idle with probability p or leaves for another vacation with probability 1−p. This pattern continues until the number of vacations taken reaches J. If the system is empty at the end of the Jth vacation, the server will wait idly for a new arrival. By using supplementary variable technique, we derive the system size distribution at arbitrary epoch, at departure epoch and at busy period initial epoch, as well as some important system characteristics. Numerical examples are provided to illustrate the influence of system parameters on several performance measures.

Stochastic Analysis of a Standby System With Waiting Repair Strategy

This paper investigates the reliability of a standby system incorporating waiting time to repair. The considered system consists of two units, namely, the main unit and the standby unit. Whenever the main unit fails, the whole load is transferred to the standby unit instantaneously by a switching-over device. As regards to the repairing of the main unit, it has to wait for repair whenever it fails due to unavailability of repair facility. When both the main and standby units fail, then the system goes to the complete failure mode. The system may also fail due to incorrect start of the system, which can occur due to an untrained and inexperience operator. The repair of the main and standby units follows general distribution, whereas repair due to human error is obtained with the help of Gumbel-Hougaard family copula. The system is analyzed by supplementary variable technique and Laplace transformation. Various reliability measures like availability, mean time to failure, and profit function have been evaluated for the considered system. A numerical example with a way to illustrate the utility of the model has also been presented.

Analysis of finite buffer renewal input queue with balking and multiple working vacations

This paper analyzes a finite buffer single server queueing system with multiple working vacations in which customers balk (i.e., refuse to join the queue). The inter-arrival times and service times during regular busy period are, respectively, arbitrarily and exponentially distributed. Vacation times and service times during vacation are exponentially distributed. Using embedded Markov chain and supplementary variable techniques, we obtain the system length distributions at pre-arrival and arbitrary epochs. Various performance measures of the model, waiting time analysis and some special cases are also presented. Finally, a variety of numerical results in the form of tables and graphs are discussed.

Numerical solution of stochastic partial differential difference equation arising in reliability engineering

In this paper, a numerical method is proposed to solve the transient state of Markovian system of equations. Such equations appear in the field of reliability engineering for systems having variable failure and repair rates. Generally, steady state behaviour of the system model is studied due to some constraint on obtaining transient state solution. The proposed method helps to determine the probability values, by utilizing finite difference scheme iteratively in conjunction with the results of integral appearing in stochastic differential difference equation obtained using supplementary variable technique. This method also uses the Lagrange’s method to interpolate the missing value of repair rates of the system wherever required in computation. Results thus obtained are found to be efficient for studying the transient state behaviour of the system.

Discrete-time renewal input bulk service queue with changeover time

In this paper, we model a discrete-time infinite buffer renewal input single server queue with changeover time under (a, c, b) policy. The service and changeover time are geometrically distributed. The server begins service if there are at least c customers in the queue and the services are performed in batches of minimum size a and maximum size b (a ≤ c ≤ b). At the instant of service completion, if the queue size is less than c but not less than a secondary limit a, the server continues to serve and it will be in the changeover period if the queue size is a − 1. Employing the supplementary variable and recursive techniques, we have derived the steady state queue length distributions at pre-arrival and arbitrary epochs. Based on the queue length distributions, some performance measures and cost functions have been discussed. Using a genetic algorithm, the optimum value of the service rate at minimum cost has been evaluated. Numerical results showing the effect of model parameters on the key performance measures are presented. Finally, we have derived the steady state queue length distributions of the continuous-time counterpart of our model.

A Two Stage Batch Arrival Queue with Reneging during Vacation and Breakdown Periods

We study a two stage queuing model where the server provides two stages of service one by one in succession. We consider reneging to occur when the server is unavailable during the system breakdown or vacation periods. We concentrate on deriving the steady state solutions by using supplementary variable technique and calculate the mean queue length and mean waiting time. Further some special cases are also discussed and numerical examples are presented.

