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Abstract Priority queues have long been used to increase revenue by exploiting the fact that time-sensitive customers are willing to pay for shorter waiting times. This fact begs the question: Can one make even more revenue by relaxing the strictness of the priority policy? This paper answers this question under the unobservable queue setting, where customers are heterogeneous in their time-sensitivity; specifically the time-sensitivity of customers is allowed to follow an arbitrary distribution. In this paper, we prove necessary and sufficient conditions under which partial priority can increase the revenue. Specifically, we find a surprising result: Although partial priority offers much more flexibility than strict priority, partial priority only increases revenue if there are two additional constraints on the service provider, one setting a maximum price and the other setting a maximum waiting time. In the absence of either of these constraints, we prove that strict priority maximizes revenue. Finally, in situations where partial priority increases the revenue, we analytically characterize the amount of improvement. 

Abstract In practice, the cost of delaying a job can grow as the job waits. Such behavior is modeled by the time-varying holding cost (TVHC) problem, where each job’s instantaneous holding cost increases with its current age (a job’s age is the time since it arrived). The goal of the TVHC problem is to find a scheduling policy that minimizes the time-average total holding cost across all jobs. However, no optimality results are known for the TVHC problem outside of the asymptotic regime. In this paper, we study a simple yet still challenging special case: A two-class M/M/1 queue in which class 1 jobs incur a non-decreasing, time-varying holding cost and class 2 jobs incur a constant holding cost. Our main contribution is deriving the first optimal (non-decreasing) index policy for this special case of the TVHC problem. Our optimal policy, called LookAhead, stems from the following idea: Rather than considering each job’scurrentholding cost when making scheduling decisions, we should look at their cost someXtime into the future, where thisXis intuitively called the “lookahead amount. This paper derives that optimal lookahead amount. 

Scheduling a stream of jobs whose holding cost changes over time is a classic and practical problem. Specifically, each job is associated with a holding cost (penalty), where a job's instantaneous holding cost is some increasing function of its current age (the time it has spent in the system since its arrival) and its class. The goal is to schedule the jobs to minimize the time-average total holding cost across all jobs. 

We consider the problem of scheduling to minimize asymptotic tail latency in an M/G/1 queue with unknown job sizes. When the job size distribution is heavy-tailed, numerous policies that do not require job size information (e.g. Processor Sharing, Least Attained Service) are known to be strongly tail optimal, meaning that their response time tail has the fastest possible asymptotic decay. In contrast, for light-tailed size distributions, only in the last few years have policies been developed that outperform simple First-Come First-Served (FCFS). The most recent of these is γ-Boost, which achieves strong tail optimality in the light-tailed setting. But thus far, all policies that outperform FCFS in the light-tailed setting, including γ-Boost, require known job sizes. In this paper, we design a new scheduling policy that achieves strong tail optimality in the light-tailed M/G/1 with unknown job sizes. Surprisingly, the optimal policy turns out to be a variant of the Gittins policy, but with a novel and unusual feature: it uses a negative discount rate. Our work also applies to systems with partial information about job sizes, covering γ-Boost as an extreme case when job sizes are in fact fully known. 

Priority queues are well understood in queueing theory. However, they are somewhat restrictive in that the low-priority customers suffer far greater waiting times than the highpriority customers. In this short paper, we introduce a novel generalization of a two-class priority queue, which we call Hybrid. We prove that Hybrid has a much broader achievability region than strict priority, allowing for a much greater range of waiting time pairs. We demonstrate settings where this new flexibility can increase the revenue obtained by a service system (like airport TSA) selling priority. 

Preemptive scheduling policies, which allow pausing jobs mid-service, are ubiquitous because they allow important jobs to receive service ahead of unimportant jobs that would otherwise delay their completion. The canonical example is Shortest Remaining Processing Time (SRPT), which preemptively serves the job with least remaining work at every moment in time [9]. There is a robust literature analyzing response time (elapsed time between a job's arrival and completion) in the M/G/1 queue under many preemptive policies [6, 10, 11], shedding light on questions such as how preemption affects the mean and tail of response time, and whether preemption is unfair towards low-priority jobs. 

Service level objectives (SLOs) for queueing systems typically relate to the tail of the system's response time distribution T. The tail is the function mapping a time t to the probability P[T > t]. SLOs typically ask that high percentiles of T are not too large, i.e. that P[T > t] is small for large t. 

We study the problem of scheduling jobs in a queueing system, specifically an M/G/1 with light-tailed job sizes, to asymptotically optimize the response time tail. This means scheduling to make P[T > t], the chance a job's response time exceeds t, decay as quickly as possible in the t \to \infty limit. For some time, the best known policy was First-Come First-Served (FCFS), which has an asymptotically exponential tail: P[T > t] ~ C e^-γ t . FCFS achieves the optimal decay rate γ, but its tail constant C is suboptimal. Only recently have policies that improve upon FCFS's tail constant been discovered. But it is unknown what the optimal tail constant is, let alone what policy might achieve it. In this paper, we derive a closed-form expression for the optimal tail constant C, and we introduce γ-Boost, a new policy that achieves this optimal tail constant. Roughly speaking, γ-Boost operates similarly to FCFS, but it pretends that small jobs arrive earlier than their true arrival times. This significantly reduces the response time of small jobs without unduly delaying large jobs, improving upon FCFS's tail constant by up to 50% with only moderate job size variability, with even larger improvements for higher variability. While these results are for systems with full job size information, we also introduce and analyze a version of γ-Boost that works in settings with partial job size information, showing it too achieves significant gains over FCFS. Finally, we show via simulation that γ-Boost has excellent practical performance. 

We consider scheduling in the M/G/1 queue with unknown job sizes. It is known that the Gittins policy minimizes mean response time in this setting. However, the behavior of the tail of response time under Gittins is poorly understood, even in the large-response-time limit. Characterizing Gittins’s asymptotic tail behavior is important because if Gittins has optimal tail asymptotics, then it simultaneously provides optimal mean response time and good tail performance. In this work, we give the first comprehensive account of Gittins’s asymptotic tail behavior. For heavy-tailed job sizes, we find that Gittins always has asymptotically optimal tail. The story for light-tailed job sizes is less clear-cut: Gittins’s tail can be optimal, pessimal, or in between. To remedy this, we show that a modification of Gittins avoids pessimal tail behavior, while achieving near-optimal mean response time.