Analysis of Bulk Queueing Model with Non Reliable Server

OPSEARCH, 2011

This paper deals with the optimal operation of a single removable and unreliable server in an N-policy two-phase M X /M/1 queueing system with gating, server startups and unpredictable breakdowns. Arrivals occur in batches according to a compound Poisson process and waiting customers receive batch service all at a time in the first phase and proceed to the second phase to receive individual service. Customers who arrive during the batch service are not allowed to enter the same batch. As soon as the system becomes empty the server leaves for a vacation of random length. When the queue length reaches a threshold value N (N 1 ≥) the server is immediately turned on but is temporarily unavailable to serve the waiting batch of customers. The server needs a startup time before providing batch service in the first phase. The server is subject to breakdowns during individual service according to a Poisson process and repair times of the server follow an exponential distribution. Explicit expressions for the steady state distribution of the number of customers in the system and hence the expected system length is derived. The total expected cost function is developed to determine the optimal threshold of N at a minimum cost. Numerical experiment is performed to validate the analytical results. The sensitivity analysis has been carried out to examine the effect of different parameters in the system.

Journal of physics, 2018

In this paper, we examine a two-stage queueing system where the arrivals are Poisson with rate depends on the condition of the server to be specific: vacation, pre-service, operational or breakdown state. The service station is liable to breakdowns and deferral in repair because of non-accessibility of the repair facility. The service is in two basic stages, the first being bulk service to every one of the customers holding up on the line and the second stage is individual to each of them. The server works under N-policy. The server needs preliminary time (startup time) to begin batch service after a vacation period. Startup times, uninterrupted service times, the length of each vacation period, delay times and service times follows an exponential distribution. The closed form of expressions for the mean system size at different conditions of the server is determined. Numerical investigations are directed to concentrate the impact of the system parameters on the ideal limit N and the minimum base expected unit cost.

—In this paper we consider an M/M/1 queueing system with non-reliable server. When the server is in normal state, the service error (or failure) occurs according to a Poisson process. In the error state, the server needs to be repaired at a repair facility with exponential repair time according to the threshold policy. The repair starts only when the number of customers in the system reaches some prespecified threshold level q ≥ 1. We perform a steady-state analysis of the continuous-time Markov chain describing the system behavior and calculate optimal threshold level to minimize the long-run average losses given by the cost structure.

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.

2017

The objective of this paper is to analyze M/Hk/1 queueing system with setup and unreliable server for both finite and infinite capacity under N-policy. The customers arrive to the system in accordance with Poisson process. The service times follow the k-type hyper exponential distribution. The breakdown and repair times are assumed to follow a negative exponential distribution. The arriving customers may balk, depending upon the number of customers present in the system. The generating function method is used to derive queue size distribution of the number of the customers present in the system. The cost function has been derived in term of cost elements related to different situations in order to determine the optimal operating policy. To explore the effects of various parameters on cost and other indices, the sensitivity analysis is carried out by taking numerical illustrations. MSC: 60J05.

This paper studies an M/G/1 repairable queueing system with multiple vacations and N-policy, in which the service station is subject to occasional random breakdowns. When the service station breaks down, it is repaired by a repair facility. Moreover, the repair facility may fail during the repair period of the service station. The failed repair facility resumes repair after completion of its replacement. Under these assumptions, applying a simple method, the probability that the service station is broken, the rate of occurrence of breakdowns of the service station, the probability that the repair facility is being replaced and the rate of occurrence of failures of the repair facility along with other performance measures are obtained. Following the construction of the long-run expected cost function per unit time, the direct search method is implemented for determining the optimum threshold N* that minimises the cost function.

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.

Computers & Operations Research, 2010

A repairable queueing model with a two-phase service in succession, provided by a single server, is investigated. Customers arrive in a single ordinary queue and after the completion of the first phase service, either proceed to the second phase or join a retrial box from where they retry, after a random amount of time and independently of the other customers in orbit, to find a position for service in the second phase. Moreover, the server is subject to breakdowns and repairs in both phases, while a start-up time is needed in order to start serving a retrial customer. When the server, upon a service or a repair completion finds no customers waiting to be served, he departs for a single vacation of an arbitrarily distributed length. The arrival process is assumed to be Poisson and all service and repair times are arbitrarily distributed. For such a system the stability conditions and steady state analysis are investigated. Numerical results are finally obtained and used to investigate system performance.