Analysis of queue with two phases of service and m phases of repair for server breakdown under N-policy

The present paper deals with a single server queueing system wherein the customers join the system in bulk with varying arrival rates. The first stage regular service and second stage optional service are provided to the customers. The server may breakdown at any instant during any stage of service. There is a provision of essential and optional repair in m phases. To discuss the steady state behaviour of the system, the assumptions of probability reasoning and supplementary variable technique is employed. Various performance indices are obtained using the generating function approach. To verify the validity of these indices, the numerical illustrations are provided. The technique of principle of maximum entropy is also considered to find the approximate values of the system probabilities for the queue length distribution.

M/G/1 Feedback Queue with Two Stage Heterogeneous Service with Time Homogeneous Breakdowns and Deterministic Repair Times

We analyze a single server feedback queue with Poisson arrivals, two stages of heterogeneous service with different (arbitrary) service time distributions subject to random breakdowns and just after a breakdown the server undergoes repairs of a fixed (constant) duration. After first-stage service, the server must provide the second stage service. However after the completion of second stage of service, if the customer is dissatisfied with his service, he can immediately join the tail of the queue as a feedback customer with probability p: Otherwise the customer may depart forever from the system with probability q = 1-p. The supplementary variable technique is employed to find explicitly the probability generating function of the number in the system and the mean number in the system. Some particular cases of interest are discussed as special cases.

A Single Server Compulsory Vacation Queue with Two Type of Services and with Restricted Admissibility

A single server queue with compulsory vacation has been considered. In addition admission to queue is based on a Bernoulli process and the server gives two types of services. For this model the probability generating functions of number of customers in the queue for different server states are obtained using supplementary variable technique. Some performance measures are calculated. Particular cases are deduced and some numerical examples are presented.

Steady State Solution of a Catastrophic-cum-Restorative M/G/1 Queuing System Using Supplementary Variable Technique

Non- Markovian queuing models have their place in modeling the real life phenomena. In fact, their utility get enhanced when one is not able to get a particular probability distribution for either the inter-arrival times or for the service times. The service times are assumed to have general service time distribution in case of computer communication modeling. Recently, the emphasis is put on the catastrophe modeling and its applications in real situations. Keeping this in view, an M/G/1 queuing model has been developed with catastrophic and restorative effects. The steady-state solution of the model has been obtained using supplementary variable technique. Some queuing models have been obtained as particular cases of this model.

Cost Benefit Analysis of Series Systems with Mixed Standby Components and <i>k</i>-stage Erlang Repair Time

In this paper, we compare the availability characteristics between three different series system configurations with mixed (cold and warm) standby components and k-stage Erlang repair times. Three configurations are studied under the assumption that the time-to-failure for each of the operative and warm standby components are assumed to be exponentially distributed with parameters λ andα , respectively. We present a recursive method, using the supplementary variable technique; we develop the explicit expressions for the steady-state availability. Under the cost/benefit criterion, comparisons are made based on assumed numerical values given to the distribution parameters, and to the cost of the operative and standby units.

Analysis for Parallel Repairable System with Degradation Facility

<b> </b><b> </b>This paper presents a mathematical model for performing availability and reliability analysis of a parallel repairable system consisting of<i> n </i>identical components with degradation facility and common-cause failures. In addition, system repair time is assumed to be arbitrarily distributed. Markov and supplementary variable techniques are used to develop equations for the model. As an illustration, system of four-identical/repairable components is analysed.

Analysis of ${\it GI/M(n)/}1{\it /N}$ queue with state-dependent multiple working vacations

This paper analyzes a finite buffer renewal input queueing system with state-dependent services and state-dependent multiple working vacations. Service times during a service period and service times during a vacation period are exponentially distributed with state-dependent rates. We provide a recursive algorithm using the supplementary variable technique and treating the remaining inter-arrival time as the supplementary variable to compute the stationary system length distributions during vacations and regular busy period. Various performance measures and the computational algorithm of the model have been discussed. The computational complexity of the algorithm is O(N3). Some queueing models discussed in the literature are derived as special cases of our model. We present some numerical results to examine the effect of different parameters on the system performance characteristics.

