A Discrete-time Vacation Queueing System With Negative And Feedback

As computer networks, communications systems is the basic unit of binary code,discrete-time queuing model more intuitive to describe the specific model of such systems.Negative arrives have been interpreted as inhibitor and synchronization signals. So theexistence of negative customers provides a mechanism to control an excessive congestionat the system. Generally negatives are constraints of the system. Negative customerqueuing system more conducive to in-depth study provides a theoretical basis for thesystem optimization. The feedback related to the network, production and all aspects oflife, the study of the model with feedback to improve the network protocols, reduce energyconsumption and improve the mechanism has an important significance.The paper deals with Geom/Geom/1feedback queue with working vacations inwhich customers are either positive or negative. Using Markov chain and Matrix-geometri-c solution method, we obtian the transition probability matrix, and then derive thesteady-state distribution for queue length and the mean of the system size of positivecustomers and the result of stochastic decomposition of the queue length. In addition,based on the alternative renewal process theory, the busy cycle and the mean values ofbusy period are derived. The paper analysis of the special circumstances of the systemmodel. And then, simulation numerical experiment to analysis of several parametersinfluence on the performance of the system.Firstly, the paper studies the Geom/Geom/1queue with negative customers, Bernoullifeedback and vacation. Using QBD (quasi birth and death) process and Matrix-geometricsolution, we obtain the steady-state distributions for the number of customers in thesystem and the stochastic decomposition of the steady state queue length.Secondly, we extend the model above, we study the discrete-time Geom/Geom/1queue with negative customers, Bernoulli feedback and multiple working vacation and thediscrete-time Geom/Geom/1queue with server set-up time, Bernoulli feedback andmultiple working vacation G-line. We describe the model, by means of the transitionprobability matrix, we get the steady-state distribution for queue length and the stochastic decomposition of the queue length and the busy cycle and the mean values of busy period.Finally, the paper analyzes of the special case corresponding to each model, theaccuracy of the theority is validated by several numerical examples.