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

Transient behavior of M[x]/G/1 Retrial Queueing Model with Non Persistent Customers, Random break down, Delaying Repair and Bernoulli Vacation

In this paper we consider a single server batch arrival non-Markovian retrial queueing model with non persistent customers. In accordance with Poisson process, customers arrive in batches with arrival rate  and are served one by one with first come first served basis. The server is being considered as unreliable that it may encounter break down at any time. In order to resume its service the server has to be sent for repair, but the repair does not start immediately so that there is a waiting time before the repair process. The customer, who finds the server busy upon arrival, can either join the orbit with probability p or he/she can leave the system with probability 1-p. More details can be found in the full paper. Key words: Batch size, break down, delay time, transient solution, steady solution,  reliability indices

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

Analysis of an M[X]/G/1 Feedback Retrial Queue with Two Phase Service, Bernoulli Vacation, Delaying Repair and Orbit Search

In this paper, we considered a batch arrival feedback retrial queue with two phase of service under Bernoulli vacation schedule and orbit search. At the arrival epoch of a batch, if the server is busy, under repair or on vacation then the whole batch joins the orbit. Where as if the server is free, then one of the arriving customers starts his service immediately and the rest join the orbit. At the completion epoch of each service, the server either goes for a vacation or may wait for serving the next customer. While the server is working with any phase of service, it may breakdown at any instant and the service channel will fail for a short interval of time. The repair process does not start immediately after a breakdown and there is a delay time for repair to start. After vacation completion, the server searches for the customers in the orbit (i.e. customer in the orbit, if any taken for service immediately) or remains idle. The probability generating function of the number of customers in the system and orbit are found using the supplementary variable technique. The mean numbers of customers in the system/orbit and special cases are analyzed. The effects of various parameters on the performance measure are illustrated numerically. Keywords: Feedback, retrial queue, Bernoulli vacation, delaying repair, orbit searc

Analysis of repairable M[X]/(G1,G2)/1 - feedback retrial G-queue with balking and starting failures under at most J vacations

In this paper, we discuss the steady state analysis of a batch arrival feedback retrial queue with two types of services and negative customers. Any arriving batch of positive customers finds the server is free, one of the customers from the batch enters into the service area and the rest of them get into the orbit. The negative customer, is arriving during the service time of a positive customer, will remove the positive customer in-service and the interrupted positive customer either enters the orbit or leaves the system. If the orbit is empty at the service completion of each type of service, the server takes at most J vacations until at least one customer is received in the orbit when the server returns from a vacation. While the busy server may breakdown at any instant and the service channel may fail for a short interval of time. The steady state probability generating function for the system size is obtained by using the supplementary variable method. Numerical illustrations are discussed to see the effect of the system parameters

Analysis of repairable M[X]/(G1,G2)/1 - feedback retrial G-queue with balking and starting failures under at most J vacations

In this paper, we discuss the steady state analysis of a batch arrival feedback retrial queue with two types of service and negative customers. Any arriving batch of positive customers finds the server is free, one of the customers from the batch enters into the service area and the rest of them join into the orbit. The negative customer, arriving during the service time of a positive customer, will remove the positive customer in-service and the interrupted positive customer either enters into the orbit or leaves the system. If the orbit is empty at the service completion of each type of service, the server takes at most J vacations until at least one customer is received in the orbit when the server returns from a vacation. The busy server may breakdown at any instant and the service channel will fail for a short interval of time. The steady state probability generating function for the system size is obtained by using the supplementary variable method. Numerical illustrations are discussed to see the effect of system parameters

Analysis of batch arrival bulk service queue with multiple vacation closedown essential and optional repair

The objective of this paper is to analyze an queueing model with multiple vacation, closedown, essential and optional repair. Whenever the queue size is less than , the server starts closedown and then goes to multiple vacation. This process continues until at least customer is waiting in the queue. Breakdown may occur with probability when the server is busy. After finishing a batch of service, if the server gets breakdown with a probability , the server will be sent for repair. After the completion of the first essential repair, the server is sent to the second optional repair with probability . After repair (first or second) or if there is no breakdown with probability , the server resumes closedown if less than ` \u27 customers are waiting. Otherwise, the server starts the service under the general bulk service rule. Using supplementary variable technique, the probability generating function of the queue size at an arbitrary time is obtained for the steady-state case. Also some performance measures and cost model are derived. Numerical illustrations are presented to visualize the effect of various system parameters

Analysis of M[X1],M[X2]/G1,G2/1 retrial queueing system with priority services, working breakdown, collision, Bernoulli vacation, immediate feedback, starting failure and repair

This paper considers an M[X1] , M[X2] /G1,G2/1 general retrial queueing system with priority services. Two types of customers from different classes arrive at the system in different independent compound Poisson processes. The server follows the non-pre-emptive priority rule subject to working breakdown, Bernoulli vacation, starting failure, immediate feedback, collision and repair. After completing each service, the server may go for a vacation or remain idle in the system. The priority customers who find the server busy are queued in the system. If a low-priority customer finds the server busy, he is routed to orbit that attempts to get the service. The system may become defective at any point of time while in operation. However, when the system becomes defective, instead of stopping service completely, the service is continued to the interrupted customer only at a slower rate. Using the supplementary variable technique, the joint distribution of the server state and the number of customers in the queue are derived. Finally, some performance measures are obtained

Unreliable Server Retrial Queue with Optional Service and Multi-Phase Repair

In this paper, the retrial unreliable server queue with batch arrivals is considered. The arrival rates of the units are different and dependent upon the joining probabilities according to the server status. On arrival, if unit finds the busy server, he may retry for the service after a random duration of time. The server facilitates the essential service and optional service, if opted after essential service. Moreover, the server is unreliable and subject to the breakdown while rendering essential/optional service. The failed server may immediately undergo for the compulsory multiphase repair or may wait to start the repair due to any technical reasons. The server can also avail the optional vacation under the Bernoulli schedule after finish the service of each unit or may continue to serve the next unit. The variables corresponding to elapsed times of general distributed service process, retrial process, repair process and vacation duration, as supplementary variables and used to frame the governing equations. By using the probability generating functions of joint distributions of the units at different states of the server, the performance characteristics of the system are derived. To validate the results, the sensitivity analysis has been performed by taking the numerical illustration