Mx G 1 Queueing System With Breakdowns and Repairs

Malaya Journal of Matematik

In this paper, we consider the queueing system where the batch of customers arrive at the system according to the compound Poisson process and two types of service, each of which has an optional reservice is provided to the server under Bernoulli vacation. After completion of each type of service, the customer may go for reservice of the same type of service without joining the tail of the queue or they may depart the system. An unpredictable breakdown may occur at any moment during the functioning of the server with any type of service or re-service and at that situation, the service channel will breakoff for a short period of time. A breakdown in a busy server is represented by the arrival of a negative customer which consequently leads to the loss of the customer who is in service. Delay time is referred to as the waiting time of the server for the two phase of repair to start. By considering elapsed service time as the supplementary variable, the PGF of the number of customers in the queue at a random epoch is derived and this PGF is further used to establish explicitly some of the following performance measures namely various states of the system, the mean queue length, and the mean waiting time in the queue. At last, some particular cases are discussed and the numerical illustrations are provided.

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...

Quality Technology and Quantitative Management, 2018

This paper deals with a batch arrivals queue with general retrial time, breakdowns, repairs and reserved time. Here we assume that customers arrive according to compound Poisson processes. Any arriving batch of primary customers finds the server free, one of the customers from the batch enters into the service area and the rest of them join into the orbit. The primary customers who find the server busy or failed are allowed to balk or are queued in the orbit in accordance with FCFS retrial policy. The customer whose service is interrupted can stay at the server waiting for repair or enter into service orbit. After the repair is completed, the server resumes service immediately if the customer in service has remained in the service position. This model has a potential applications in various fields, such as in the cognitive radio network and the manufacturing systems. By using supplementary variables technique, we carry out an extensive analysis of the considered model. Then, we obtain some important performance measures, stochastic decomposition property of the system size distribution and the reliability indices. Next, by setting the appropriate parameters, some special cases are discussed. Finally, some numerical examples and cost analysis are presented.

2019

This paper deals with a batch arrival queueing system prepared with a single server providing service to a batch of customers with dissimilar service rate in two fluctuating modes of service. By using the supplementary variable technique, we derive the probability generating function of the number of customers in the queue at a random epoch under the steady state conditions. Performance measures like mean queue size has been obtained explicitly.

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.

2014

In this paper, we study the optimal operation of a single removable and non-reliable server in a Markovian queuing system under steady-state conditions. The system is in idle state before the arrival of customer and after the arrival of customer it is in working state, it may breakdown and systems goes to vacation during that period. Here, there is bulk arrival of customer and bulk service provided to them. Inter-arrival and service time distributions of customers are assumed to be exponentially distribution. Breakdown and repair time distributions of the server are assumed to be exponentially distributed. Key word: Inter-arrival, service time, Break down, repair time, Bulk service, Non reliable server

IAEME PUBLICATION, 2020

We analyze an M/G/1 queue with Poisson arrivals, three optional services, random breakdowns and deterministic repair. The first service is essential and other three services are optional. All arriving customers demand first essential service and after completion of first service, the customer has the option of choosing any one of the three optional services with probability Otherwise the customer can leave the system without choosing any of the three optional services with probability . The server is subject to random breakdowns and just after a breakdown the server undergoes repairs of fixed (constant) duration ( ) The supplementary variable technique is used to find explicitly the probability generating function of the number in the system and the mean number in the system

International Journal of Internet and Enterprise Management, 2012

This paper studies a general retrial M X /G/1 queue with an additional phase of second optional service and Bernoulli vacation where breakdowns occur randomly at any instant while servicing the customers. If an arriving batch finds that the server is busy in providing either first essential service (FES)/second optional service (SOS) or on vacation then arriving batch enters an orbit called retrial queue. Otherwise, one customer from arriving batch starts to be served by the server while the rest join the orbit. The vacation times and service times of both first essential and second optional services are assumed to be general distributed while the retrial times are exponential distributed. Introducing supplementary variables and by employing embedded Markov chain technique, we derive some important performance measures of the system such as average orbit size, average queue size, mean waiting time, expected lengths of busy period, etc. Numerical results have been facilitated to illustrate the effect of different parameters on several performance measures.

