Equilibrium in vacation queueing system with complementary services

This paper studies a queue with server vacations where customers need complementary services. The main service provider or the queue server may become absent for a random period of time called a vacation. More specifically, the server adopts a multiple vacation policy which means that the server keeps taking vacations until at least a customer waiting at the vacation completion instant. The secondary service provider offers an instantaneous service (no delay or no queue). The two services are complementary and a customer must receive both services if he or she joins the system. We investigate the equilibriums of such a system in a time-based fee model under competition and monopoly cases, respectively. Furthermore, a fixed or flat fee model is also analysed.