The GI/Geom/1Queue With Negative Customers

In daily life, discrete time queues have expensive application background. Alongwith the development of the digital information system and the communication system, itfurther promotes researches and applications for discrete time queueing system.Meanwhile vacation queueing models also have expensive application background in thecomputer system, the communication system, the manufacturing system and themanagement engineering and so on. However, due to the existence of vacation, morenumbers of waiting customers, a longer waiting time for customers, customer drains andso on, these problems cause that systems endure too much load. And then, workingvacation policy is introduced to actual queueing issues. In the past21years, queueingsystems with negative customers have become a hot topic and applied to some areas, suchas computer network system and communication system and sales system and dynamicsystem. Therefore, the study has important theoretical significance and application valueabout queueing systems with negative customers.In recent years, many scholars mainly apply two kinds of removal policies ofnegative customers in the research process. One is RCH that negative customers removalcustomers at the head, another is RCE that negative customers removal customers at theend.Based on queuing models with negative customers and discrete time queueingmodels, the paper discusses two new models which are the GI/Geom/1queue withnegative customers and multiple working vacations and the N-policy GI/Geom/1queuewith negative customers and multiple vacations. In this paper, we arrange that negativecustomers only offset customers who are accepting service at the head. Usingmatrix-geometric solution, we give the steady-state distributions for the number ofcustomers in the system at arrival epochs. Furthermore, we gain the generating function ofdistributions for the number of customers, the average value of the number of customersand stochastic decomposition result of the steady-state number of customers in the system.At the last, we make programs using the Matrix Laboratory, and derive some numerical examples. Then we observe that the influence of system performance is showed byparameters in the system.