Fluid Model Driven By The M/M/c Queue With Vacation Policies

Owing the research of fluid model can be widely used in transportation, financialrisks, environmental engineering, computer, remote communications and other productionfields, the fluid model become a very active research topic in the application probabilityfield. By analyzing the influence of various parameters to the performance indicators offluid model, we can do our best to optimize and design, so as to solve practical problemsmore effectively, and obtain greater economic benefits. In a word, the fluid model hasgreat research value.Based on the fluid model driven by the queue with single server, we extending thesigle server to multiple server, and taking the negative customer and bernoulli feedbacktactics into the fluid model, then, we analyze the fluid mode driven by M/M/c multiplevacation queue with above policies, making the vacation queuing fluid model system morecompletely. Its subject matter is broadly divided into the following sections:Firstly, considering the fluid model driven by an M/M/c queue with multiplevacations, using quasi birth and death process and matrix-geometric solution method, weget the distributions of the steady state for queue length, then analyze the fluid modeldriven by this system, the differential equations satisfied by the steady-state jointdistribution function of the fluid model is obtained. Furthermore, on the basis of theLaplace transform (LT) and Laplace-Stieltjes transform (LST), we get the LST of stabledistribution and mean of the buffer content. At last, we through numerical examples toverify the results.Secondly, based on the M/M/c queue with multiple vacations, we introduce thenegative customer with RCH (Removal of Customers in the Head) offset policy, anddiscuss the fluid model driven by M/M/c queue with multiple vacations and negativecustomer. This kind of customer just offset the positive customer who is being served, andleft directly after the holiday arrive. By analyzing the fluid model, and resort to the LT andLST, the LST of buffer content is obtained, on the basis of mean and LST, the expressionof mean also can be got. Next, through numerical simulation,we show the effect of various parameters to the system performance indexes.Finally, we study the fluid model driven by M/M/c queue with multiple vacations andBernoulli feedback policy. In this system, after one service, the customer into the queuetail to waiting for another service with probability p and left the system directly withprobability of1p. After getting the generator of driven system, we talking thedistributions of queue length under steady state. Furthermore, using the differentialequations satisfied by the distribution function of fluid model, the LT and the LST, theprobability of empty buffer and the mean of buffer content are obtained. At last, we resortto numerical examples to confirm the correctness of the conclusions.