Multiple Vacation Queues With N-policy

Multiple working vacations system is an important emerging research in queueing system.The M/M/c queues with multiple working vacations and N-policy are researched in this paper.In other words,when a vocation is over,waiting customers are less than N in queueing system,the servers begin another vacation.The servers begin serving customers till waiting customers are not less than N at vacation's end.Firstly,consider an M/M/1 Bernoulli feedback queue with negative customers and N-policy and working vacations,we study the queue length distribution and other stable indices are given.The results of conditional stochastic decompositions are also proved.Secondly,an M/M/c queue with two-phase service,N-policy and multiple vacation discipline is considered.We study the queue length distribution and other stable indices are given and the results of conditional stochastic decompositions are proved.Finally,consider an M/M/c queue with multiple working vocations and N-policy,and we have M/M/c(N-WV) in short.The server works at a lower rate rather than completely stops during a vacation period.Using quasi birth and death process and matrix-geometric solution method,we gain concise expression of the steady-state distributions for queue length and conditional waiting time.Furthermore,we indicate the conditional stochastic decomposition structures of queue length and waiting time in the stationary state and obtain the distributions for additional queue length and additional delay.