Threshold start-up control policy for polling systems

A threshold start-up policy is appealing for manufacturing (service) facilities that incur a cost for keeping the machine (server) on, as well as for each restart of the server from its dormant state. Analysis of single product (customer) systems operating under such a policy, also known as the N-policy, has been available for some time. This article develops mathematical analysis for multiproduct systems operating under a cyclic exhaustive or globally gated service regime and a threshold start-up rule. It pays particular attention to modeling switchover (setup) times. The analysis extends/unifies existing literature on polling models by obtaining as special cases, the continuously roving server and patient server polling models on the one hand, and the standard M/G/1 queue with N-policy, on the other hand. We provide a computationally efficient algorithm for finding aggregate performance measures, such as the mean waiting time for each customer type and the mean unfinished work in system. We show that the search for the optimal threshold level can be restricted to a finite set of possibilities. This revised version was published online in June 2006 with corrections to the Cover Date.     

Polling systems with state-dependent setup times

We analyze a polling system with multiple stations (queues) attended by a cycling server, in which a setup occurs only when the queue that is polled by the server has one or more customers present. Although such systems are appropriate for modeling numerous manufacturing and telecommunication systems, their analysis is not well developed in the literature. We provide an exact analysis for the 2 station model and present two approximation schemes to determine the mean station waiting times for models with 3 or more stations. We show that some approximate models which have been proposed in the literature for providing upper bounds on the mean station waiting times do not always yield upper bounds. Extensive numerical tests indicate that a simple average of the two approximation schemes yields a close estimate of the true mean station waiting time. This average-of-approximations procedure appears to be robust for a large range of parameter values.Research supported in part by the Natural Sciences and Engineering Research Council of Canada under grant OGP0045904.Research supported in part by the National Science Foundation under grant DMI-9500471.     

一般对角同余式的解的个数

设Ｐ为素数,ｎ,ｒ以及１ｌ２…,ｌｒ均为正整数,ａ１,ａ２,…,ａｒ及ｂ均为整数,诸ａｉ,ｌｉ均与ｐ互素．本文给出一般对角同余式ａ１ｘ１ｌ１＋ａ２ｘ２ｌ２＋…＋ａｒｘｒｌｒ＝ｂ（ｍｏｄｐｎ）的解数公式．