Linearizability of concurrent data structures is usually proved by monolithic simulation arguments relying on the identification of the so-called linearization points. Regrettably, such proofs, whether manual or automatic, are often complicated and scale poorly to advanced non-blocking concurrency patterns, such as helping and optimistic updates. In response, we propose a more modular way of checking linearizability of concurrent queue algorithms that does not involve identifying linearization points. We reduce the task of proving linearizability with respect to the queue specification to establishing four basic properties, each of which can be proved independently by simpler arguments. As a demonstration of our approach, we verify the Herlihy and Wing queue, an algorithm that is challenging to verify by a simulation proof.
To the best of our knowledge, there exist only two earlier published proofs of the HW queue: (1) the original pencil-and-paper proof by Herlihy and Wing @cite_8 , and (2) a mechanized backward simulation proof by @cite_7 .
@cite_7 @cite_8
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It is well known that head-of-line blocking limits the throughput of an input-queued switch with first-in-first-out (FIFO) queues. Under certain conditions, the throughput can be shown to be limited to approximately 58.6 . It is also known that if non-FIFO queueing policies are used, the throughput can be increased. However, it has not been previously shown that if a suitable queueing policy and scheduling algorithm are used, then it is possible to achieve 100 throughput for all independent arrival processes. In this paper we prove this to be the case using a simple linear programming argument and quadratic Lyapunov function. In particular, we assume that each input maintains a separate FIFO queue for each output and that the switch is scheduled using a maximum weight bipartite matching algorithm. We introduce two maximum weight matching algorithms: longest queue first (LQF) and oldest cell first (OCF). Both algorithms achieve 100 throughput for all independent arrival processes. LQF favors queues with larger occupancy, ensuring that larger queues will eventually be served. However, we find that LQF can lead to the permanent starvation of short queues. OCF overcomes this limitation by favoring cells with large waiting times. This paper is a survey of certain known results concerning networks of queues. The choice of the topics presented has been made with special emphasis on mathematical results which can be applied to the analysis and synthesis of mathematical or simulation models of complex computer systems in which an ensemble of resources is shared among a set of user programs. The subjects covered include the Jackson, and Gordon and Newell theorems; the work-rate theorems of Chang, Lavenberg and Traiger; the Baskett, Chandy, Muntz, Palacios results. We also discuss in a unified manner problems related to Poisson processes in queueing networks. Companion papers (Parts II, III) will present results concerning approximations to queueing networks and some of their applications to computer system performance evaluation. In this paper, we study discrete-time priority queueing systems fed by a large number of arrival streams. We first provide bounds on the actual delay asymptote in terms of the virtual delay asymptote. Then, under suitable assumptions on the arrival process to the queue, we show that these asymptotes are the same. As an application of this result, we then consider a priority queueing system with two queues. Using the earlier result, we derive an upper bound on the tail probability of the delay. Under certain assumptions on the rate function of the arrival process, we show that the upper bound is tight. We then consider a system with Markovian arrivals and numerically evaluate the delay tail probability and validate these results with simulations. In this paper we study a fundamental open problem in the area of probabilistic checkable proofs: What is the smallest s such that NP ⊆ naPCP1,s[O(log n),3]? In the language of hardness of approximation, this problem is equivalent to determining the smallest s such that getting an s-approximation for satisfiable 3-bit constraint satisfaction problems ("3-CSPs") is NP-hard. The previous best upper bound and lower bound for s are 20 27+µ by Khot and Saket [KS06], and 5 8 (assuming NP subseteq BPP) by Zwick [Zwi98]. In this paper we close the gap assuming Khot's d-to-1 Conjecture. Formally, we prove that if Khot's d-to-1 Conjecture holds for any finite constant integer d, then NP naPCP1,5 8+ µ[O(log n),3] for any constant µ > 0. Our conditional result also solves Hastad's open question [Has01] on determining the inapproximability of satisfiable Max-NTW ("Not Two") instances and confirms Zwick's conjecture [Zwi98] that the 5 8-approximation algorithm for satisfiable 3-CSPs is optimal.