Research On Queuing Strategy In The M/M/1 Queue With Vacation Times And Complementary Service

Queuing theory, as a branch of operational research, also called stochastic service system theory. No matter in the classic queuing system or vacation queuing system, based on their own interests every customer can determine if join the queue system or not. So studying the strategy from the perspective of customer can give the optimal suggestions to the customer and provide theoretical foundation for operators to the problem of service strategy and product price. In the actual life, sometimes customers need the complementary service, and some scholars have studied the complementary service, but these studies are all queuing model of continuous time. To simulate the real queuing system better, it is necessary to study customers’ equilibrium problem of the complementary service of discrete time.Firstly in the perfect competitive environment, the paper studies service strategy of the M/M/1 queue with vacations from both continuous and discrete time aspects. Based on the profit function of firms, we obtain expected waiting time of customers, then obtain the optimal product price of firms and the optimal output rate. Based on this, we study the profit maximizing service capacity, which obtain that the optimal strategy of firms must follow differs depending on the functional form of service capacity cost. Secondly, the paper studies customers’equilibrium of Geom/Geom/1 queue with complementary service. We obtain equilibrium based on analysis of customer behavior from two aspects, and give the numerical analysis of customer benefit and expenditure, which obtain that the equilibrium joining rate is smaller than the social optimal joining rate. Finally, we give customers’equilibrium of the monopoly case, which obtain the profit optimal price set by a monopolist also maximize the social welfare.