Sojourn times in polling systems with various service disciplines

We consider a polling system of N queues Q1,…,QN, cyclically visited by a single server. Customers arrive at these queues according to independent Poisson processes, requiring generally distributed service times. When the server visits Qi, i=1,…,N, it serves a number of customers according to a certain polling discipline. This discipline is assumed to belong to the class of branching-type disciplines, which includes gated and exhaustive service. The special feature of our study is that, within each queue, we do not restrict ourselves to service in order of arrival (FCFS); we are interested in the effect of different service disciplines, like Last-Come-First-Served, Processor Sharing, Random Order of Service, and Shortest Job First, on the sojourn time distribution of a typical customer that arrives to the system during steady-state. After a discussion of the joint distribution of the numbers of customers at each queue at visit epochs of the server to a particular queue, we determine the Laplace-Stieltjes transform of the cycle-time distribution, viz., the time between two successive visits of the server to, say, Q1. This yields the transform of the joint distribution of past and residual cycle time, w.r.t. the arrival of a tagged customer at Q1. Subsequently concentrating on the case of gated service at Q1, we use that cycle-time result to determine the (Laplace-Stieltjes transform of the) sojourn-time distribution at Q1, for each of the scheduling disciplines mentioned above. Next to locally gated polling disciplines, we also consider the globally gated discipline. Again, we consider various non-FCFS service disciplines at the queues, and we determine the (Laplace-Stieltjes transform of the) sojourn-time distribution at an arbitrary queue.