We investigate in this paper the properties of multimodular functions. In doing so we give elementary proofs for properties already established by Hajek, and we generalize some of his results. In particular, we extend the relation bewteen convexity and multimodularity to some convex subsets of the integers numbers. We also obtain general optimizatnio results for average costs related to a sequence of multimodular functions rather than to a single functions. Under this general context, we show that the expected average cost problem is optimized by ising balanced sequences. We finally illustrate the usefulness of this theory in admission control into a D/D/1 queue with fixed bathch arrivals, with no state information. We show that the balanced policy minimizes the average queue length for the case of an intinite queeu, but not for the case of a finite queue. When further adding a constraint on the losses, it si show that a balanced policy is also optimal for the finite queue case.
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机译：我们在本文中研究多模函数的性质。通过这样做，我们对Hajek已经建立的属性给出了基本的证明，并概括了他的一些结果。特别地，我们将凸性和多模性之间的关系扩展到整数的一些凸子集。我们还获得了与一系列多模函数而不是单个函数相关的平均成本的一般优化结果。在此一般情况下，我们表明，通过分配平衡序列可以优化预期的平均成本问题。最后，我们说明了该理论在具有固定浴池到达，没有状态信息的D / D / 1队列中的准入控制中的有用性。我们表明，对于无限队列，平衡策略将平均队列长度最小化，而对于有限队列，则没有。当进一步增加对损失的约束时，它表明平衡策略对于有限队列情况也是最佳的。
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