Analysis of Single Server Queueing System With Batch Service Under Multiple Vacations With Loss And Feedback

Consider a single server queueing system with foxed batch service under multiple vacations with loss and feedback in which the arrival rate ? follows a Poisson process and the service time follows an exponential distribution with parameter ?. Assume that the system initially contain k customers when the server enters the system and starts the service in batch. The concept of feedback is incorporated in this model (i.e) after completion of the service, if this batch of customers dissatisfied then this batch may join the queue with probability q and with probability (1-q) leaves the system. This q is called a feedback probability. After completion of the service if he finds more than k customers in the queue then the first k customers will be taken for service and service will be given as a batch of size k and if he finds less than k customers in the queue then he leaves for a multiple vacation of exponential length ?. The impatient behaviour of customer is also studied in this model (i.e) the arriving customer may join the queue with probability p when the server is busy or in vacation. This probability p is called loss probability. This model is completely solved by constructing the generating function and Rouche’s theorem is applied and we have derived the closed form solutions for probability of number of customers in the queue during the server busy and in vacation. Further we are providing the analytical solution for mean number of customers and variance of the system. Numerical studies have been done for analysis of mean and variance for various values of ?, µ, ?, p, q and k and also various particular cases of this model have been discussed. Keywords : Single Server , Batch Service, Loss and Feedback,  Multiple vacations, Steady state distribution

(R2051) Analysis of MAP/PH1, PH2/2 Queueing Model with Working Breakdown, Repairs, Optional Service, and Balking

In this paper, a classical queueing system with two types of heterogeneous servers has been considered. The Markovian Arrival Process (MAP) is used for the customer arrival, while phase type distribution (PH) is applicable for the offering of service to customers as well as the repair time of servers. Optional service are provided by the servers to the unsatisfied customers. The server-2 may get breakdown during the busy period of any type of service. Though the server- 2 got breakdown, server-2 has a capacity to provide the service at a slower rate to the current customer who is receiving service when the moment of server-2 struck with breakdown. In the period of vacation/closedown of server-1 and the server-2 is in working breakdown or under repair process, the arrival of customers may balk the system due to the impatient. Stability conditions has derived for our system and the stationary probability vector was evaluated by using the matrix analytical method. This model also examined at the analysis of busy period,waiting time distribution and system performance measures. The numerical illustrations are provided with the aid of two dimensional and three dimensional graphs

(R1975) MAP/PH(1), PH(2)/2 Queue with Multiple Vacation, Optional Service, Consultations and Interruptions

Two types of services are explored in this paper: regular server and main server, both of which provide both regular and optional services. Customers arrive using the Markovian Arrival Process (MAP), and service time is allocated based on phase type. The regular server uses the main server as a resource. Customers’ service at the primary server is disrupted as a result. When the queue size is empty, the main server can take several vacations. This system has been represented as a QBD Process that investigates steady state with the use of matrix analytic techniques, employing finite-dimensional block matrices. Our model’s waiting time distribution has been examined in more detail during the busy times. The system’s key parameters are assessed, and a few graphs and numerical representations are constructed

Priority Queueing System with a Single Server Serving Two Queues M[X1],M[X2]/G1,G2/1 with Balking and Optional Server Vacation

In this paper we study a vacation queueing system with a single server simultaneously dealing with an M[x1] /G1/1 and an M[x2] /G2/1 queues. Two classes of units, priority and non-priority, arrive at the system in two independent compound Poisson streams. Under a non-preemptive priority rule, the server provides a general service to the priority and non-priority units. We further assume that the server may take a vacation of random length just after serving the last customer in the priority unit present in the system. If the server is busy or on vacation, an arriving non-priority customer either join the queue with probability b or balks(does not join the queue) with probability (1 - b). The time dependent probability generating functions have been obtained in terms of their Laplace transforms and the corresponding steady state results are obtained explicitly. Also the average number of customer in the priority and the non-priority queue and the average waiting time are derived. Numerical results are computed

Transient Solution of M[X1],M[X2]/G1,G2/1 with Priority Services, Modified Bernoulli Vacation, Bernoulli Feedback, Breakdown, Delaying Repair and Reneging

This paper considers a queuing system which facilitates a single server that serves two classes of units: high priority and low priority units. These two classes of units arrive at the system in two independent compound Poisson processes. It aims to decipher average queue size and average waiting time of the units. Under the pre-emptive priority rule, the server provides a general service to these arriving units. It is further assumed the server may take a vacation after serving the last high priority unit present in the system or at the service completion of each low priority unit present in the system. Otherwise, he may remain in the system. Also, if a high priority unit is not satisfied with the service given it may join the tail of the queue as a feedback unit or leave the system. The server may break down exponentially while serving the units. The repair process of the broken server is not immediate. There is a delay time to start the repair. The delay time to repair and repair time follow general distributions. We consider reneging to occur for the low priority units when the server is unavailable due to breakdown or vacation. We concentrate on deriving the transient solutions by using supplementary variable technique. Further, some special cases are also discussed and numerical examples are presented

