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Reports (Research Report) Year : 1992

Some qualitative properties in polling systems


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 set-up times form sequences of independent and identifically distributed random variables. For such a system, we provide sufficient 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 first moment. The ergodicity implies thev weak concergence of station times, intervisit times and cycles 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 set-up 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 set-up times, we prove that the weighted sum of mean waiting times increases in the arrival rate, in the first and second moments of 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.


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inria-00074964 , version 1 (24-05-2006)


  • HAL Id : inria-00074964 , version 1


Eitan Altman, Panagiotis Konstantopoulos, Zhen Liu. Some qualitative properties in polling systems. [Research Report] RR-1596, INRIA. 1992. ⟨inria-00074964⟩
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