Diffusion Approximations For Multiclass G/PH/n+GI Systems With Non-Preemptive Priority Discipline

This paper studies the asymptotic behavior of multi-server queues with non-preemptive priority discipline and customer abandonment, i.e., this paper proves the diffusion approximations of a pair of diffusion-scaled waiting-queue-length and server-allocation process (Qn, Zn)={(Qn(t), Zn(t)), t≥ 0}. Firstly, this paper studies the overloaded G/PH/n+M system, where the patience times are independent, iden-tically and exponentially distributed; Secondly, the paper studies the critically loaded G/PH/n+GI system, where the patience time distribution is generally distributed.More precisely, the paper first describes a perturbed system, which is equivalent to the original system, and then develops the dynamical equations that the perturbed system must satisfy. These equations allow one to prove the limit theorems by apply-ing the standard continuous-mapping theorem and the standard random-time-change theorem. Furthermore, when the abandonment rates are equal, (Qn, Zn) converges in distribution to a diffusion process for overloaded G/PH/n+M system and converges in distribution to a continuous Markov process for critically loaded G/PH/n+GI system. In addition, the paper discusses what will happen if the abandonment rates are not equal, and the same result is obtained if the higher priority customer’s traffic intensity is bigger than 1 for overloaded G/PH/n+M system.