Single Server Queue Single Server Queue with Server Breakdown Including Priority and Varying Rates (original) (raw)
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Single Server Queue with Server Breakdown Including Priority and Varying Rates
Journal of King Abdulaziz University-Engineering Sciences, 1999
A single server queueing system with service interruption due to failure of the major/minor service unit has been studied. The job's inter-arrival time as well as the failure time and the repair time of the major/minor service unit are assumed to be exponentially distributed. The arrival of the jobs to the service facility consisting of major and minor units depends upon the state of the server which may be either operational (partially operational) mode with both units (only major unit) functioning well or in breakdown mode due to failure of the major unit or both units. The failure of service units may occur individually or due to some common cause. The repair rate of the major service unit is also affected with the state of the minor service unit. The repair of the major unit is given pre-emptive priority over the repair of the minor unit. The steady state queue size distribution for various states has been obtained by using generating function method. The average number of jobs in various states, server availability etc. have been derived explicitly.
An M/G/1 queue with two phases of service subject to the server breakdown and delayed repair
Applied Mathematical Modelling, 2009
This paper deals with the steady-state behaviour of an M/G/1 queue with an additional second phase of optional service subject to breakdowns occurring randomly at any instant while serving the customers and delayed repair. This model generalizes both the classical M/G/1 queue subject to random breakdown and delayed repair as well as M/G/1 queue with second optional service and server breakdowns. For this model, we first derive the joint distributions of state of the server and queue size, which is one of chief objectives of the paper. Secondly, we derive the probability generating function of the stationary queue size distribution at a departure epoch as a classical generalization ofPollaczek-Khinchin formula. Next, we derive Laplace Stieltjes transform of busy period distribution and waiting time distribution. Finally, we obtain some important performance measures and reliability indices of this model. Choudhury and Paul [5] investigated such a model under Bernoulli feedback mechanism. In this context Krishnakumar and Arivudainambi in [6] obtained the explicit expression for transient probabilities for this type of finite capacity model M/G/1/1 Bernoulli feedback queue and M/G/1/1 queue with unreliable server . Recently, Wang [8] investigated such a model with the assumption that the server is subject to breakdowns and repairs, and some critical reliability indices are obtained. More recently, Ke [9] extended the result for a multi-optional service system where concept of setup time is also introduced.
Mathematics
In this paper, we discuss a non-Markovian batch arrival general bulk service single-server queueing system with server breakdown and repair, a stand-by server, multiple vacation and re-service. The main server's regular service time, re-service time, vacation time and stand-by server's service time are followed by general distributions and breakdown and repair times of the main server with exponential distributions. There is a stand-by server which is employed during the period in which the regular server remains under repair. The probability generating function of the queue size at an arbitrary time and some performance measures of the system are derived. Extensive numerical results are also illustrated.
Quality Technology & Quantitative Management, 2017
In this paper, we study the steady state behaviour of an M/G/1 queue with two types of general heterogeneous service and optional repeated service subject to server's breakdowns occurring randomly at any instant while serving the customers and delayed repair. We assume that customers arrive to the system according to a Poisson process with rate 'λ' and the server provides two types of general heterogeneous service. At the beginning of a service, a customer has the option to choose any one type of service. After completion of either type of service, the customer has the further option to repeat the same type of service. For this model, we first derive the joint distribution of state of the server and queue size by considering both elapsed and remaining time, which is one of the objectives of this paper. Secondly, we derive the probability generating function of the stationary queue size distribution at departure epoch. Next, we derive Laplace-Stieltjes transform of busy period distribution and waiting time distribution. Finally, we obtain some important performance measure and reliability indices of this model.
Transient Analysis of a Repairable Single Server Queue with Working Vacations and System Disasters
Communications in Computer and Information Science, 2019
This study investigates the repairable single server queue with working vacations and system disasters. The server allows to take a working vacation if there is no any customers in the system. There is a possibility of breakdowns happening in a system. When the system occurs server breakdowns, the server goes to the failure state and all customers in the queue are flushed away. The repairing process starts immediately, when the server comes to the failure state. The explicit expression for system size probabilities of the queueing system is derived in terms of the modified Bessel function of first kind using the probability generating function method, Laplace transform and continued fractions. Additionally, the mean and variance for number of jobs in the system at time t are derived as the performance measures. Finally, a numerical example is presented to study the behavior of the system.
Communications in Statistics - Theory and Methods, 2019
This article deals with the N-policy with setup time of an unreliable M X /G/1 queue provides two types of general heterogeneous service under optional repeated service and delayed repair. The server is turned off each time as soon as the system becomes empty and waits until the queue size becomes exactly. As soon as exactly N (! 1) customers accumulate in the system, the server has to undertake a set up period before starting the busy period. The steady state queue size distributions by considering both elapsed and remaining times as well as various system characteristics has been derived for this model.
We study the behavior of a batch arrival queuing system equipped with a single server providing general arbitrary service to customers with different service rates in two fluctuating modes of service. In addition, the server is subject to random breakdown. As soon as the server faces breakdown, the customer whose service is interrupted comes back to the head of the queue. As soon as repair process of the server is complete, the server immediately starts providing service in mode 1. Also customers waiting for service may renege (leave the queue) when there is breakdown or when server takes vacation. The system provides service with complete or reduced efficiency due to the fluctuating rates of service. We derive the steady state queue size distribution. Some special cases are discussed and numerical illustration is provided to see the effect and validity of the results.
International Journal of Reliability and Safety, 2022
In this paper, we investigate the performance of batch service queue model with second optional service, repairable breakdown and warm standby server. Both primary operating and warm standby servers provide First Essential Service (FES) and Second Optional Service (SOS) to customers, wherein FES is all arriving customers and only some of them may further request the SOS. The service times, failure times and repair times of both primary operating and warm standby server are assumed to follow exponential distributions. We use Runge-Kutta method to obtain the transient state probabilities and matrix-decomposition method to obtain the steady-state probabilities of the model. Also, a cost model is presented to determine the optimal service rates so that the expected cost is minimised. Finally, the effect of the model parameters on the system behaviour is demonstrated through numerical results and discussions.
Server Breakdown and Delayed Repair in Three Phases of Service for an M/G/1 Retrial Queueing System
2020
This paper deals with an unreliable server having three phases of heterogeneous service on the basis of M/G/1 queueing system. We suppose that customers arrive and join the system according to a Poisson’s process with arrival rate λ. When the server is working with any phase of service, it may breakdown at any instant. After breakdown, when the server is sent for repair then server stops its service and arrival customers are waiting for repair, which we may called as waiting period of the server. This waiting time stands for delay time/delay repair. In this model, first we derive the joint probability distribution for the server. Secondly, we derive the probability generating function of the stationary queue size distribution at a departure epoch as a classical generalisation of Pollaczek Khinchin formula. Third, we derive Laplace Stieltjes transform of busy period distribution and waiting time distribution. Finally, we obtain some important performance measures and reliability anal...