The finite-capacity polling system analysed in the paper consists of N queues that are attended by a single server in cyclic order. Each queue i is characterized by its maximum capacity \(0<K_ i<\infty\), a Poisson arrival process with rate \(\lambda_ i\), and a general service time distribution. The time the server needs to switch from queue i to the next is assumed to follow a given distribution.
Three different policies are distinguished that differ in the way the server switches from one queue to the next. In exhaustive service the server leaves a queue when there are no more customers requesting service in this queue. In gated service the server leaves a queue when all customers have been served that were already present when it started serving the queue (customers arriving during the service period are not served). In limited service, only one customer is served before switching. In all three cases, the server immediately switches over to the next queue if there is no customer in queue i. For all three policies, the distribution of the waiting time (excluding service time) at each queue i is derived.
The results are obtained by building on existing results for a single M/G/1/K queue with multiple vacations. For each particular queue of the polling system, the time the server needs for switching or for serving another queue can be regarded as a vacation period. In this paper it is shown how the distribution of vacation periods can be derived by solving state transition equations of the polling system. Given this distribution, the distribution of the waiting time can be calculated for each individual queue using the known results for M/G/1/K queues with multiple vacations.
The organisation of the paper is excellent. The relevant results for M/G/1/K queues with multiple reservations are summarized for easy reference. A detailed analysis of a system with two queues \((N=2)\) and single buffers \((K_ 1=1\), \(K_ 2=1)\) serves as illustration of the derivations.