Bulk Queueing Models With Unreliable Server, Restricted Admissions and Threshold Policies For Vacations

Abstract

1. INTRODUCTION Queueing theory is concerned with the development of Mathematical models to find the behavior of a system that provides service for randomly arising demands. Since demands for service are assigned to be governed by some probability law, the theory of queues has been developed within the framework of the theory of stochastic processes. Queueing models with vacations can be found in manufacturing industries, production assembly line systems, call centers with multi-task employees, designing of local area networks and data communication systems, etc. The abstracts of the models proposed in the thesis are given below. 2. ANALYSIS OF A BATCH ARRIVAL SINGLE SERVICE QUEUEING SYSTEM WITH UNRELIABLE SERVER AND SINGLE VACATION ABSTRACT: In this chapter, the operating characteristics of an queueing system with unreliable server and single vacation are analyzed. Customers arrive according to the Poisson process with random arrival size. Arrival rate varies according to the server’s status: up or down. If there are one or more customers waiting in the queue, then the server will start the service and continue to do the service until the system becomes empty. At a service completion epoch, if there are no customers in the queue, then the server avails a vacation of random length. The server on his return from the vacation finds an empty system, remains in the system for the first one to arrive. Otherwise, he starts the service. The service is interrupted if break down occurs, and the server is immediately repaired. When the repair is completed, the server immediately returns for service. Breakdown times are exponentially distributed and the repair times follow general distribution. The model is studied by the embedded Markov chain technique and level crossing analysis. The probability generating function of the steady state system size at an arbitrary time is obtained. Various performance measures are obtained. Particular cases are also discussed. A cost model for the queueing system is discussed with a numerical illustration. 3. ANALYSIS OF A BATCH ARRIVAL GENERAL BULK SERVICE QUEUEING SYSTEM WITH MULTIPLE VACATIONS, SETUP TIME AND SERVER’S CHOICE OF ADMITTING RE-SERVICE ABSTRACT: This chapter concentrates on a bulk queueing system with multiple vacations, setup time and server’s choice of admitting re-service. At a service completion epoch, the leaving batch may request for re-service with probability π and it is not mandatory to accept it; the server admits this request with a probability . After the re-service or service completion without request for re-service, if the queue length is less than a, the server leaves for a vacation of random length. After this vacation, if the queue length is still less than ‘a’, the server leaves for another vacation and so on, until he finally finds at least ‘a’ customers waiting for service. At a vacation completion epoch, if the server finds at least ‘a’ customers waiting for service, he requires a setup time to start the service. After a setup time or on service completion or on re-service completion, if the server finds at least ‘a’ customers waiting for service say , he serves a batch of min ( ,b ) customers, where . The model is studied by the supplementary variable technique. The probability generating function (PGF) of queue length distributions at an arbitrary time epoch is obtained. Particular cases are also discussed. A cost model for the proposed queueing system has been developed. Various performance measures are derived. The effects of several parameters on the total average cost for the proposed queueing system and to optimize the cost, a numerical illustration is provided. 4. ANALYSIS OF A BATCH ARRIVAL GENERAL BULK SERVICE QUEUEING SYSTEM WITH VARIANT THRESHOLD POLICY FOR VACATIONS ABSTRACT: This chapter analyses a bulk arrival general bulk service queueing system with variant threshold policies for vacations. The server starts the service only if at least ‘a’ customers are waiting in the queue, and renders the service according to the bulk service rule with minimum of ‘a’ customers and maximum of ‘b’ customers until the queue length reaches less than ‘a’. On completion of a service, if the queue length is less than ‘a’, then the server performs a secondary job of type one, repeatedly, until the queue length reaches the threshold value ‘a’. At a secondary job of type one completion epoch, if the queue length is at least ‘a’, then the server performs another secondary job of type two in a faster way repeatedly, until the queue length reaches the threshold value ‘N’ ( ), and serves a batch of ‘b’ customers. Analytical treatment of this model is obtained by the supplementary variable technique. The primary focus of this chapter is optimizing the overall cost when the server is allotted for different secondary jobs of different lengths depending on the queue size. The model is studied by the supplementary variable technique. The probability generating function of the steady state queue size at an arbitrary time is obtained. A cost model for the proposed queueing system has been developed. Various performance measures are derived. The effects of several parameters on the total average cost for the proposed queueing system and to optimize the cost, a numerical illustration is provided. 