Supplementary Variable Method Research Articles

Non-Markovian bulk queueing system with state dependent arrivals and multiple vacations - a simulation approach

We considered a non-Markovian bulk arrival general bulk service queueing system with state dependent arrivals and multiple vacations in this paper. The arrivals occur in bulk with a rateλ, when the server is busy and with a rateλ0, when the server is on vacation. Using a supplementary variable technique and difference differential equations method we derive the queue size distribution, expected length of idle and busy periods and expected queue size of a Mx/G(a, b)/1 queueing system with state dependent arrivals and multiple vacations. We study the same through simulation using ARENA, a powerful manufacturing system simulation software and obtained some performance measures, which are mandatory for the study and analysis of the said queueing model. The simulated results for a few cases are presented.

Performance Analysis of a Repairable Robot Safety System with Standby, Imperfect Coverage and Reboot Delay

The present study deals with a robot safety system composed of standby robot units and inbuilt safety unit. When the main operative unit fails, it is replaced by the standby robot unit available in the system. The concept of reboot delay is also incorporated in this study according to which the robot unit is rebooted if it is not successfully recovered. The recovery and reboot times of failed units, life time of the operative as well as standby units and the repair time are assumed be exponentially distributed. Furthermore, the repair time of partially-failed unit of total system failure is assumed to be arbitrarily distributed. The expressions for the state probabilities, availability, reliability and mean time to failure are derived with the help of Markovian and supplementary variable methods. The occurrence of standby units, imperfect coverage and reboot demonstrates the significant impact on the robot system reliability, availability and mean time to failure. A numerical illustration has been provided to validate the present model as well as to demonstrate the effects of various parameters on the performance measures of the robot safety system.

An M[X]/G/1 retrial g-queue with single vacation subject to the server breakdown and repair

An M[X]/G/1 retrial G-queue with single vacation and unreliable server is investigated in this paper. Arrivals of positive customers form a compound Poisson process, and positive customers receive service immediately if the server is free upon their arrivals; Otherwise, they may enter a retrial orbit and try their luck after a random time interval. The arrivals of negative customers form a Poisson process. Negative customers not only remove the customer being in service, but also make the server under repair. The server leaves for a single vacation as soon as the system empties. In this paper, we analyze the ergodical condition of this model. By applying the supplementary variables method, we obtain the steady-state solutions for both queueing measures and reliability quantities.

QUEUEING ANALYSIS FOR TRAFFIC CONTROL WITH COMBINED CONTROL OF DYNAMIC MMPP ARRIVALS AND TOKEN RATES

We analyze the queueing model for leaky bucket (LB) scheme with dynamic arrivals and token rates. In other words, in our LB scheme the arrivals and token rates are changed according to the buffer occupancy. In telecommunication networks, the LB scheme has been used as a policing function to prevent congestion. By considering bursty and correlated properties of input traffic, the arrivals are assumed to follow a Markov-modulated Poisson process (MMPP). We derive the distribution of system state, and obtain the loss probability and the mean waiting time. The analysis is done by using the embedded Markov chain and supplementary variable method. We also present some numerical examples to show the effect of our proposed model.

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.

ANALYSIS OF QUEUEING MODEL WITH PRIORITY SCHEDULING BY SUPPLEMENTARY VARIABLE METHOD

We analyze queueing model with priority scheduling by supplementary variable method. Customers are classified into two types (type-1 and type-2 ) according to their characteristics. Customers of each type arrive by independent Poisson processes, and all customers regardless of type have same general service time. The service order of each type is determined by the queue length of type-1 buffer. If the queue length of type-1 customer exceeds a threshold L, the service priority is given to the type-1 customer. Otherwise, the service priority is given to type-2 customer. Method of supplementary variable by remaining service time gives us information for queue length of two buffers. That is, we derive the differential difference equations for our queueing system. We obtain joint probability generating function for two queue lengths and the remaining service time. Also, the mean queue length of each buffer is derived.

Discrete-Time Geo x /G/1 Queue with Geometrically Working Vacations and Vacation Interruption

This paper treats a discrete-time batch arrival queue with geometrically working vacations and vacation interruption. The main purpose of this paper is to present a performance analysis of this system. For this purpose, we first derive the probability generating function (PGF) of the stationary system length at the departure epochs based on the embedded Markov chain technique. Next, we present the stationary distributions of the system lengths as well as some performance measures at random epochs by using the supplementary variable method. Thirdly, still based on the supplementary variable method we give a distribution for the number of the customers at the beginning of a busy period and give a stochastic decomposition formula for the PGF of the stationary system length at the departure epochs. Additionally, we investigate the relation between our discrete-time system and its continuous counterpart. Finally, some numerical examples show the influence of the parameters on some crucial performance characteristics of the system.

