Hysteresis control by the number of active servers in queueing system MMAP/PH/NMMAP/PH/N with priority service

• Multi-server queueing system with hysteresis strategy of control by the number of active servers is analyzed.
• Ergodicity condition is derived.
• A procedure for computation of the steady state distribution of the system states is presented.
• The problem of optimal choice of hysteresis parameters is discussed.
• Presented results are numerically illustrated.

The problem of choosing the optimal hysteresis strategy of control by the number of active servers in the multi-server queue is considered. Customers of two types arrive to the system according to the marked Markovian arrival process (MMAPMMAP). Type 1 customers have a non-preemptive priority, but the buffer for these customers is finite. The buffer for type 2 customers is infinite. The service time distribution is of phase-type (PHPH) depending on the type of customers. Some servers are always active. The rest of servers can be switched on or off depending on the number of customers in the system. The strategy of control by the number of active servers is of hysteresis type. Such a strategy is defined by two sets of thresholds. The servers are activated or switched off depending on the relation of the number of customers and the thresholds. The main contribution of the paper is development of a procedure for computation of the stationary distribution of the system states and the value of economical cost criterion under any fixed thresholds. Numerical results show effectiveness of the hysteresis control and importance of account of correlation in the arrival process and variance of service times.