An M^x/G(a,b)/1 queue with breakdown and delay time to two phase repair under multiple vacation

In this paper, we consider an Mx /G(a,b)/1 queue with active breakdown and delay time to two phase repair under multiple vacation policy. A batch of customers arrive according to a compound Poisson process. The server serves the customers according to the “General Bulk Service Rule” (GBSR) and the service time follows a general (arbitrary) distribution. The server is unreliable and it may breakdown at any instance. As the result of breakdown, the service is suspended, the server waits for the repair to start and this waiting time is called as „delay time‟ and is assumed to follow general distribution. Further, the repair process involves two phases of repair with different general (arbitrary) repair time distributions. Immediately after the repair, the server is ready to start its remaining service to the customers. After each service completion, if the queue length is less than \u27a\u27, the server will avail a multiple vacation of random length. In the proposed model, the probability generating function of the queue size at an arbitrary and departure epoch in steady state are obtained using the supplementary variable technique. Various performance indices, namely mean queue length, mean waiting time of the customers in the queue etc. are obtained. In order to validate the analytical approach, we compute numerical results

Time Dependent Solution of Batch Arrival Queue with Second Optional Service, Optional Re-Service and Bernoulli Vacation

This paper deals with an M[X]/G/1 queues with second optional service, optional re-service and Bernoulli vacations. Each customer undergoes first phase of service after completion of service, customer has the option to repeat or not to repeat the first phase of service and leave the system without taking the second phase or take the second phase service. Similarly after the second phase service he has yet another option to repeat or not to repeat the second phase service. After each service completion, the server may take a vacation with probability 1-theta or may continue staying in the system with probability . The service and vacation periods are assumed to be general. The time dependent probability generating functions have been obtained in terms of their Laplace transforms and the corresponding steady state results have been obtained explicitly. Also the average number of customers in the queue and the waiting time are also derived. Keywords: Batch arrival, Second optional service, Optional re-service, Average queue size, Average waiting time

Batch arrival bulk service queue with unreliable server, second optional service, two different vacations and restricted admissibility policy

This paper is concerned with batch arrival queue with an additional second optional service to a batch of customers with dissimilar service rate where the idea of restricted admissibility of arriving batch of customers is also introduced. The server may take two different vacations (i) Emergency vacation-during service the server may go for vacation to an emergency call and after completion of the vacation, the server continues the remaining service to a batch of customers. (ii) Bernoulli vacation-after completion of first essential or second optional service, the server may take a vacation or may remain in the system to serve the next unit, if any. While the server is functioning with first essential or second optional service, it may break off for a short period of time. As a result of breakdown, a batch of customers, either in first essential or second optional service is interrupted. The service channel will be sent to repair process immediately. The repair process presumed to be general distribution. Here, we assumed that the customers just being served before server breakdown wait for the server to complete its remaining service after the completion of the repair process. We derived the queue size distribution at a random epoch and at a departure epoch under the steady state condition. Moreover, various system performance measures, the mean queue size and the average waiting time in the queue have been obtained explicitly. Some particular cases and special cases are determined. A numerical result is also introduced

Analysis of Two Stage M[X1],M[X2]/G1,G2/1 Retrial G-queue with Discretionary Priority Services, Working Breakdown, Bernoulli Vacation, Preferred and Impatient Units

In this paper, we study M[X1] , M[X2] /G1 ,G2 /1 retrial queueing system with discretionary priority services. There are two stages of service for the ordinary units. During the first stage of service of the ordinary unit, arriving priority units can have an option to interrupt the service, but, in the second stage of service it cannot interrupt. When ordinary units enter the system, they may get the service even if the server is busy with the first stage of service of an ordinary unit or may enter into the orbit or leave the system. Also, the system may breakdown at any point of time when the server is in regular service period. During the breakdown period, the interrupted priority unit will get the fresh service at a slower rate but the ordinary unit can not get the service and the server will go for repair immediately. During the ordinary unit service period, the arrival of negative unit will interrupt the service and it may enter into an orbit or leave the system. After completion of each priority unit’s service, the server goes for a vacation with a certain probability. We allow reneging to happen during repair and vacation periods. Using the supplementary variable technique, the Laplace transforms of time-dependent probabilities of system state are derived. From this, we deduce the steady-state results. Also, the expected number of units in the respective queues and the expected waiting times, are computed. Finally, the numerical results are graphically expressed

Non Markovian Queue with Two Types service Optional Re-service and General Vacation Distribution

We consider a single server batch arrival queueing system, where the server provides two types of heterogeneous service. A customer has the option of choosing either type 1 service with probability p1 or type 2 service with probability p2 with the service times follow general distribution. After the completion of either type 1 or type 2 service a customer has the option to repeat or not to repeat the type 1 or type 2 service. As soon as the customer service is completed, the server will take a vacation with probability θ or may continue staying in the system with probability 1 -θ: The re-service periods and vacation periods are assumed to be general. Using supplementary variable technique, the Laplace transforms of time dependent probabilities of system state are derived and thus we deduce the steady state results. We obtain the average queue size and average waiting time. Some system performance measures and numerical illustrations are discussed