.m / M / C Queuing System Degenerative

In the literature of queuing theory,most of the celebrated results are obtained through the stationary analysis of the queues system in the steady state.However,most of the practical queues system are running over a finite-time horizon;their performance depends on the systematical time and initial conditions.Under this circumstance,the appropriate analysis of measures of the performance would be transient instead.This gap exists because the transient results are usually difficult to get.In this paper,we study the decay property of M/M/c queues system based on the theory of continuous time Markov chains.We consider decay properties including decay parameter,invariant measures;invariant vectors and quasi-stationary distributions of a M/M/c queues system which is transient on state E.Investigating such behavior is crucial in analyzing the busy period and some other related properties of M/M/c queues.In chapter three,not only we present the exact value of the decay parameterλ_c,but also we obtained the clear geometric interpretation of its.We then prove that M/M/c queues system are alwaysλ_c-transient through two different methods.At the same time,we find a family of invariant measures index throughλ∈[0,λ_c].There is another important result in this paper,that is,quasi-stationary distributions of a M/M/c queues system does not exist.We firstly presentλ_c-invariant measures of M/M/c queues system,whose specific expression is presented by a generating function,then we prove that the sum ofλ_c-invariant measures with parameter m₀ always diffuse.Because of this and the definition of quasi-stationary distribution,we show that there is not quasi-stationary distributions of M/M/c queues system which is under the condition ofλ_c-transient on the state E.Lastly,two examples are provided to illustrate the results obtained in this paper.