Analysis Of The Retrial Queueing System With An Inventory

In this paper, some retrial queueing systems with an inventory are investigated, which are based on the classic continuous time review (s, S) policy to replenish the related inventory systems. To take the retrial schedule into the inventory management, is an innovative method to deal with the condition that the inventory is stock out. The whole system is consisted with the server, retrial orbit and inventory. When the positional customer arrives at the server, if the server are available to work, he will launch his service immediately and departure the system after a random service time, and simultaneously take away one item from the on-hand inventory; if the server is unable to serve him, his demand will be recorded in the server, in other words, he will go in the orbit, he can retry his request to be served in a stochastic time. When the inventory is out of stock, in order to cut down the cost, we could let the server closed or take a vacation. The inventory is replenished by the supplier or producer. There is a random lead-time in the replenishment. We will mainly consider these independent stochastic variables:the interval time between the successively arriving customers, the service time, the retrial time, the lead-time, and so on. For researching conveniently, we assume that all the variables are exponentially distributed.We have investigated the following three models:first, the retrial rate of the customer in the orbit is a constant; second, the retrial rate is liner with the number of customer in the orbit, the customers in the orbit are assumed not only to retry for their demands, but also may renege from the orbit at random time, each item in the inventory has a random perishing period; third, when the inventory is lack of stock, the server will take a vacation which last a random time. All these three models can be concluded as the QBDs, using the matrix analysis method, the steady-state condition are investigated; the joint probability distribution of the number of customers in the orbit and the inventory level is obtained in the steady state case. Further more, various system performance measures in the steady-state are derived and the long-run total expected cost rate is calculated. At last, several instances of numerical examples, which provide insight into the behavior of the system, are presented respectively in each of the three models.