Analysis Of The Geo/Geo/1/SWV Queue With Set-Up Period

First of all, we consider a classic Geo/Geo/1 queuing system with set-up period. After modeling, using the matrix geometric method, we get the probability distribution of the number of customers, the mean number of customers and the mean waiting-time of a customer in the steady-state.Secondly, based on the above-mentioned system, we get a Geo/Geo/1/SWV system with set-up period. During a working vacation period, a server doesn't completely stop services immediately, but serves customers at a lower rate.The operation of this system: While the system is in turned-off period, a customer arrives, a set-up period begins. Just as the end of it, a server begins the normal services. When a service ends, the server begins the next service immediately until there are no customers in the system. Now the working vacation period begins. The server serves a customer at a lower rate. After the working vacation, the system will switch into the normal service period if there are customers in the system, or will shut down (This is the single working vacation. The multiple working vacations will get into a next one at the same lower service rate. There is no shut-down period. The latter is relatively simple).Until the arrival of a customer the system enters a new cycle. In this model, the server serves at a lower rate when there are very few customers, and shut down when there are no customers in the system. Thus we can achieve the purpose of saving energy and reducing cost.We establish the model, get the state space. It is a Markov chain. Sorting according to the dictionary, we get the transition probability matrix and find that the chain is a quasi birth and death chain. Using the matrix geometric method we get the probability distribution of the number of customers, the mean number of customers and the mean sojourn time of a customer in the steady-state. In the end we verify it through numerical examples.Finally, according to the queuing system Geo/Geo/1/SWV with set-up time, we give several groups of data for the mean number of customers changing with the probability of the set-up,the costomer's coming and the serves of the woking vacation time.We give the graphs of them. From the graphs, we can verify the correctness of the results of this article.