Srinivas R. Chakravarthy

We consider a priority polling system consisting of two queues attended by a single server. Type i customers arrive to queue i according to a homogeneous Poisson process with rate λ _i . Queue i has its own buffer of finite capacity K _i for i= 1 2. Customers are served in groups, with the group service time distributed according to distribution with mean . The distribution is assumed to be independent of the group size. The service discipline is as follows: at the completion of a service (of any type) the server polls queue 1 and serves all waiting customers at the same time. If there are no customers waiting in queue 1, the server polls queue 2 and serves the group that it finds upon its arrival to queue 2. If the system is empty, the server waits for the first arrival. Customers who arrive during a service wait until the server becomes free. The steady-state analysis of the model is carried out by deriving expressions for the distribution of the number of type i customers in the queue and the waiting time distribution of a type 1 customer, who has a non-preemptive priority over a type 2 customer. In the case where both service times are of phase type, efficient algorithms for computing various performance measures, such as the throughput, the proportion of idle time, the fraction of type i customers served, and the mean and the standard deviation of the number of type i customers in the queue, are developed. Furthermore, the waiting time distribution of a type 1 customer is shown to be of phase type. Illustrative examples are presented and further work on extensions of the model is discussed

A discrete time queueing model with Markovian arrival process in which jobs require primary and possibly secondary service, is analysed. Under the assumptions of infinite waiting room for the primary queue and no waiting room for the secondary queue we show that the steady state queue length density is of the matrix-geometric type. Stationary waiting time distribution of an arriving job is derived explicitly. Efficient algorithmic procedures for the computation of the rate matrix and the waiting time distribution are given