Analysis Of PH Type Queueing System With Second Optional Vacation Policy

As an effective mechanism to make full use of service resources,vacation strategy can be used for other auxiliary work in the idle period to reduce the waste of resources.Based on the queue model of the secondary optional vacation strategy,the arrival time of customers or service time is generalized to PH distribution and the relevant mathematical model is established and analyzed.The more practical queuing index is obtained by taking advantage of the PH distribution and the algorithm of the approximation program provided by Neuts is optimized.That is to say,the method of solving the rate matrix in high dimensions improves the algorithm,which greatly simplifies the workload of late data operation.Finally,combining with specific values,the approximate solution of arbitrary precision of the rate matrix is obtained by using Matlab software to continuously iterate the rate matrix and the corresponding performance indexes are obtained.The main contents of this paper are as follows:Firstly,based on the queue model with the secondary optional vacation strategy,the arrival time follows a 3-order PH distribution and the service time follows the exponential distribution.According to the assumptions of the model.The corresponding quasi birth and death process and its infinitesimal generator are obtained.By using the theory of quasi birth-death process and matrix-geometric method,the performance indexes of the system are obtained,such as the steady-state distribution and average length.Secondly,based on the queue model with the secondary optional vacation strategy,the customer arrives in Poisson distribution and the customer service time is follows a 3-order PH distribution.The quasi birth-death process and its infinitesimal generator are obtained according to the assumptions of the model.The steady state distribution,average length and other performance indexes of the system are obtained by using the theories of quasi birthdeath process and matrix-geometric method.Due to the vacation strategy in the model,the service rate of the system appears two types of 3-order PH representation due to busy period and work vacation,which increases the complexity of solving the model to a certain extent.Finally,we study the queue model in which both the arrival time and service time of the customer are follows a 3-order PH distribution.The high dimensional quadratic matrix equation in the model is complicated.Taking the approximate program provided by Neuts as a reference,the algorithm of matrix quadratic equation is improved and the matrix expression of corresponding index is also obtained.At last,through the analyzing of the examples of numerical value,the system performance index changes according to the parameter by the analysis of Matlab.