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We consider a system consisting of a single server serving a fixed number of stations. At each station, there is an infinite queue of customers that have to undergo a preparation phase before being served. This model is connected to layered queueing networks, to an extension of polling systems and surprisingly to random graphs. We are interested in the waiting time of the server. For the case where the server polls the stations cyclically, we give a sufficient condition for the existence of a limiting waiting-time distribution and we study the tail behavior of the stationary waiting time. Furthermore, assuming that preparation times are exponentially distributed, we describe in depth the resulting Markov chain. We also investigate a model variation where the server does not necessarily poll the stations in a cyclic order, but always serves the customer with the earliest completed preparation phase. We show that the mean waiting time under this dynamic allocation never exceeds that of the cyclic case, but that the waiting-time distributions corresponding to both cases are not necessarily stochastically ordered. Finally, we provide extensive numerical results investigating and comparing the effect of the system's parameters to the performance of the server for both models. [ABSTRACT FROM AUTHOR]

The so called “Israeli Queue” is a single server polling system with batch service of an unlimited size, where the next queue to be visited is the one in which the first customer in line has been waiting for the longest time. The case with finite number of queues (groups) was introduced by Boxma, Van der Wal and Yechiali [3]. In this paper we extend the model to the case with a (possibly) infinite number of queues. We analyze the M/M/1, M/M/c, and M/M/1/N—type queues, as well as a priority model with (at most) M high-priority classes and a single lower priority class. In all models we present an extensive probabilistic analysis and calculate key performance measures. [ABSTRACT FROM PUBLISHER]