Some Qualitative Properties in Polling SystemsEitan ALTMAN

Architectures parall eles, Bases de donn ees, R eseaux et Syst emes distribu es Abstract Consider a polling system with K 1 queues and a single server that visits the queues in a cyclic order. The polling discipline in each queue is of general gated-type or exhaustive-type. We assume that in each queue the arrival times form a Poisson process, and that the service times, the walking times as well as the setup times form sequences of independent and identically distributed random variables. For such a system, we provide suucient conditions under which the vector of queue lengths is stable. We treat several criterions for stability: the ergodicity of the process, the geometric ergodicity, and the geometric rate of convergence of the rst moment. The ergodicity implies the weak convergence of station times, intervisit times and cycle times. Next, we show that the queue lengths, station times, intervisit times and cycle times are stochastically increasing in arrival rates, in service times, in walking times and in setup times. The stability conditions and the stochastic monotonicity results are extended to the polling systems with additional customer routing between the queues, as well as bulk and correlated arrivals. For polling systems with mixed limited, gated and exhaustive service disciplines and without setup times, we prove that the weighted sum of mean waiting times increases in the arrival rate, in the rst and second moments of the the service times and in the variances of the walking times. This monotonicity also holds with respect to the mean walking times if they are deterministic or if there are \many queues" or the system is heavily loaded. Finally, we prove that the mean cycle time, the mean intervisit time and the mean station times are invariant under general service disciplines and general stationary arrival and service processes.