We consider scheduling to minimize mean response time of the M/G/k queue with unknown job sizes. In the single-server case, the optimal policy is the Gittins policy, but it is not known whether Gittins or any other policy is optimal in the multiserver case. Exactly analyzing the M/G/k under any scheduling policy is intractable, and Gittins is a particularly complicated policy that is hard to analyze even in the single-server case.
In this work we introduce monotonic Gittins (M-Gittins), a new variation of the Gittins policy, and show that it minimizes mean response time in the heavy-traffic M/G/k for a wide class of finite-variance job size distributions. We also show that the monotonic shortest expected remaining processing time (M-SERPT) policy, which is simpler than M-Gittins, is a 2-approximation for mean response time in the heavy traffic M/G/k under similar conditions. These results constitute the most general optimality results to date for the M/G/k with unknown job sizes. Our techniques build upon work by Grosof et al., who study simple policies, such as SRPT, in the M/G/k; Bansal et al., Kamphorst and Zwart, and Lin et al., who analyze mean response time scaling of simple policies in the heavy-traffic M/G/1; and Aalto et al. and Scully et al., who characterize and analyze the Gittins policy in the M/G/1.
Multiserver-job systems, where jobs require concurrent service at many servers, occur widely in practice. Much is known in the dropping setting, where jobs are immediately discarded if they require more servers than are currently available. However, very little is known in the more practical setting where jobs queue instead. In this paper, we derive a closed-form analytical expression for the stability region of a two-class (non-dropping) multiserver-job system where each class of jobs requires a distinct number of servers and requires a distinct exponential distribution of service time, and j…
Multiserver-job systems, where jobs require concurrent service at many servers, occur widely in practice. Essentially all of the theoretical work on multiserver-job systems focuses on maximizing utilization, with almost nothing known about mean response time. In simpler settings, such as various known-size single-server-job settings, minimizing mean response time is merely a matter of prioritizing small jobs. However, for the multiserver-job system, prioritizing small jobs is not enough, because we must also ensure servers are not unnecessarily left idle. Thus, minimizing mean response time re…
Queues with setup time are extensively studied because they have application in performance evaluation of power-saving data centers. In a data center, there are a huge number of servers which consume a large amount of energy. In the current technology, an idle server still consumes about 60\% of its peak processing a job. Thus, the only way to save energy is to turn off servers which are not processing a job. However, when there are some waiting jobs, we have to turn on the OFF servers. A server needs some setup time to be active during which it consumes energy but cannot process a job. Theref…
The Shortest Remaining Processing Time (SRPT) scheduling policy and its variants have been extensively studied in both theoretical and practical settings. While beautiful results are known for single-server SRPT, much less is known for multiserver SRPT. In particular, stochastic analysis of the M/G/k under multiserver SRPT is entirely open. Intuition suggests that multiserver SRPT should be optimal or near-optimal for minimizing mean response time. However, the only known analysis of multiserver SRPT is in the worst-case adversarial setting, where SRPT can be far from optimal. In this paper, w…
Multiserver queueing systems are found at the core of a wide variety of practical systems. Many important multiserver models have a previously-unexplained similarity: identical mean response time behavior is empirically observed in the heavy traffic limit. We explain this similarity for the first time. We do so by introducing the work-conserving finite-skip (WCFS) framework, which encompasses a broad class of important models. This class includes the heterogeneous M/G/k, the limited processor sharing policy for the M/G/1, the threshold parallelism model, and the multiserver-job model under a n…
Recently it was shown that, contrary to expectations, the First-Come-First-Served (FCFS) scheduling algorithm can be stochastically improved upon by a scheduling algorithm called {\it Nudge} for light-tailed job size distributions. Nudge partitions jobs into 4 types based on their size, say small, medium, large and huge jobs. Nudge operates identical to FCFS, except that whenever a {\it small} job arrives that finds a {\it large} job waiting at the back of the queue, Nudge swaps the small job with the large one unless the large job was already involved in an earlier swap. In this paper, we sho…
A queue is required when a service provider is not able to handle jobs arriving over the time. In a highly flexible and dynamic environment, some jobs might demand for faster execution at run-time especially when the resources are limited and the jobs are competing for acquiring resources. A user might demand for speed up (reduced wait time) for some of the jobs present in the queue at run time. In such cases, it is required to accelerate (directly sending the job to the server) urgent jobs (requesting for speed up) ahead of other jobs present in the queue for an earlier completion of urgent j…
Deep neural networks training jobs and other iterative computations frequently include checkpoints where jobs can be canceled based on the current value of monitored metrics. While most of existing results focus on the performance of all jobs (both successfully completed and canceled), in this work we explore scheduling policies that improve the sojourn time of successful jobs, which are typically more valuable to the user. Our model assumes that each job has a known discrete size distribution (e.g., estimated from previous execution logs) where the largest size value indicates a successful co…
When parallelizing a set of jobs across many servers, one must balance a trade-off between granting priority to short jobs and maintaining the overall efficiency of the system. When the goal is to minimize the mean flow time of a set of jobs, it is usually the case that one wants to complete short jobs before long jobs. However, since jobs usually cannot be parallelized with perfect efficiency, granting strict priority to the short jobs can result in very low system efficiency which in turn hurts the mean flow time across jobs. In this paper, we derive the optimal policy for allocating servers…
Scheduling to minimize mean response time in an M/G/1 queue is a classic problem. The problem is usually addressed in one of two scenarios. In the perfect-information scenario, the scheduler knows each job's exact size, or service requirement. In the zero-information scenario, the scheduler knows only each job's size distribution. The well-known shortest remaining processing time (SRPT) policy is optimal in the perfect-information scenario, and the more complex Gittins policy is optimal in the zero-information scenario. In real systems the scheduler often has partial but incomplete information…
This report contains the description of two novel job shop scheduling benchmarks that resemble instances of real scheduling problem as they appear in industry. In particular, the aim was to provide large-scale benchmarks (up to 1 million operations) to test the state-of-the-art scheduling solutions on problems that are closer to what occurs in a real industrial context. The first benchmark is an extension of the well known Taillard benchmark (1992), while the second is a collection of scheduling instances with a known-optimum solution.
