Performance Analysis Of The Queueing System With Working Vacations And Reneging

Vacation queue has been widely used in the fields of system design, performance analysis for communication network and system reliability. Working vacation queue generalizes vacation queue and plays a key role in the modeling and analysis for routers in optical communication system, which brings broad research on diversified working vacation queues. The queue models with reneging are comprehensively studied for being used in communication system, in which the waiting time of the customers is limited. In this paper, using the idea of probability, matrix, queueing theory and stochastic process, the multi-server queue models with working vacations and the discrete-time queue model with reneging are considered. The main results are as follows.1. The GI/M/c queue model with multiple working vacations is researched. With the method of embedded Markov chain, a two-dimension Markov chain is established, and its transition probability matrix is block-structured matrix of GI/M/1 type. By using the matrix-geometric solution, the sufficient and necessary condition of the stable queuing system is given. Combined with the UL-type RG-factorization, the stationary probability distribution is calculated. Based on this distribution, the probability distribution of queue length and its conditional stochastic decomposition are analyzed. Moreover, the cumulative distribution function of waiting time is obtained and a result about conditional waiting time is represented. Using the conditional stochastic decomposition structures of queue length and waiting time, it is convenient for comparing working vacation queue with classical queue. Consequently, the influence of this working vacation policy on the classical queueing system is distinct.2. The discrete-time GI/Geo/c queue with working vacations is analyzed, and early arrival system is assumed. Through establishing two-dimension embedded Markov chain, using the matrix-geometric solution and UL-type RG-factorization, the stability condition of queueing system and the stationary probability vector are enumerated. Accordingly, the probability distribution of queue length and the probability generated function of waiting time are gained.3. The discrete-time MMAP[K]/PH[K]/1 queueing system with reneging is discussed, where there is K types of the customers, patient time of different customers is a discrete random variable and follows differently general distribution. Through choosing appropriate random variables, four dimensional Markov chain, which is proved that it is a homogeneous Markov chain with finitely-many levels of GI/M/1 type, is constructed. Under the condition that this Markov chain is irreducible, the stationary probability distribution is given by using the UL-type RG-factorization. Therefore, some performance measures, such as the loss probability of the customers, the probability distribution of waiting queue length and the probability distributions of waiting queue length of k (1≤k≤K) type of the customers, are acquired.