Research Of The Queue With Impatient Customers And Working Breakdown

This paper introduce working breakdown strategy,impatient customers strategy,vacation strategy and setup time strategy on the basis of the M/M/1 queueing system,and analyze two kinds of new queueing systems.Some explicit expressions of system performance measures are derived.Meanwhile,we model and analyze the operation mechanism of Bitcoin breakdown mining pool which is based on the Blockchain technology,and provide effective and theoretical basis for solving the power saving problem of Bitcoin breakdown mining pool.The paper is composed of three parts as follows:Firstly,we investigate the M/M/1 queueing system with working breakdown and impatient customers.The two-dimensional Markov chain is constructed to analyze the queueing system,and the state transition rate matrix is obtained.The balance equations in steady state are solved by the method of generating function,then the probability generating function of the queue length of system is derived and a series of accurate expressions of performance measures are obtained.We carry out Matlab software to examine the influence of the system parameters on system performance measures,and simulate the queuing process.What is more,the utility function is constructed from individual and social perspectives respectively,the optimal strategy is analyzed and the optimal parameters are obtained by means of numerical experiments.Secondly,we analyze the M/M/1 queueing model with working breakdown,impatient customers and multiple vacations.Assuming that it is breakdown that the server takes place when there are customers in the system,and the server will be in the multiple vacations state when there are not customers in the system.Meanwhile,we consider the abandon of the customers in the queue when the server is in working breakdown.The state transition diagram of queueing system is drawn,and the balance equations which are constructed in steady state are solved by the method of generating function,then a series of accurate expressions of performance measures are obtained,such as the queue length in steady state.The relation which is between the parameters and performance measures is examined by numerical experiments.Furthermore,the equilibrium strategy of the customer and the social optimal strategy are analyzed in detail.Finally,we put forward the new power saving operation mechanism for the operation mode of Bitcoin breakdown mining pool which is based on the Blockchain technology.The power saving operation mechanism of Bitcoin mining pool with breakdown is modeled as the working breakdown queueing system with sleep and wake-up strategy,the competition of computing power between the mining pools is considered.The state transition rate matrix is obtained and a series of accurate expressions of performance measures about the operation mechanism of the mining pool are derived by analyzing the queueing system.The sensitivity analyses of the performance measures are presented by means of the numerical experiments.Furthermore,we construct the power consumption function and the power saving rate function to analyze the power consumption of the mining pool.