We study the performance of non-adaptive scheduling policies in computing systems with multiple servers. Compute jobs are mostly regular, with modest service requirements. However, there are sporadic data intensive jobs, whose expected service time is much higher than that of the regular jobs. Forthis model, we are interested in the effect of scheduling policieson the average time a job spends in the system. To this end, we introduce two performance indicators in a simplified, only-arrival system. We believe that these performance indicators are good predictors of the relative performance of t…
Research Center of Smart Networks and Systems, Fudan University（复旦大学智慧网络与系统研究中心）；School of Operations Research and Information Engineering, Cornell University（康奈尔大学运筹学与信息工程学院）
Minimizing job scheduling time is a fundamental issue in data center networks that has been extensively studied in recent years. The incoming jobs require different CPU and memory units, and span different number of time slots. The traditional solution is to design efficient heuristic algorithms with performance guarantee under certain assumptions. In this paper, we improve a recently proposed job scheduling algorithm using deep reinforcement learning and extend it to multiple server clusters. Our study reveals that deep reinforcement learning method has the potential to outperform traditional…
The job-shop scheduling is one of the most studied optimization problems from the dawn of computer era to the present day. Its combinatorial nature makes it easily expressible as a constraint satisfaction problem. In this paper, we compare the performance of two constraint solvers on the job-shop scheduling problem. The solvers in question are: OR-Tools, an open-source solver developed by Google and winner of the last MiniZinc Challenge, and CP Optimizer, a proprietary IBM constraint solver targeted at industrial scheduling problems. The comparison is based on the goodness of the solutions fou…
It is well known that size-based scheduling policies, which take into account job size (i.e., the time it takes to run them), can perform very desirably in terms of both response time and fairness. Unfortunately, the requirement of knowing a priori the exact job size is a major obstacle which is frequently insurmountable in practice. Often, it is possible to get a coarse estimation of job size, but unfortunately analytical results with inexact job sizes are challenging to obtain, and simulation-based studies show that several size-based algorithm are severely impacted by job estimation errors.…
We study the conditional sojourn time distributions of processor sharing (PS), foreground background processor sharing (FBPS) and shortest remaining processing time first (SRPT) scheduling disciplines on an event where the job size of a customer arriving in stationarity is smaller than exactly k>=0 out of the preceding m>=k arrivals. Then, conditioning on the preceding event, the sojourn time distribution of this newly arriving customer behaves asymptotically the same as if the customer were served in isolation with a server of rate (1-ρ)/(k+1) for PS/FBPS, and (1-ρ) for SRPT, respectively, wh…
We address single machine problems with optional jobs - rejection, studied recently in Zhang et al. [21] and Cao et al. [2]. In these papers, the authors focus on minimizing regular performance measures, i.e., functions that are non-decreasing in the jobs completion time, subject to the constraint that the total rejection cost cannot exceed a predefined upper bound. They also prove that the considered problems are ordinary NP-hard and provide pseudo-polynomial-time Dynamic Programming (DP) solutions. In this paper, we focus on three of these problems: makespan with release-dates; total complet…
For job scheduling systems, where jobs require some amount of processing and then leave the system, it is natural for each user to provide an estimate of their job's time requirement in order to aid the scheduler. However, if there is no incentive mechanism for truthfulness, each user will be motivated to provide estimates that give their job precedence in the schedule, so that the job completes as early as possible. We examine how to make such scheduling systems incentive compatible, without using monetary charges, under a natural queueing theory framework. In our setup, each user has an esti…
Queuing systems with redundant requests have drawn great attention because of their promise to reduce the job completion time and variability. Despite a large body of work on the topic, we are still far from fully understanding the benefits of redundancy in practice. We here take one step towards practical systems by studying queuing systems with bi-modal job service time distribution. Such distributions have been observed in practice, as can be seen in, e.g., Google cluster traces. We develop an analogy to a classical urns and balls problem, and use it to study the queuing time performance of…
