Perturbation Of Regulatrized Operator Semigroups And Their Applications To M/M/1 Queueing Model

A C- regularized semigroup | T( (t)}t≥0 is of contractions if || T(t) || ≤ || Cx || for every ≥ 0 and x ∈ X. In this paper, We first study the perturbation problem of regularized operator semigroups. We have me following results:Theorem 2.1.1 Suppose that A and B are densely defined linear operators in X such that D (B) D(A) and A + tB is C - dissipative for 0≤t≤1, CA AC; CB BC if || Bx || < a || Ax|| + b || x|| for every x ∈ D(A)Where 0≤a<1,b≥0 and for some t0∈ [0,1],R(I- (A + t0B)) =X. Then R(I- (A + tB)) = X, far all t∈[0,l].Theorem 2.1.2 Suppose that A generates a contractive C - regularized Semigroup, p( A) 7= 0, D( A) = X and CA AC. Let B is C- dissipative such mat D(B) D(A), CB BC and || Bx || ≤ a || Ac || + b||x || for every x∈D(A)Where 0≤a< 1, B≥0. Then A+ B generates a contractive C - regularized Semigroup.Theorem 2.1.3 Suppose that A generates a contractive C - regularized Semigroup, p( A) = 0, D(A)=X and CA AC. Let B is C- dissipative such that D(B) D(A), CB BC and || Bx || ≤ H AX || +B||x || for every x ∈ D(A)Where B≥0 is a constant. If B* the adjoint of B, is densely defined then the closure A+ B of A+ B is the infinitesimal generator of a contractive C - regularized semigroup.Secondly, we discuss the M/M/1 queueing model described by ordinary differential equations:Where,p0(t) represents the probability that the system is empty at time t, Pn(t) represents the probability that there are n customers in the system at timet t. A >0 is the arrival rate of customer, u>0 is the service rate of me server.By using the operator semigroup theory we prore the existence of a unique positive solution of this model on c0 and study the spectral properties of the corresponding operator. We have the following results:Theorem 3.3.2 Qco generates a positive contractive CQ - semigroup T(t) on GO-Theorem 3.3.3 This model has a unique nonnegative solution p(t) on c0, which satisfiesTheorem 4.1.1 The spectrum radius of Qc0 is r(Qc0 ) which satistiesr(Qc0) = 2(A + u) 'Theorem 4.1.2 At last, we apply the perturbation theorem of contractive C - regularized semigroups to the M/M/1 queueing model. We have the following result:Theorem 5.1.1 Suppose that A generates a contractive C - regularized semigroup, p (A) = 0, D(A) = X and CA AC. Then A + Qc generates a contractive C - regularized semigroup.