Supplementary Variable Technique Research Articles

Analysis of an retrial queueing system with priority services, working breakdown, Bernoulli vacation, admission control and balking

A single server serves two classes of customers under non-preemptive priority service rule are studied. In addition to this working breakdown, admission control, balking and Bernoulli vacation are also investigated. Here, the customers arrive according to two independent compound Poisson processes and the service is in general distribution. Once the system gets a breakdown, the server proceeds the service at a slower speed for the current customer, after which, the repair will commence. Moreover, the server can not allow all the customers to enter into the system, there is an admission control policy to allow customers. Applying the supplementary variable technique, the Laplace transforms of time-dependent probabilities of system state are determined and further deduced the steady state results. Also, the average number of customers in the corresponding queues and the average waiting times are estimated. Finally, the numerical results are represented.

Stationary queue and server content distribution of a batch-size-dependent service queue with batch Markovian arrival process:

Queueing systems with batch Markovian arrival process (BMAP) have paramount applications in the domain of wireless communication. The BMAP has been used to model the superposition of video sources and to approximate the super-position of data, voice and video traffic. This article analyzes an infinite-buffer generally distributed batch-service queue with BMAP, general bulk service (a, b) rule and batch-size-dependent service time. In this proposed analysis, we mainly focus on deriving the bivariate vector generating function of queue and server content distribution together at departure epoch using supplementary variable technique. The mathematical procedure for the complete extraction of distribution at departure epoch has been discussed and using those extracted probabilities, we achieve the queue and server content distribution at arbitrary epoch. Finally, numerical illustrations have been carried out in order to make a deep insight to the readers which contains deterministic as well as phase-type service time distributions.

Analysis of $ D $-$ BMAP/G/1 $ queueing system under $ N $-policy and its cost optimization

<p style='text-indent:20px;'>This article studies an infinite buffer single server queueing system under <inline-formula><tex-math id="M4">\begin{document}$ N $\end{document}</tex-math></inline-formula>-policy in which customers arrive according to a discrete-time batch Markovian arrival process. The service times of customers are independent and obey a common general discrete distribution. The idle server begins to serve the customers as soon as the queue size becomes at least <inline-formula><tex-math id="M5">\begin{document}$ N $\end{document}</tex-math></inline-formula> and serves the customers until the system becomes empty. We determine the queue length distribution at post-departure epoch with the help of roots of the associated characteristic equation of the vector probability generating function. Using the supplementary variable technique, we develop the system of vector difference equations to derive the queue length distribution at random epoch. An analytically simple and computationally efficient approach is also presented to compute the waiting time distribution in the queue of a randomly selected customer of an arrival batch. We also construct an expected linear cost function to determine the optimal value of <inline-formula><tex-math id="M6">\begin{document}$ N $\end{document}</tex-math></inline-formula> at minimum cost. Some numerical results are demonstrated for different service time distributions through the optimal control parameter to show the key performance measures.</p>

Reliability and Availability analysis of a cloud computing transition system under multiple failures

PurposeTo analyze the performance of multistate cloud computing transition system through the various reliability measures is the purpose of this paper.Design/methodology/approachIn this article, a mathematical model for a multistate cloud computing transition system with various types of failures has been analyzed by using the Markov process, supplementary variable technique and Laplace transformation.FindingsVarious reliability measures such that reliability, availability, mean time to failure (MTTF), mean time to repair and cost analysis have also been analyzed. This article presents some geographic illustrations for the practical utility of the model.Originality/valueThe authors developed a mathematical model to analyze the reliability of the cloud computing transition system by considering the possible failures.

Performance Analysis of Multi-computer System Consisting of Three Subsystems in Series Configuration Using Copula Repair Policy

The present paper discuss the reliability characteristics of computer network system consisting of the devices load balancer, distributed database servers and a centralized server configured as series parallel system containing three subsystems. Subsystem 1: load balancer (LB) a device that act as a medium of channeling request to servers and receiving request or response from servers. Subsystem 2: identical distributed database servers (DDS) I, II and III connected in parallel, working under 2-out-of-3: G: policy, which manages and store some part of application data and Subsystem 3: centralized distributed database server (CDDS), which takes care of all the application data in DDS. Through the transition diagram, system of first-order partial differential equations are derived and solved using supplementary variable techniques, Laplace transforms and copula method. All failures are assumed to be exponentially distributed, whereas repairs follow two types of distributions; general distribution and Gumbel–Hougaard family copula, partial failures are repaired by general distribution while complete failures are repaired by copula distribution. The objective is to obtain the expressions for availability, reliability, mean time to failure (MTTF) and cost function. Computations for reliability measures are taken as particular case by evaluating: reliability, availability, MTTF and cost analysis, so as to capture the effect of both failure and repair rates to reliability measures. The results of the computations are presented in tables and graphs. The analysis of the results obtained with respect to the availability and expected profit indicates that copula repair policy is reliable than general repair policy.