Availability, MTTF and cost analysis of a system having two units in series configuration with controller

The paper deals with the availability analysis of a system, which consists of two subsystems namely subsystem-1 and subsystem-2. Subsystem-1 is working under k-out of n: good configuration while subsystem-2 has two identical units connected in parallel configuration. A controller is attached with each subsystem for proper functioning of the system. All failure rates are constant but repairs follow general and exponential distributions. The transitional state probabilities, asymptotic behavior and some characteristics such as reliability, availability, MTTF and the cost effectiveness of the system have been evaluated with the help of supplementary variable technique, Laplace transformations and copula methodology. At last, some particular cases and numerical examples have been taken to describe the model.

Optimal System for Warm Standby Components in the Presence of Standby Switching Failures, Two Types of Failures and General Repair Time

different system configurations with warm standby components, standby switching failures, two types of failures common cause and hardware failure and general repair are compared based on the availability. The time-to- for each of the primary and warm standby components are assumed to follow the exponential distribution. Laplace transforms of state probability equations are developed by using the supplementary variable technique. We develop the explicit expressions for the steady-state availability, Av, for two configurations. For all configurations, comparisons are made for specific values of distribution parameters and of the cost of the components. The configurations are ranked based on Av and cost/benefit, for three various repair time distributions: Gamma (G), Weibull (W) and Lognormal (L), where benefit is Av. KeywordsStandby switching failures; Common cause CCF; Supplementary variable; General repair times.

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

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 1-p. 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.

RELIABILITY AND SENSITIVITY ANALYSIS OF AN OPERATING SYSTEM WITH INSPECTION IN DIFFERENT WEATHER CONDITIONS

This paper deals with the reliability analysis of a system which consists of two subsystems namely main and standby with two weather conditions — normal and abnormal (turbulent). The subsystems are nonidentical and each has two modes — normal and total failure. The main subsystem is k-to-l-out-of-n and standby subsystem is k-out-of-n configurations. Single repairman is available for repair who plays the dual role of inspection and repair. Repairman inspects main subsystem and both the subsystems at its failure to see the feasibility of repair. If repair of the (sub) system is not feasible, it is replaced by new one. Inspection activities are stopped in abnormal weather conditions but system remains operative. Failure rates of both subsystems and the rate of change of weather conditions are constant, whereas the repair rates are general. With the help of Supplementary variable technique, Laplace transformations and copula methodology the transition state probabilities, asymptotic behavior and some characteristics such as reliability, availability, M.T.T.F., cost effectiveness and sensitivity of the system have been evaluated. At last some particular cases and numerical examples have been taken to describe the model.

RELIABILITY MEASURES AND SENSITIVITY ANALYSIS OF A COMPLEX MATRIX SYSTEM INCLUDING POWER FAILURE

This paper investigates the reliability characteristics of a complex system having nine subsystems arranged in the form of 3x3 matrix in which each row contains three subsystems. The configuration of the row is of the type 2-out-of-3: F. Each subsystem has n units connected in series. The system fails if any one row containing three subsystems fails. The considered system analyzed incorporating different types of power failure which also leads to failure of the system. With the help of Supplementary variable technique, Laplace transformations and copula methodology the transition state probabilities, asymptotic behavior, availability, reliability, M.T.T.F., busy period, sensitivity analysis and cost effectiveness of the system have been evaluated. At last some particular cases and numerical examples have been taken to describe the model.

Optimal repairable M&lt;SUP align="right"&gt;X&lt;/SUP&gt;/G/1 queue with Bernoulli feedback and setup

We consider an MX/G/1 queue with Bernoulli feedback under N-policy. According to N-policy, the server starts the service only if N customers are accumulated in the system. The server undergoes breakdown during the busy state, which is then repaired by the repairman in k number of phases. Both the server and the repairman take start-up and set-up times, respectively. Using the supplementary variable technique and the generating function method, we establish the stationary distribution of the queue length and various other queueing characteristics. Using the maximum entropy approach. We determine the approximate results for the steady-state probability distributions of the queue length and the waiting time of an arbitrary customer in the queue.