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 indicate that 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？在可抢占式M/G/1队列中，已知最优策略是吉廷斯策略，该策略利用作业剩余服务时间的可用信息动态调整优先级。对于比M/G/1更复杂的模型，最优调度通常难以处理。这引出一个问题：在M/G/1之外，吉廷斯策略是否仍能保持良好性能？最新结果表明，吉廷斯策略在M/G/k队列中表现优异——其加性次优性差距受限于一个在重流量条件下可忽略的表达式。但允许多服务器仅是扩展M/G/1的一个方向，其他扩展方式大多尚未研究。当到达过程非泊松时，或服务器从空闲到繁忙状态需要设置时间时，吉廷斯策略是否仍有效？本文首次给出吉廷斯策略的分析结果，可同时处理以下任意组合：(a)多服务器、(b)非泊松到达、(c)设置时间。因此，我们的结论覆盖含/不含设置时间的G/G/1与G/G/k队列，并在每种情况下限定了吉廷斯策略的次优性差距。上述(a)(b)(c)三类因素各自为界限增添一个项，但所有项在重流量条件下均可忽略，从而证明吉廷斯策略在我们考虑的所有系统中均具有重流量最优性。结论的另一推论是：在含设置时间的M/G/1队列中，吉廷斯策略在所有负载条件下均为最优策略。