ANALYSIS AND COMPUTATIONAL ALGORITHM FOR QUEUES WITH STATE-DEPENDENT VACATIONS I： G/M（n）/1/K

In this paper we study a queueing system with state-dependent services and state-dependent vacations, or simply G/M（n）/1/K. Since the service rate is state-dependent, this system includes G/M/c and G/M/c/K queues with various types of station vacations as special cases. We provide a recursive algorithm using the supplementary variable technique to numerically compute the stationary queue length distribution of the system. The only input requirement is the Laplace-Stieltjes transform of the interarrival distribution as well as the state-dependent service rate and state-dependent vacation rate. In a subsequent companion paper, we study its dual system M（n）/G/1/K queue with statedependent vacations.     

ANALYSIS AND COMPUTATIONAL ALGORITHM FOR QUEUES WITH STATE-DEPENDENT VACATIONS Ⅱ:M(n)/G/1/K

We study a single-server queueing system with state-dependent arrivals and general service-distribution.or simply M(n)/G/1/K.where the server follows an N policy and takes multiple vacationswhen the system is empty.We provide a recursive algorithm using the supplementary variable tech-nique to munerically compute the stationary queue length distribution of the system.The only inputrequirements are the Laplace-Stieltjes transforms of the service time distribution and the vacation timedistribution.and the state-dependent arrival rate.The computational complexity of the algorithm isO(K~3).