2014

Here we will study bulk service to customer under optimal operation of a single removable and non-reliable server in Markovian queueing system under steady-state conditions. The decision maker can turn a server on at customer’s arrival or off at service completion. Here it is assumed that the server may breakdown only if working and requires repair at repair facility. Inter-arrival and service time distributions of the customers are assumed to be exponentially distributed. Breakdown and repair time distributions of the server are assumed to be exponentially distributed. The following cost structure is incurred to be system; a holding cost for each customer in the system per unit time, cost per unit time when a server fails, and fixed costs for turning the server on or off. The expected cost function per unit time is developed to obtain the optimal operating policy at minimum cost.

International Journal of Scientific and Innovative Mathematical Research

The retrial queueing systems are characterized as a customer, arriving when all servers are busy, leaves the system, but after some time makes a demand to the service facility again. These models play a vital role in computer and telecommunication networks. For example, in a telephone system, a customer might receive a busy signal due to a lack of capacity. Such a customer is not allowed to queue, but will try their luck again after some random time. Between trials, the blocked customers join a pool of unsatisfied customers called 'orbit'. Queueing models with negative customers, otherwise known as G-queues, were first introduced by Gelenbe(1989). In simple words, the arrival of a negative customer has the effect of removing a positive (ordinary) customer from the system. The characteristics of negative arrivals are, (i) arrival of a negative customer eliminates all the customers in the system (catastrophe), (ii) arrival of a negative customer that removes the customer in service, (iii) arrival of a negative customer that deletes the customer at the end of queue. Negative arrivals have been interpreted as virus, orders or inhibitor signals. In this paper we consider the negative arrival of type (ii). Retrial queues can be applied in telecommunication networks, switching systems and computer networks, and there has been an increasing interest in the analysis of retrial systems in recent years.

arXiv (Cornell University), 2015

Efficient use of call center operators through technological innovations more often come at the expense of added operation management issues. In this paper, the stationary characteristics of an M/G/1 retrial queue is investigated where the single server, subject to active failures, primarily attends incoming calls and directs outgoing calls only when idle. The incoming calls arriving at the server follow a Poisson arrival process, while outgoing calls are made in an exponentially distributed time. On finding the server unavailable (either busy or temporarily broken down), incoming calls intrinsically join the virtual orbit from which they re-attempt for service at exponentially distributed time intervals. The system stability condition along with probability generating functions for the joint queue length distribution of the number of calls in the orbit and the state of the server are derived and evaluated numerically in the context of mean system size, server availability, failure frequency and orbit waiting time.

2013

This paper investigates a single server queue with Poisson arrivals, general (arbitrary) service time distributions and Bernoulli vacation subject to random breakdowns. However, after the completion of a service, the server will take Bernoulli vacation, that the server make take a vacation with probability θ or may continue to stay in the system with probability 1 − θ for serving the next customer, if any. In addition to this, the vacation period of the server has two phases in which first phase is compulsory followed by the second phase in a such way a that the server may choose second phase with probability p or may return back to the system with probability 1−p and the vacation time follows general (arbitrary) distribution. The system may breakdown at random with mean break down rate α and repair process starts immediately in which the repair time follows exponential distribution with mean repair rate β. We obtain the time dependent probability generating functions in terms of th...

International Journal of Operational Research, 2010

International Journal of Difference Equations (IJDE), 2022

The present paper describes a Mx/M/Gƞ/1 queueing system in which the units arrive in batches of variable size and the arriving units are served one by one in order of their arrival. The service times of the server follows exponential distribution. The service may be interrupted due to random breakdowns by the occurrence of catastrophe. When the service channel fails due to catastrophe, it stops providing service to the units and wait for the repairs to be started. Assume that the delay times, due to repair, are distributed according to Gamma distribution with parameter . Once repair is completed the service channel instantaneously starts service. A set of differential-difference equations have been framed for this model through probability generating function and Laplace transform and steady-state probabilities have been derived. The expression for average number of units in the system has also been derived.

In this paper, we study the strong stability in the M=G=1 queueing system with breakdowns and repairs after perturbation of the breakdown's parameter. Using the approximation conditions in the classical M=G=1 system, we obtain stability inequalities with exact computation of the constants. Thus, we can approximate the characteristics of the M=G=1 queueing system with breakdowns and repairs by the classical M=G=1 corresponding ones.