The Shortest Remaining Processing Time (SRPT) scheduling policy and variants thereof have been deployed in many computer systems, including web servers [5], networks [9], databases [3] and operating systems [1]. SRPT has also long been a topic of fascination for queueing theorists due to its optimality properties. In 1966, the mean response time for SRPT was first derived [11], and in 1968 SRPT was shown to minimize mean response time in both a stochastic sense and a worst-case sense [10]. However, these beautiful optimality results and the analysis of SRPT are only known for single-server systems. Almost nothing is known about SRPT in multiserver systems, such as the M/G/k, even for the case of just k = 2 servers.

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 singleserver 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 worstcase adversarial setting, where SRPT can be far from optimal. In this paper, we give the first stochastic analysis bounding mean response time of the M/G/k under multiserver SRPT. Using our response time bound, we show that multiserver SRPT has asymptotically optimal mean response time in the heavy-traffic limit. The key to our bounds is a strategic combination of stochastic and worst-case techniques. Beyond SRPT, we prove similar response time bounds and optimality results for several other multiserver scheduling policies.

This paper proves that push-pull block puzzles in 3D are PSPACE-complete to solve, and push-pull block puzzles in 2D with thin walls are NP-hard to solve, settling an open question

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 (nondropping) multiserver-job system where each class of jobs requires a distinct number of servers and requires a distinct exponential distribution of service time, and jobs are served in first-come-first-served (FCFS) order. This is the first result of any kind for an FCFS multiserver-job system where the classes have distinct service distributions. Our work is based on a technique that leverages the idea of a "saturated" system, in which an unlimited number of jobs are always available.Our analytical formula provides insight into the behavior of FCFS multiserver-job systems, highlighting the huge wastage (idle servers while jobs are in the queue) that can occur, as well as the nonmonotonic effects of the service rates on wastage. 1 The data was published in a scaled form [27]. We rescale the data so the smallest job in the trace uses one normalized CPU.

We consider scheduling to minimize mean response time of the M/G/k queue with unknown job sizes. In the singleserver k = 1 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.