Discrete-Time Geox/G/1 Retrial Queue with General Retrial Times, Working Vacations and Vacation Interruption

We consider a discrete-time GeoX/G/1 retrial queue with general retrial times, and introduce working vacations and vacation interruption policy into the retrial queue. Firstly, we analyze the stationary condition for the embedded Markov chain at the departure epochs. Secondly, using supplementary variable method, we obtain the stationary probability distribution and some performance measures. Furthermore, we prove the conditional stochastic decomposition for the queue length in the orbit. Finally, some numerical examples are presented.

A single-server retrial G-queue with priority and unreliable server under Bernoulli vacation schedule

We consider an M/G/1 retrial queue with negative customers and priority under Bernoulli vacation schedule subject to the server breakdowns and repairs. Arrivals of both positive customers and negative customers are two independent Poisson processes. Positive customers receive service immediately if the server is idle upon their arrivals. Otherwise, they may either with probability p join the priority queue or with complementary probability p¯ enter a retrial orbit. A breakdown at the busy server is represented by the arrival of a negative customer which causes the the customer being in service to be lost. The server takes Bernoulli vacation after a service or a repair completion. It is assumed that the server has arbitrary repair time and vacation time distributions. With the help of Lyapunov functions we have obtained the necessary and sufficient condition for ergodicity of embedded Markov chain. By applying the supplementary variables method, we obtain the steady-state solutions for both queueing measures and reliability quantities. Moreover, we investigate the stochastic decomposition law. Besides, some special cases of interest are discussed. Finally, the effects of various parameters on the system performance are analyzed numerically.

On a BMAP/G/1 G-queue with setup times and multiple vacations

In this paper, we consider a BMAP/G/1 G-queue with setup times and multiple vacations. Arrivals of positive customers and negative customers follow a batch Markovian arrival process (BMAP) and Markovian arrival process (MAP) respectively. The arrival of a negative customer removes all the customers in the system when the server is working. 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 also obtain the mean of the busy period based on the renewal theory.

Reliability Analysis of $M$-for-$N$ Shared Protection Systems With General Repair-Time Distributions

In this paper, we investigate the user perceived reliability of M-for-N shared protection systems. The M-for-N shared protection system is a general type of the fault-tolerant configuration with shared backup units. We assume that there are N working units each serving a single group of users and M shared protection (spare) units in the system. We also assume that the time to failure of the unit is subject to an exponential distribution, and the time to repair it is subject to a general distribution. Under these assumptions, we derive the Laplace transform of the reliability function (The cdf of the failure time, i.e. the cumulative distribution function of the time beyond which the system will survive) for each user, as well as the user-perceived Mean Time to First Failure (MTTFF) by combining the state transition analysis and the supplementary variable method. We also show the effect of the repair time distribution of the units through the case study of the cloud computing systems in which N service-independent servers share M spares for backup purpose. The analysis reveals that there exist optimal values for M and N in terms of the reliability and the cost for the given conditions of failures and repairs.

On the asymptotic behaviour of [formula omitted] retrial queues with batch arrivals and impatience phenomenon

In this work, we consider an M / G / 1 retrial queue with batch arrivals and impatient customers. By using the method of supplementary variables, we obtain the partial generating functions of the steady state joint distribution of the server state and the number of customers in the retrial group. To complete the analysis of the considered model, we find the steady state distribution of the embedded Markov chain. Although the generating function of the steady state distribution of the number of customers in the retrial group can be obtained in explicit form, it is cumbersome and does not reveal the nature of the distribution in question. Therefore, we investigate the asymptotic behaviour of the random variable representing the number of customers in the retrial group under limit values of various parameters.

An M/G/1 retrial G-queue with non-exhaustive random vacations and an unreliable server

This paper deals with an M/G/1 retrial queue with negative customers and non-exhaustive random vacations subject to the server breakdowns and repairs. Arrivals of both positive customers and negative customers are two independent Poisson processes. A breakdown at the busy server is represented by the arrival of a negative customer which causes the customer being in service to be lost. The server takes a vacation of random length after an exponential time when the server is up. We develop a new method to discuss the stable condition by finding absorb distribution and using the stable condition of a classical M/G/1 queue. By applying the supplementary variable method, we obtain the steady-state solutions for both queueing measures and reliability quantities. Moreover, we investigate the stochastic decomposition law. We also analyse the busy period of the system. Some special cases of interest are discussed and some known results have been derived. Finally, an application to cellular mobile networks is provided and the effects of various parameters on the system performance are analysed numerically.