Research Square (Research Square), 2022

A system with two heterogeneous servers with impatient customers and finite capacity is considered. The admission control of customers is based on F-policy. This paper is based on M/M/2 queues, and presents the study of two heterogeneous servers where one server is the primary server and other is secondary. Customers are primarily served by the main server, who also takes customer's feedback into account. If customers are unsatisfied and opt for further service, the second server will serve them. The main server considered is unreliable, that is, it may breakdown at any time. The main server will go for vacation whenever there is no customer to serve. To analyze the performance of the model, we calculated the measures of performance and did the cost analysis. Impact of F-policy on system's estimated profit is shown. At last, conclusions are drawn.

Fourth International Conference on Advances in Information Processing and Communication Technology - IPCT 2016, 2016

We consider an M X /G/1 queuing system with breakdown and repairs, where batches of customers are assumed to arrive in the system according to a compound poisson process. While the server is being repaired, the customer in service either remains the service position or enters a service orbit and keeps returning, after repair the server must wait for the customer to return. The server is not allowed to accepte new customers until the customer in service leaves the system. We find a stability condition for this system. In the steady state the joint distribution of the server state and queue length is obtained, and some performance mesures of the system, such as the mean number of customers in the retrial queue and waiting time, and some numerical results are presented to illustrate the effect of the system parameters on the developed performance measures. Keywords-batch arrival, break down, repair. I. Introduction Retrial queuing systems have been widely used to model many practical problems arising in telephone switching systems, telecommunication networks, and computer systems. The main characteristic of these queues is that a customer who find the sever busy upon arrival joins the retrial group called orbit to repeat his request for service after some random time. For a systematic account of the fundamental methods and results on this topic the reader can refer to the survey papers of (

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...

International Journal of Mathematics in Operational Research, 2012

In this paper, we study the modelling and analysis of unreliable M/M/2/K queuing system under N-and F-policy with multi-optional phase repair and start-up. According to the (N, F) policy, the first server is turned on only when there are 'N' or more customers are available in the system. The second server is turned on when the queue length of the customers reaches its system capacity K. Furthermore, no more customers are allowed in the system till the number of the customers again drops up to a certain threshold level 'F'. The servers may break down during busy period. There is a provision of 'l' phase repairs to restore the servers. The first-phase repair is necessary but remaining (l-1) phases are optional. The lifetime, start-up time and repair time of the servers are assumed to be exponentially distributed. Using matrix method, transient probabilities of the system states are determined. To examine the effect of different parameters on various performance indices, the numerical results are provided by taking an illustration.

2016

The single server queueing model with control policies has a wide range of applications in the quality control of telephone switching systems and production systems. As a result, several variations of the controlled queueing model have been studied extensively in the literature. These studies give rise to several types of control policies. To name a few, Yadin and Naor [1] analyzed the control policy in which the server doesn't start service in idle period unless there are N customers in queue, which is called the N-policy. The T-policy is another control policy, introduced by Heyman [2]. The T-policy suggests turning the server into active state after a time interval of length T when the server is in the idle state. Later, the D-policy is proposed by Balachendran [3] which is the focus of this research. In the D-policy, the server remains inactive until a total of D service times are accumulated in the queue. Many variations of these control policies were suggested in the literature. A comprehensive survey paper in this topic is found in Crabill et al. [4] and Doshi [5,6]. In this research, we derive the steady state distribution of a bulk-arrival queueing model with an unreliable single server under D-policy. In the queueing model under consideration, the server is switched to the inactive mode upon the completion of each busy period. Later, the server is switched to active mode when the total service times of all customers waiting in the queue exceed some pre-specified value D. Further, it is assumed that the server is subject to unpredictable breakdowns. The server is repaired immediately when a breakdown occurs and then put back into service. The D-policy M/G/1 queueing system first introduced by Balachandran [3] , Balachandran and Tijms [7] , Gakis et al. [8] and many others in the literature, assume that the server has no breakdowns. Balachandran and Tijms proposed the optimal D-policy

International Journal of Mathematics Trends and Technology, 2018

This paper is concerned with the transient and steady state analysis of unreliable server batch arrival general bulk service queueing system with multiple vacation under a restricted admissibility policy of arriving batches. Arrivals occur in batches according to compound Poisson process. Unlike the usual batch arrival queueing system, the restricted admissibility policy differs during a busy period and a vacation period and hence all arriving batches are not allowed to join the system at all times. The service is done in bulk with minimum of 'a' customers and maximum of 'b' customers. The service time follows a general (arbitrary) distribution. In addition, the server subject to active breakdown. As soon as the breakdown occurs the server is sent for repair and the customer who was just being served before server breakdown waits for the remaining service to complete. In the proposed model, the transient and steady state results for queue size distribution by applying the supplementary variable technique are derived. Some performance measures, special and particular cases are also discussed. Numerical illustration is provided to see the effect and validity of the results.

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.