%0 Journal Article
%T An M/M/2 Queueing System with Heterogeneous Servers Including One with Working Vacation
%A A. Krishnamoorthy
%A C. Sreenivasan
%J International Journal of Stochastic Analysis
%D 2012
%I Hindawi Publishing Corporation
%R 10.1155/2012/145867
%X This paper analyzes an queueing system with two heterogeneous servers, one of which is always available but the other goes on vacation in the absence of customers waiting for service. The vacationing server, however, returns to serve at a low rate as an arrival finds the other server busy. The system is analyzed in the steady state using matrix geometric method. Busy period of the system is analyzed and mean waiting time in the stationary regime computed. Conditional stochastic decomposition of stationary queue length is obtained. An illustrative example is also provided. 1. Introduction Queueing models with vacation have gained significance in the last three decades due to their wide range of applications, especially in the communication and the manufacturing systems. Doshi [1] provides an excellent survey of related works prior to 1986. Takagi [2] and Tian and Zhang [3] provide a good account of developments in this field since then. The literature on the vacation queueing models is growing rapidly. In multiserver queueing models, we come across two classes of vacation mechanisms: station vacation and server vacation. In the first case, all servers take vacation simultaneously whenever the system becomes empty and they return to the system all together. Thus, station vacation is group vacation for all servers. For example, when a system consists of a number of machines operated by a single individual this scenario occurs. In such a situation, the whole station needs to be treated as a single unit for vacation, when the system is utilized for a secondary task. In the second case, each server takes its own vacation whenever it completes a service and finds no customers waiting for service. This phenomenon also occurs in practice. For example, in a post office or bank, when a clerk completes a service and finds no customer waiting, he or she might go to attend another task. This is what we refer to as the server vacation model. Analysis of a server vacation model is more complicated than that of a station vacation model. This is because at any time point, in the latter, we may have any number of servers between and on vacation. We need to track individual servers going on vacation and completing their vacation. Upon returning from a vacation some servers may find no customers waiting for service. These servers take another vacation. But if any server finds a waiting customer on returning from a vacation, it immediately starts service. For further details on queues with station and server vacations, we refer the reader to Chao and Zhao [4]. Most of the
%U http://www.hindawi.com/journals/ijsa/2012/145867/