M/G/1处理机与永久作业共享队列逗留时间矩的显式公式

On the sojourn time of a batch in the $M^{[X]}/M/1$ Processor Sharing Queue

In this paper, we analyze the sojourn of an entire batch in a processor sharing $M^{[X]}/M/1$ processor queue, where geometrically distributed batches arrive according to a Poisson process and jobs require exponential service times. By conditioning on the number of jobs in the systems and the number of jobs in a tagged batch, we establish recurrence relations between conditional sojourn times, which subsequently allow us to derive a partial differential equation for an associated bivariate generating function. This equation involves an unknown generating function, whose coefficients can be com…

Asymptotic analysis of the sojourn time of a batch in an $M^{[X]}/M/1$ Processor Sharing Queue

In this paper, we exploit results obtained in an earlier study for the Laplace transform of the sojourn time $Ω$ of an entire batch in the $M^{[X]}/M/1$ Processor Sharing (PS) queue in order to derive the asymptotic behavior of the complementary probability distribution function of this random variable, namely the behavior of $P(Ω>x)$ when $x$ tends to infinity. We precisely show that up to a multiplying factor, the behavior of $P(Ω>x)$ for large $x$ is of the same order of magnitude as $P(ω>x)$, where $ω$ is the sojourn time of an arbitrary job is the system. From a practical point of view, t…

Approximate Formulas for Characteristics of Multichannel LIFO Preemptive-Resume Priority Queueing System

This paper considers a multichannel preemptive-resume priority queueing system with a Poisson input and an arbitrary service time distribution depending on the priority of job. Jobs of the same priority are serviced according to the LIFO rule. If, at moment of job arrival, all servers are busy, and at least one server is busy with the service of a job of a not higher than the priority of arriving job, then the service of a job is preempted such that the priority of preempted job is lowest from the priorities of the jobs in service. The service of a preempted job is resumed later. The paper pro…

Analysis of Load Balancing in Large Heterogeneous Processor Sharing Systems

We analyze randomized dynamic load balancing schemes for multi-server processor sharing systems when the number of servers in the system is large and the servers have heterogeneous service rates. In particular, we focus on the classical power-of-two load balancing scheme and a variant of it in which a newly arrived job is assigned to the server having the least instantaneous Lagrange shadow cost among two randomly chosen servers. The instantaneous Lagrange shadow cost at a server is given by the ratio of the number of unfinished jobs at the server to the capacity of the server. Two different a…

具有软截止期的处理器共享队列的高负荷服务限制

Heavy traffic limit for a processor sharing queue with soft deadlines

本文研究了一个GI/GI/1处理机共享队列，其中的作业具有软截止期。在每个时间点，剩余的服务时间和最后期限的收集建模使用右半平面上的随机计数措施。在扩散标度和高负荷服务条件下得到了该测度值过程的极限，并刻画为极限队列长度过程的确定性函数.作为特殊情况，得到了提前时间曲线和排队时间曲线的扩散近似。还得到了逗留时间的快拍原理。

具有服务重置的队列

Queues with service resetting

针对服务时间由服务端减速与任务固有大小共同决定的S&X队列，构建服务重置的完整理论，分析两种主流重置策略下的系统性能，证明服务重置可有效提升该类排队系统性能。