How should we schedule jobs to minimize mean queue length? In the preemptive M/G/1 queue, we know the optimal policy is the Gittins policy, which uses any available information about jobs' remaining service times to dynamically prioritize jobs. For models more complex than the M/G/1, optimal scheduling is generally intractable. This leads us to ask: beyond the M/G/1, does Gittins still perform well? Recent results show Gittins performs well in the M/G/k, meaning that its additive suboptimality gap is bounded by an expression which is negligible in heavy traffic. But allowing multiple servers is just one way to extend the M/G/1, and most other extensions remain open. Does Gittins still perform well with non-Poisson arrival processes? Or if servers require setup times when transitioning from idle to busy? In this paper, we give the first analysis of the Gittins policy that can handle any combination of (a) multiple servers, (b) non-Poisson arrivals, and (c) setup times. Our results thus cover the G/G/1 and G/G/k, with and without setup times, bounding Gittins's suboptimality gap in each case. Each of (a), (b), and (c) adds a term to our bound, but all the terms are negligible in heavy traffic, thus implying Gittins's heavy-traffic optimality in all the systems we consider. Another consequence of our results is that Gittins is optimal in the M/G/1 with setup times at all loads.

翻译：如何调度作业以最小化平均队列长度？在可抢占的M/G/1队列中，已知最优策略为Gittins策略，该策略利用作业剩余服务时间的可用信息动态分配优先级。对于比M/G/1更复杂的模型，最优调度通常难以求解。这引出一个问题：在M/G/1模型之外，Gittins策略是否仍能保持良好性能？最新研究表明，Gittins策略在M/G/k队列中表现优异，其加性次优性间隙的上界在重流量条件下可忽略不计。但引入多服务器仅是M/G/1的扩展方向之一，大多数其他扩展仍属开放问题。当到达过程非泊松时，或服务器在空闲转忙碌状态需要启动时间时，Gittins策略是否仍能保持良好性能？本文首次对Gittins策略进行能同时处理以下三种情境的分析：(a)多服务器、(b)非泊松到达过程、(c)启动时间。因此，我们的结果涵盖含/不含启动时间的G/G/1与G/G/k队列，并在每种情况下界定了Gittins策略的次优性间隙。(a)(b)(c)每个因素都会为我们的界增加一项，但所有项在重流量条件下均可忽略，从而证明在考虑的所有系统中Gittins策略均具有重流量最优性。研究结果的另一推论是：Gittins策略在所有负载条件下的含启动时间M/G/1队列中均为最优策略。