Analysis of stationary queue-length distributions of the BMAP/R(a,b)/1 queue

This paper investigates an infinite waiting space single-server queueing system in which customers arrive according to a batch Markovian arrival process. The server serves the customers in batches of maximum size ‘b’ with a minimum threshold value ‘a’. The service time of each batch follows R-type distribution, where R-type distribution represents a class of distributions whose Laplace-Stieltjes transform is rational or approximated rational function, which is independent of service batch size and the arrival process. We determine the queue-length distribution at departure epoch in terms of roots of the associated characteristic equation of the vector probability generating function. To show the strength and advantage of roots method, we also provide a comprehensive analysis of the queue-length distribution at departure epoch using the matrix-analytic method (MAM). To determine the queue-length distribution at random epoch, we obtain a relation between the queue-length distributions at departure and random epochs using the supplementary variable technique with remaining service time of a batch in service as the supplementary variable. We also derive the queue-length distributions at pre-arrival epoch of an arrived batch and post-arrival epoch of a random customer of an arrived batch. Some numerical results are demonstrated for different service-time distributions to show the key performance measures of the system.

Stochastic Analysis of Complex System with Auto Changeover Switch and Advert Environment Employing Copula Approach

This paper studies the reliability measures of a system consisting of two subsystems in a series configuration for different types of failure and two types of repair. The subsystem-1 has four identical units in a parallel configuration operating under 3-out-of-4: G policy and this has connected to subsystem-2 which has three identical units arranged in a parallel configuration and working under 2-out- of-3: F, scheme. The units of subsystem-1 are controlled by a controller for preventing failure effect and safety purposes. It is assumed that units of each subsystem have different types of failure and repair rates. The unsuitability of the environmental conditions such as overheating as a well-known natural cause of failure of any system and also weather conditions like heavy rain, thunderstorm, and catastrophic shakeups, etc. have treated as a complete failure of the system. This study considers the environmental causes of failure in the proposed repairable system as complete failure by which the system stops functioning. Human failure in the system is trickled as complete failure and repair employing copula (Gumbel- Hougaard family copula distribution) like another complete failed states of subsystems. To analyze the proposed system, the supplementary variable technique are used and some measures of system reliability like availability, reliability; MTTF and incurred profit function for different values of parameters are derived. Some particular cases for different values of failure rates that have explicit are also studied.

Cost Benefit Analysis of a k-out-of-n: G Type Warm Standby Series System Under Catastrophic Failure Using Copula Linguistics

This paper deals with the study of reliability measures of a complex engineering system consisting three subsystems namely L, M, and N in series configuration. The subsystem-L has three units working under 1-out-of-3: G; policy, the subsystem-M has two units working under 1-out-of-2: G policy and the subsystem-N has one unit working under 1-out-of-1: G; policy. Moreover, the system may face catastrophic failure at any time t. The failure rates of units of all subsystems are constant and assumed to follow the exponential distribution however, their repair supports two types of distribution namely general distribution and Gumbel-Hougaard family copula distribution. The system is analyzed by using the supplementary variable technique, Laplace transformation and Gumbel-Hougaard family of copula to derive the differential equations and to obtain important reliability characteristics such as availability of the system, reliability of the system, MTTF, and profit analysis. The numerical results for reliability, availability, MTTF, and profit function are obtained by taking particular values of various parameters and repair cost using maple. Tables and figures demonstrate the computed results and conclude that copula repair is more effective repair policy for better performance of repairable systems. It gives a new aspect to scientific community to adopt multi-dimension repair in form of copula. Furthermore, the results of the model are beneficial for system engineers and designers, reliability and maintenance managers.

Availability Analysis of Machining System with Common Cause Failure, Inspection and Imperfect Repair

This investigation is concerned with the availability prediction of a non-Markovian model for the multi-component complex repairable manufacturing system. The system is prone to multiple types of failures, either due to an individual or common cause failure. The system can also fail catastrophically which may occur due to the incorrect start of the system or due to some errors by an untrained and inexperienced operator. When a system failure occurs, the recovery of the system is done by the inspection/repair process. The stochastic analysis is carried out using the supplementary variable technique and Laplace transformation for evaluating the system state probabilities. The performance indices such as availability, reliability, MTTF, MTTR, and cost–benefit analysis of the system are established analytically and then validated by taking a numerical example.

