Bulk Queueing Models with Breakdown, Control Policies on Arrival, Priority Queue and Variant Vacations

Abstract

Queueing theory requires the study of queueing models to predict the performance and behaviour of the process that try to provide services for randomly arising demands and it plays an important role in the world of industry and communications. Queueing theory, as a stochastic process, provides important information by predicting various performance measures which are required in decision making.It has a wide range of applications in computer system, communication networks and production systems and they are essential in many practical situations such as flexible assembly systems, flow lines, job shops, and transfer lines involved in a manufacturing system. Since bulk arrival and bulk service queueing systems have been gaining importance in recent years, this work considers bulk queueing system. The abstracts of the proposed models are given below for your kind perusal: I Maximum entropy analysis of a bulk queueing system with breakdown, controlled arrival and multiple vacations In this chapter, a bulk arrival bulk service queueing system with multiple vacations, controlled arrival of batches and breakdown is analyzed. The server assigned for secondary jobs (vacations) repeatedly when the number of customers is inadequate to process. Moreover, the arrivals are accepted with some probability during the busy, vacation and renovation period. During a batch service, if the server breaks down, the service for the particular batch is processed without interruption. Upon completion of batch service, the renovation of service station will be considered. II Maximum entropy analysis of queueing system with balking, startup and vacation interruption In this chapter, a batch arrival single server queueing system with a startup, balking, and vacation interruption is considered. The server provides service one by one to the primary arriving customers. During the busy period of the server, the customers may join the queue or may balk from the system. On completion of service, the server goes for vacation when there are no customers available in the queue, otherwise, it resumes service. After a vacation, the server stays idle until the customer arrives and when an arrival occurs, the server begins the startup work and resumes service. Furthermore, vacation is interrupted suddenly when a customer arrives during vacation and server goes for startup and then starts service after completing the startup work. III Analysis of a bulk queueing system with server breakdown and vacation interruption In this chapter, the mathematical model for the server breakdown with an interrupted vacation in a MX/G(a,b)/1 queueing system is considered. After completing a batch of service, if the server breaks down, then the renovation of service station will be considered. After completing the renovation of service station or there is no breakdown of the server, if the server finds at least ‘a’ customers waiting for service say ξ, then the server serves a batch of min (ξ, b) customers, where b ≥ a. On the other hand, if the queue length is less than ‘a’, the server leaves for a secondary job (vacation) of random length. Furthermore, the secondary job is interrupted abruptly and the server resumes primary service, if the queue size reaches ‘a’ during the secondary job. On completion of the secondary job, the server remains in the system until the queue length reaches ‘a’. IV Analysis of a queueing system with two phases of bulk service, closedown and interrupted vacation In this chapter, batch arrival single server two phases of service with vacation interruption and closedown is considered. The server provides first essential service (FES) (bulk service) to the arriving customers, and after completing FES, the server must provide the second essential service (SES). Upon completion of SES, if the queue length is ξ, where ξ<a, then the server performs closedown work; otherwise it resumes FES. After completing the closedown work, if the queue length is still less than a, the server leaves for single vacation (secondary job); otherwise the server provides FES. It is assumed that during a vacation, if the queue length reaches ‘a’, the secondary job is interrupted abruptly and resumes the FES. On completion of the secondary job, the server remains in the system until the queue length reaches ‘a’. V Analysis of a bulk queueing system with controlled arrival and multiple vacations: a simulation approach In this chapter, system with a limited number of admissions and multiple vacations is analyzed. The service is done in bulk to a minimum of ‘a’ customers and a maximum of ‘b’ customers. If the queue length is greater than ‘b’, then ‘b’ customers are considered for service and the rest is kept in a queue. On the other hand, when the number of customers is inadequate (less than ‘a’), then the operator goes for secondary job (vacation) during its idle period. On completion of a single vacation, if the queue length is still less than ‘a’, the operator goes for another vacation (multiple vacations) and so on, until it finds ‘a’, customers in the queue. Furthermore, the arrivals are accepted with some probability during the service and vacation period. Simulation is carried out for the proposed model and justified through numerical illustration. VI Analysis of queueing system with priority queue and multiple vacations: a simulation approach In this chapter, queueing system with two queues and multiple vacations is considered. The server provides service one by one to the arriving customers in the priority queue which has finite capacity ‘N’. On each service completion epoch of priority customers, the server checks the priority queue and if there are no customers in the priority queue, then the server provides service to the customers if any in non-priority queue. Upon completion of service of non-priority customer, the server checks for the customers in a priority queue and provide service if customers available, otherwise serves customers in non-priority queue. If both queues are empty, the server goes for multiple vacations until customers arrive to any of the two queues. Simulation is carried out for the proposed model. CONCLUSION Real time applications in production and manufacturing industries for the proposed models have been taken up for the research. Maximum entropy principle is deployed to determine the approximate results for steady state probability distribution of queue size and expected waiting time in the queue. It is inferred that the maximum entropy principle can be applied for bulk queueing systems to obtain the explicit results of probability distribution based upon the available information. In addition, using simulation technique, the effects of changes in few parameters on a system are studied easily. The method proposed provides a convenient way to determine results in less time, less computer resources and reduces the manual work. The analytical results are therefore compared with the maximum entropy and simulated results to find the variation. It is worth pointing out that the findings are of direct practical relevance and can be successfully used for a number of manufacturing applications. Future research can be carried out to incorporate the concept of blocking, bulk failure, negative customers, stochastic decomposition, and discrete time queueing models, etc.