Bilevel hysteretic service rate control for bulk arrival queue

This paper concerns M[X] / M / 1 queue. The number of customers in each arriving unit is a random variable. There is two control threshold values K and N, K is smaller than N. Service rate is switched from u to tu whenever the system size increases to N. The tu rate is switched to u when the system size drops to the value K. We derive the steady-state probabilities of the number of customers in system and the expected number of customers in system. A cost model is introduced for the service cost, queuing cost, and switching cost.