Stationary Theory Of The M/M/c Queue With (e,d)-Policy Vacation And Its Applications

Along with popularization of internet and development of wireless communication technology, requirement for bandwidth and quality of service (QOS) is increasing. It is important to make analysis and optimization on performance indices of internet in order to improve QOS and decrease production cost. Therefore, the research on theory and application of wireless communication network is a realistic and urgent task. Based on the multiserver vacation queue theory together with the operation mechanism of wireless communication network, a set of M/M/c queues with (e, d) -policy vacation are parallelly studied and distributions of stationary performance indices and conditional stochastic decomposition results are presented. Furthermore, performance analysis and optimization of systems are shown by numerical examples.Firstly, the research status of vacation queue with multiserver is summarized and a set of systemic methods are given. However, due to variety of application background in a vacation queue with multiserver, part of idle servers can take vacation and the other servers do not take vacation even if they are idle in order to avoid delaying service requirement of arriving customers. Furthermore, servers can take vacation continuously if the vacation condition is satisfied. Hence, the M/M/c queue with multiple (e, d)-policy vacations is built. Based on the transition probability matrix of model, with quasi birth-and-death process and matrix-geometric solution method, homogeneous linear set of equations that stationary queue length satisfies are listed. By recursive method, the distribution of stationary queue length is derived. Furthermore, the new property of conditional Erlang distribution is proved, by which the distribution of stationary waiting time is obtained. Moreover, the conditional stochastic decomposition results of stationary performance indices are found.Secondly, in an M/M/c queue with multiple (e, d)-policy vacations, servers' continuous vacation can delay the service requirement of arriving customers, consequently, the policy that server can take only one vacation is presented. Similar to the analysis of the M/M/c queue with multiple (e, d)-policy vacations, the M/M/c queue with single (e, d)-policy vacation is investigated. With matrix-geometric solution method, the distribution of stationary queue length is given;By the property of conditional Erlang distribution, the distributionof stationary waiting time is gained;Moreover, the conditional stochastic decomposition results of model are shown.Thirdly, considering energy expend of servers' operating, the system can close part idle servers when the number of customers decreases and these closed servers can be set up if the number of customers increases. Therefore, the M/M/c queue with (e, d)-policy setup times is proposed. With matrix-geometric solution method, the distribution of stationary queue length is given;By the property of conditional Erlang distribution, the distribution of stationary waiting time is obtained;Moreover, the conditional stochastic decomposition results of model are shown.Finally, to decrease the switch cost from taking vacation to serving customers, introducing a threshold N, the M/M/c queue with (e, d, ./V)-policy is studied. With matrix-geometric solution method, the distribution of stationary queue length is given;Moreover, the conditional stochastic decomposition results of stationary performance indices are shown.Furthermore, application and optimization of the above models in wireless communication network are given here by numerical examples.