Performance assessment of Linux operating system employing copula distribution approach

This paper focused on studying the Linux operating system connected to a LAN (local area network). The STAR topology (subsystem-1) and BUS topology (subsystem-2) are taken into account, which is placed at two different locations and connected to a server through a hub. Under consideration, both topologies BUS topology and the STAR topology, each has n clients. The system has two types of failures, i.e., partial failure and complete failure. Partial failure has categorized as minor and major partial failure. It expected that the minor partial failure degrades the sub-systems, and the major partial failure brings the subsystem to break down mode. The system may fail due to the failure of the server, such as hacking and blocking. The system has analyzed by changing to none-Markov model to the Markov model with the implication of the supplementary variable technique and Laplace transform. The various measures of reliability, such as the availability of the system, the reliability of the system, MTTF, profit function for different parametric values, have evaluated.

Retrial queue with multiple repairs, multiple services and non preemptive priority

The study presented here deals with the steady state analysis of bulk arrival retrial queue with unreliable server and multi phase essential services. The server renders service to two types of customers viz. priority (type 1) and ordinary (type 2) customers. The ordinary customers are forced to join the orbit on nonavailability of the server whereas priority customers are served by the server as soon as they arrive. The service is provided in ‘l’ essential phases for both type of customers. Moreover, the server is not an ideal server and hence may stop working in between. As soon as the server fails or stops working, it is sent for repair immediately. The broken down server is also repaired in ‘d’ essential phases. In the present investigation, the supplementary variable technique and the method of generating function have been used. The sensitivity of various parameters on the system performance has been examined numerically by taking an illustration.

Reliability analysis of repairable network system of three computer labs connected with a server under 2- out- of- 3 G configuration

This paper presents the study of reliability measures of a complex system consisting of three computer labs connecting to a server in the parallel configuration under 2-out-of-3: G operation policy. The system is assumed to work if at least two labs are in good operative condition. The server failure brings the system to a complete failed state. The failure rates of all labs are different and assumed to follow an exponential distribution, but their repair supports two types of distribution, namely general distribution and Gumbel–Hougaard family copula distribution. The system of three labs with the server studied using the supplementary variable technique, laplace transform, expressions of reliability measures such as system availability, system reliability, MTTF, and cost analysis as incurred profit by operations have been computed. The numerical results for relevant measures like availability, reliability, MTTF, and profit function are also obtained by taking particular values of various failure and repair rate as stated. Tables and figures demonstrate the computed results.

Operational availability analysis of generators in steam turbine power plants

The main aim of this study is to explore the availability and profitability of generators, which are key units of steam turbine power plants, considering exponential failure and arbitrary repair time distributions. A Markov birth–death process and supplementary variable technique are used to evaluate system effectiveness measures. Such a generator mainly consists of seven repairable components arranged in a series configuration. A stochastic model of the generator is developed, and the Chapman–Kolmogorov differential difference equation of the system derived. Finally, the results for the steady-state availability and profitability of the system are obtained for a particular case that will be helpful to system designers to enhance the performance of plants to some extent.

Transient Analysis of M[X1],M[X2]/G1,G2/1 retrial queueing system with priority services, working breakdown, start up/close down time, Bernoulli vacation, reneging and balking

This paper considers M[X1],M[X2]/G1,G2/1 general retrial queueing system with priority services. Two types of customers from different classes arrive at the system in different independent compound Poisson processes. The server follows the pre-emptive priority rule subject to working breakdown, startup/closedown time and Bernoulli vacation with general (arbitrary) vacation periods. After completing the service, if there are no priority customers present in the system the server may go for a vacation or close down the system. On completion of the close down, the server needs some time to set up the system. The priority customers who find the server busy are queued in the system. A low-priority customer who find the server busy are routed to a retrial (orbit) queue that attempts to get the service. The system may breakdown at any point of time when it is in operation. However, when the system fails, instead of stopping service completely, the service is continued only to the high priority customers at a slower rate. We consider balking to occur to the low priority customer while the server is busy or idle, and reneging to occur at the high priority customers during server’s vacation, start up/close down time. Using the supplementary variable technique, we derive the joint distribution of the server state and the number of customers in the system. Finally, some performance measures and numerical examples are presented.

Comparison of Three Availability Systems with Warm Standby Components, Detection Delay, and General Repair Times

This study analyzes three availability systems with warm standby units, fault detection delay, and general repair times. The failure times and repair times of failed components were assumed to follow exponential and general distributions, respectively. The detection delay times were assumed to be exponentially distributed. This study exploited the supplementary variable technique to develop a recursive method for deriving the steady-state availability for three systems. By using extensive numerical computations, we compared three systems in terms of system availability based on specific values given to the system parameters. The state transition rate diagrams of the three systems revealed the symmetry property approximately. The three systems were ranked based on the system availability and the cost/benefit for the three various repair time distributions: exponential, three-stage Erlang, and deterministic, where the benefit was system availability.

