The Queue With Working Vacation Under Bernoulli Schedule

Servi and Finn introduced the working vacation queue that become a hot topic in the research of queuing theory.The feature of the working vacation queue is that the server continues to work with a lower speed,rather than completely stop service during the vacation time.This means that some servers still serve customers,while others may take vacations or do other work.Compared with the ordinary vacation queue,working vacation can ensure the operation of the system more effectively and achieve the dual effect of reducing system loss and load.In other words,when the main work is relatively less in the system,other auxiliary work can be done.If there are no service(service rate is zero)during the working vacation time,the working vacation queue becomes an ordinary vacation queue.Therefore,the working vacation queue is a promotion of the ordinary vacation queue.The working vacation queue allows the server to flexibly switch between high and low speed services,making the service system more practical and more widely used.This paper mainly studies several M/M/1 working vacation queue with Bernoulli schedule controlled.During the normal period,when a customer service is completed,if there are no customers in the system,the server begins an ordinary vacation with probability p(0?p?1)or a working vacation with probability p(p=1-p).Obviously,ordinary vacation and working vacation respectively correspond to p=1 and p=0.At this point,we can assume that working vacation is controlled by Bernoulli Therefore,we can put ordinary vacation and working vacation together to discuss.On this basis,we mainly studied the following three parts:First of all,we consider negative customers and set-up(close-down)period:in the system,the negative customer will remove the first positive customer in line who is serving.If there is no positive customer in the system,the negative customer will have no influence.When there are arriving customers(no customers)in the system,we usually set a period of set-up(close-down)time for the service equipment,it can avoid the damage caused by the instantly turning on(turning off)of the service equipment.Second,we not only consider the negative customers,but also consider feedback and N-policy:when a customer completes his service,he may leave the system with probability q(0 ? q ?:1)or return to the back of the line for another service with probability q(p=1-q).During the vacation period,when the number of customers is greater than or equal to N,the vacation ends.If not,the server will continue on vacation.Finally,we mainly consider retrial customers and impatient customers:If the arriving customer finds the server is idle,the customer accepts the service immediately.On the other hand,if the server is busy,the customer is forced to wait in the orbit of infinite size.In the working vacation period,when the customer retries,if the server is found to be free,customer gets service immediately.If the server is found to be busy,on the other hand,the retrial customer becomes non-persistent(leaves the system)with probability q(0 ?q ?1)or returns into the orbit with probability q(q=1-q).