Optimal Overhaul Policy And Analysis Of M/G/1 Queueing System With Randomized Overhaul(p,Y)-Policy

In real manufacturing systems,the equipment of the system may fail due to its years old and running wear.Once the equipment of the system fails and the production has to be stopped,which must affect the progress of production.In order to ensure that there are no or fewer failures during the operation of the equipment,the manager usually arranges the repairman to overhaul it during the system’s idle period.Inspired by the facts above,this dissertation proposes random overhaul(p,Y)-policy and establishes an M/G/1 queuing model under the control of random overhaul(p,Y)-policy.Then,applying the renewal process and total probability decomposition technique,we study some queueing performance measures of the system.Moreover,the optimal overhaul policy for economizing the system cost is discussed under a given cost model and established cost objective function.This dissertation is divided into two chapters as follows.In chapter 1,based on the operation background of actual manufacturing systems,we establish an M/G/1 queuing model with random overhaul(p,Y)-policy,in which the random overhaul(p,Y)-policy means that when the system becomes empty,the manager shuts down the system immediately with probability p(0≤p≤1)and arranges the repairman to overhaul it with a random time length.During the overhaul period,customers who arrive at the system must wait until the overhaul period ends.Otherwise,the manager does not shut down the system with probability 1-p and the customers who arrive at the system will be served at once.Firstly,we study the embedded Markov chain {Nn+,n≥1} with the queue size {N(t),t≥0}of t at any time and obtain the probability generating function of the steady-state queue size distribution for the embedded Markov chain,where N(t)denotes the queue size at any time t(≥0),and Nn+ denotes the queue size at the embedded time point,i.e.the departure time point of the nth departure from the system,n≥1.Secondly,applying the renewal process and total probability decomposition technique,we study the transient behavior of the queue size at any time t in detail and obtain the expressions of the Laplace transform of the transient queuelength distribution.Furthermore,the explicit recursive formulas of the steady-state distribution of the queue size at any time t are derived by using L’Hospital’s rule.Meanwhile,some other queueing performance measures of the system,such as the expected queue size,the stochastic decomposition structure of the steady-state queue size,the probability distribution of the additional queue size,the additional expected queue size and so on,are presented.Additionally,Numerical examples are provided to investigate the sensitivity of the overhaul time parameter on the additional expected queue size and the probability that the system is empty.Finally,with the help of MAT LAB soft tool,the optimal overhaul policy T*for economizing the system cost is discussed under a given cost model and established cost objective function.In chapter 2,in order to increase the income of the system,the manager often arranges the server to do other auxiliary work during the overhaul period.Thus,we propose a more complex queueing model-an M/G/1 queuing model with a single vacation and random overhaul(p,Y)-policy,in which whenever the system becomes empty,the manager immediately shuts down the system with probability p(0≤p≤1)and arranges the maintenance worker to overhaul the system.At the same time,the manager arranges the server to do other auxiliary work to increase the income of the system or the server to go a vacation.Otherwise,the manager does not shut down the system with probability 1-p and the customers who arrive at the system will be served at once.Under the assumption that the vacation time and overhaul time follow general distributions,we adopt the total probability decomposition method and research route used in the first chapter to discuss some queueing performance measures of the system,such as the transient and steady-state distributions of the queue size at any time t.Meanwhile,we investigate the sensitivity of the vacation time parameter on the additional expected queue size and the probability that the system is empty by numerical examples.At last,the optimal overhaul policy T*for economizing the system cost per unit time is studied by establishing the cost model and cost objective function.In chapter 3,We discussed the research direction of M/G/1 queuing model with random overhaul(p,Y)-policy.