5. ANALYSIS OF A BATCH ARRIVAL GENERAL BULK SERVICE QUEUEING SYSTEM WITH MULTIPLE VACATIONS AND RESTRICTED ADMISSIBILITY OF ARRIVING BATCHES ABSTRACT: This chapter analyses a bulk queueing system with multiple vacations under a restricted admissibility policy of arriving batches. Arrivals occur in bulk according to Poisson process. But all the arrivals are not considered for service. During the busy period of the server, the arrivals are admitted with probability ‘ ’, whereas with probability ‘ ’, they are admitted when the server is idle. Such assumption is quite meaningful in many real life situations. The service is done in bulk with minimum of ‘a’ customers and maximum of ‘b’ customers. The server is assigned for secondary jobs (vacations) repeatedly when the number of waiting jobs is inadequate to the process. The model is studied by the supplementary variable technique. The probability generating function of the steady state queue size at an arbitrary time is obtained. A cost model for the proposed queueing system has been developed. Various performance measures are derived. The effects of several parameters on the total average cost for the proposed queueing system and to optimize the cost, a numerical illustration is provided. 6. ANALYSIS OF A BATCH ARRIVAL GENERAL BULK SERVICE QUEUEING SYSTEM WITH INTERRUPTED VACATION ABSTRACT: This chapter analyses a batch arrival general bulk service queueing system with interrupted secondary job (vacation). At a service completion epoch, if the server finds at least ‘a’ customers waiting for service say , he serves a batch of min ( ,b ) customers, where . On the other hand, if the queue length is at the most ‘a-1’, the server leaves for a secondary job (vacation) of random length. It is assumed that the secondary job is interrupted abruptly and the server resumes for primary service, if the queue size reaches ‘a’ during the secondary job period. On completion of the secondary job, the server remains in the system (dormant period) until the queue length reaches ‘a’. The model is studied by the supplementary variable technique. The probability generating function of the steady state queue size at an arbitrary time is obtained. A cost model for the proposed queueing system has been developed. Various performance measures are derived. The effects of several parameters on the total average cost for the proposed queueing system and to optimize the cost, a numerical illustration is provided. 7. ANALYSIS OF A BATCH ARRIVAL SINGLE SERVICE RETRIAL QUEUEING SYSTEM WITH THRESHOLD POLICY FOR MODIFIED VACATIONS ABSTRACT: This chapter considers a batch arrival single server retrial queue with modified vacations under N – policy, where the retrial follows classical retrial policy. If an arriving batch of customers finds the server busy or on vacation, then the entire batch join the orbit in order to seek service again. Otherwise one customer from the arriving batch receives the service, while the rest join the orbit. The customers in the orbit will try for service one by one when the server is idle with a classical retrial policy and the retrial rate is ‘jv’, where ‘j’ is the size of the orbit. At a service completion epoch, if the number of customers in the orbit is zero, then the server leaves for a vacation (secondary job) of random length. At a vacation completion epoch, if the orbit size is at least N, then the server remains in the system to render service for the customers (arriving primary customer tries to get the service either upon arrival or after visiting the orbit). On the other hand, if the number of customers in the orbit is less than ‘N’ at a vacation completion epoch, the server avails multiple vacations subject to maximum ‘M’ repeated vacations. After availing ‘M’ vacations, irrespective of the orbit size, the server waits in the system to render service. This retrial system has potential applications in communication networks, inventory models, production system, manufacturing industries, etc. The model is studied by the supplementary variable technique. The probability generating function of the steady state orbit size at an arbitrary time and some performance measures are obtained. A cost model for the queueing system is developed. A numerical illustration is also provided. 8. ANALYSIS OF A BATCH ARRIVAL GENERAL BULK SERVICE RETRIAL QUEUEING SYSTEM WITH CONSTANT RETRIAL RATE ABSTRACT: In this chapter, a batch arrival general bulk service retrial queueing model with constant retrial rate is analyzed. The primary customers arrive in bulk according to Poisson process and they get service under general bulk service rule with minimum of one customer and maximum of ‘b’ customers. If the arriving batch of customers, of size ‘ ’, , finds the server free , then all of them get service immediately; while, if the size of the arriving batch is more than ‘b’, then, ‘b’ customers enter the service station and the remaining customers join the orbit. However, if an arriving batch of customers finds the server busy, then the entire batch joins the orbit in order to seek service again. The customers in the orbit will try for service one by one with a constant retrial rate ‘v’ when the server is idle. The model is studied by the supplementary variable technique. The probability generating function of the steady state orbit size at an arbitrary time and some performance measures are obtained. A cost model for the queueing system is developed. A numerical illustration is also provided. 9. CONCLUSION The rapid growth of industries, communication networks, etc, demand mathematical models to justify the performance of the system in various aspects. This research work is devoted to the study of queueing models for many practical situations. The analytical treatment of this work is done with the use of embedded Markov chain and supplementary variable techniques which, in turn, results in obtaining performance measures of the system. Numerical illustrations of all models are discussed with cost analysis. Further investigation like waiting time distribution, busy period distribution, decomposition property would be quite rewarding and would be worth studying.