The M/G/1 queue with a substitute server

Abstract

In this dissertation, the M/G/1 queue with a substitute server is discussed. During the main server is out of system because of server vacation or server breakdown, the system does not stop working but provides services at a low rate. As a typical example of a queueing system with a substitute server, a one type of queueing system with working vacation is introduced. Furthermore, a queueing system with disasters and a substitute server which is the little-known system is analyzed. As a working vacation queueing model, the M/G/1 queue with a single working vacation and vacation interruption is discussed. During the working vacation period, the server does not stop working but provides services at a low rate. At the instant of service completion, if there are customers in the system during the working vacation period, the server is forced to come back to the normal working level. As a disaster queueing model, the M/G/1 queue with disasters and working breakdowns is discussed. The system consists of a main server and a substitute server, and disasters only occur while the main server is in operation. The occurrence of disasters forces all customers to leave the system and causes the main server to fail. At a failure instant, the main server is sent to the repair shop and the repair period immediately begins. During the repair period, the system is equipped with the substitute server which provides the working breakdown services to arriving customers Using the supplementary variable technique, the stationary queue length distributions of two model are obtained. The laplace-stieljes transforms of the waiting time distribution are also presented. Furthermore, the results of the cycle analysis are presented. Finally, some numerical experiments are conducted.