Reliability analysis of 1-for-2 shared protection systems with general repair-time distributions

In this paper, we investigate the reliability of a type of 1-for-2 shared protection systems. The 1-for-2 shared protection system is the most basic fault-tolerant configuration with shared backup units. We assume that there are two working units each serving a single user and one shared protection (spare) unit in the system. We also assume that the times to failure and to repair are subject to exponential and general distributions respectively. Under these assumptions, we derive the Laplace transform of the survival function (the cdf that the system will survive beyond a given time) for each user as well as the user-perceived Mean Time to First Failure (MTTFF) by combining the state transition analysis and the supplementary variable method. We also show the effect of the repair-time distribution, the failure rates and the repair rates of the units through the case study of small-sized two enterprises that share one spare device for backup purpose. The analysis reveals what is important and what should be done in order to improve the user-perceived reliability of shared protection systems.

Reliability and availability analysis of a robot‐safety system

PurposeThe purpose of this paper is to study reliability, availability, and mean time to failure of a repairable robot‐safety system composed of n robots, m safety units, and a perfect switch.Design/methodology/approachGeneralized expressions for system state probabilities, system availability, reliability, and mean time to failure are developed. Supplementary variable and Markov methods were used to develop these expressions.FindingsThis study clearly demonstrates that backup robots, safety units, and the repair process help to improve system availability.Practical implicationsThis study will help maintenance engineers and reliability practitioners to become aware of the combined effect of backup robots, safety units, and the repair process on the performance of the robot‐safety system. Consequently, they will make better maintenance related decisions in organizations that make use of robots.Originality/valueThis paper has studied the effects of having redundant robots, safety units, and the repair facilities on the performance of a robot‐safety system with perfect mechanism to turn on a safety unit. This is one of the first attempts to study the combined effects of all these factors on a robot‐safety system.

An M/G/1 Bernoulli feedback retrial queueing system with negative customers

A single server retrial queue with negative customers and two types of Bernoulli feedback is considered. A necessary and sufficient condition for the system to be stable is investigated. The system size probabilities at output epochs are obtained by using an embedded Markov chain. Further, the joint generating functions of queue length and server status are studied by using supplementary variables method. Some important system performance measures are derived. Busy period of the system is also discussed. Finally, extensive numerical illustrations are provided.

A new analytical algorithm for computing probability distribution of project completion time

An analytical algorithm was presented for the exact computation of the probability distribution of the project completion time in stochastic networks, where the activity durations are mutually independent and continuously distributed random variables. Firstly, stochastic activity networks were modeled as continuous-time Markov process with a single absorbing state by the well-know method of supplementary variables and the time changed from the initial state to absorbing state is equal to the project completion time. Then, the Markov process was regarded as a special case of Markov skeleton process. By taking advantage of the backward equations of Markov skeleton processes, a backward algorithm was proposed to compute the probability distribution of the project completion time. Finally, a numerical example was solved to demonstrate the performance of the proposed methodology. The results show that the proposed algorithm is capable of computing the exact distribution function of the project completion time, and the expectation and variance are obtained.

Analysis of the M [X]/G/1 queues with second multi-optional service and unreliable server

A bulk-arrival single server queueing system with second multi-optional service and unreliable server is studied in this paper. Customers arrive in batches according to a homogeneous Poisson process, all customers demand the first “essential” service, whereas only some of them demand the second “multi-optional” service. The first service time and the second service all have general distribution and they are independent. We assume that the server has a service-phase dependent, exponentially distributed life time as well as a servicephase dependent, generally distributed repair time. Using a supplementary variable method, we obtain the transient and the steady-state solutions for both queueing and reliability measures of interest.

Availability and maintenance scheduling of a repairable block-board manufacturing system

The main objective of this paper is to develop a reliability model of a block-board manufacturing system in the plywood industry. The system discussed here undergoes partial as well as direct total failure. The model discussed here helps in calculating both time dependent and steady state availability under idealised as well as faulty Preventive Maintenance (PM). The PM rates and corrective repair rates are taken as arbitrarily distributed. Corrective repair of units is performed on the basis of preemptive resume priority discipline. The supplementary variable method is used to develop generalised expressions of state probabilities and system availabilities. Runge-Kutta method is used to calculate the availability of the system for transient state. Long run availability is carried out making use of software package Matlab 5.1. The tables for various parameters are given which are useful to the management for improving and planning the maintenance schedule.

Maximum entropy approach to machine repair problem

For an M/G/1 machine repair problem, the steady-state probability distributions of the number of down machines are usually found by using the supplementary variable technique and recursive method. We use in this paper the required steady-state availability and the expected number of down machines to compute the maximum entropy approximation for the steady-state probability distributions of the number of down machines in the system, thus avoiding the use of supplementary variable technique. In numerical examples, comparison of the steady-state probability distributions obtained by the exact approach and the approximation procedure are provided to illustrate the effectiveness of maximum entropy approach.