Queueing system with batch arrival of heterogeneous orders, flexible limited processor sharing and dynamical change of priorities

A queueing system with the discipline of flexible limited sharing of the server is considered. This discipline assumes the admission, for a simultaneous service, of only a finite number of orders, as well as the use of a reduced service rate when the bandwidth required by the admitted orders is less than the total bandwidth of the server. The orders arrive following a batch-marked Markov arrival process, which is a generalization of the well-known $MAP$ (Markov arrival process) to the cases of heterogeneous orders and batch arrivals. The orders of different types have different preemptive priorities. The possibility of an increase or a decrease in order priority during the service is suggested to be an effective mechanism to prevent long processing orders from being pushed out of service by just-arrived higher-priority orders. Under a fixed priority scheme and a mechanism of dynamic change of the priorities, the stationary analysis of this queueing system is implemented by considering a suitable multidimensional continuous-time Markov chain with a generator that has an upper Hessenberg structure. The possibility of the optimal restriction on the number of simultaneously serviced orders is numerically demonstrated.