Randomized control of arrivals in a finite-buffer GI/M/1 system with starting failures

This paper proposes a randomized policy for the control of arrivals in a finite-bufferGI/M/1 system with a single server. When the capacity of system is full, no new arrivals are permitted to enter the system. When the number of customers in the system decreases to thresholdF, a new arriving customer is allowed to join the system with probabilityp. The system requires an exponential startup time before allowing customers to enter the system. The startup process may not be successful and is then restarted once again. Using the supplementary variable technique in a recursive process, we obtain the stationary distribution of the system size. Various performance measures of the system are developed. We also create a cost model based on the system performance measures and cost elements. The optimal threshold, optimal capacity and optimal startup rate of the system are determined to minimize the expected cost per unit time. Finally, we provide numerical examples to conduct a sensitivity analysis.

A Renewal Generated Geometric Catastrophe Model with Discrete-Time Markovian Arrival Process

Any event that results in sudden change of the normal functioning of a system may be thought of as a catastrophe. Stochastic processes involving catastrophes have very rich application in modeling of a dynamic population in areas of ecology, marketing, queueing theory, etc. When the size of the population reduces abruptly as a whole, due to a catastrophe, it is termed as the total catastrophe. However, in many real-life circumstances the catastrophes have a mild influence on the population and have a sequential effect on the individuals. This paper presents a discrete-time catastrophic model in which the catastrophes occur according to renewal process, and it eliminates each individual of the population in sequential order with probability p until the one individual survives or the entire population wipes out. The individuals arrive according to the discrete-time Markovian arrival process. Using the supplementary variable technique, we obtain the steady-state vector generating function (VGF) of the population size at various epochs. Further using the inversion method of VGF, the population size distribution is expressed in terms of the roots of the associated characteristic equation. We further give a detailed computational procedure by considering inter-catastrophe time distributions as discrete phase-type as well as arbitrary. Finally, a few numerical results in form of tables and graphs are presented. Moreover, the impact of the correlation of arrival process on the mean population size is also investigated.

Analysis of bulk queue with unreliable service station, second optional repair, N-policy multiple vacation, loss and immediate feedback in production system

In this paper an unreliable server bulk service queue with second optional repair, multiple vacation under N-policy, customer's feedback and impatience are considered. After finishing a service, the server avails a multiple vacation under N-policy only when the queue length is less than 'a'. Server vacation models are very much useful for a queueing system in which server needs to utilise his idle time for a different purpose. The server is subject to breakdown during the service and requires second optional repair for restoration. After repair completion, the server will start its remaining service to the batch of customers whose service was interrupted. Further, it is assumed that the arriving units may balk (loss) from the system when the server is on vacation and immediate feedback service was provided to the dissatisfied customers. The queue size distribution at a random and departure epoch has been derived using the supplementary variable technique. Finally, the stability condition, some performance measures, particular cases and numerical results of the proposed model are obtained.

Analysis of an <i>M</i><SUP align="right">[<i>X</i>]</SUP>/<i>G</i>(<i>a</i>, <i>b</i>)/1 queueing model with multiple vacation controllable arrival during multiple vacation and two phases of repair with delay

The objective of this paper is to analyse an M[X]/G(a, b)/1 queueing model with multiple vacation, controllable arrival during multiple vacation and two phases of repair with delay. Whenever the queue size is less than 'a', the server resumes multiple vacation and continues this until at least 'a' customers are waiting in the queue. After finishing a batch of service, if the server is breakdown with probability γ, the server will be sent for repair after a short interval of time called delay time. After this delay, two consecutive phases of repair (first phase repair, second phase repair) are considered. After the second phase of repair or when there is no breakdown with probability 1-γ, the server starts a vacation if the queue length is less than 'a'. Otherwise, the server starts a service under the general bulk service rule. Using supplementary variable technique, the probability generating function of the queue size at an arbitrary time is obtained for the steady-state case. Also some performance measures and cost model are derived. Numerical illustrations are presented to visualise the effect of various system parameters.

Performance analysis of a complex repairable system with two subsystems in series configuration with an imperfect switch

This paper presents the study of reliability measures of a complex system consisting of two subsystems, subsystem-1, and subsystem-2, in a series configuration with switching device. The subsystem-1 has five units that are working under 2-out-of-5: G policy and the subsystem-2 has two units that are working under 1-out-of-2: G policy. Moreover, the switching device in the system is unreliable, and as long as the switch fails, the whole system fails immediately. Failure rates of units of subsystems are constant and assumed to follow the exponential distribution. Still, their repair supports two types of distribution, namely general distribution and Gumbel-Hougaard family copula distribution. Using the supplementary variable technique, Laplace transformations, and copula approach differential equations developed. Important reliability characteristics such as availability of the system, reliability of the system, MTTF, profit analysis, and sensitivity analysis for MTTF have computed for fixed values of failure and repair rates. Particular cases corresponding to the switching device have also considered. Graphs demonstrate results, and consequently, conclusions have done.