Approximating GI/GI/1 Queue With Vacations And Queueing Network With Vacations

Queue with vacations system is an important area of queueing theory. It is the development and extension research of classic queueing system. And it has been extensively used for the performance analysis of manufacturing systems, transportation systems, and computer and communication networks.In this paper, firstly, we consider a GI/GI/1 queue with vacations. To be more precise, the station will leave for a vacation once it completes service of customers in buffer. And on the completion of a vacation, it will begin to work if there are some customers waiting for service, and start another vacation otherwise. Unlike a GI/GI/1 queue, an idleness may occur in a GI/GI/1 queue with vacations even when there are customers waiting, which is because some customers may come into the system when the station is in vacation. It is the difference between them. And in this paper, by Functional Strong-Law-of-Large-Numbers (known as fluid limit or fluid approximation) and Functional Central Limit Theorem (known as diffusion approximation), we develop an approximating technique on max function of independent and identically distributed (i.i.d.) random variables, that is, max{η_i, 1≤i≤n}. The approximating technique is used to obtain the fluid and diffusion approximations for the queue length, workload and busy time processes. Secondly, we consider a queueing network that generalizes the models studied above. Similarly, in this queueing network with vacations, by Functional Strong-Law-of-Large-Numbers (known as fluid limit or fluid approximation) and Functional Central Limit Theorem (known as diffusion approximation), we develop an approximating technique on max function of independent and identically distributed random variables, that is, max {η_i, 1≤i≤n}. Then, similarly, the approximating technique is used to obtain the fluid and diffusion approximations for the queue length, workload and busy time processes.