(n, k) fork-join queues are prevalent in popular distributed systems, erasure coding based cloud storages, and modern network protocols like multipath routing, estimating the sojourn time of such queues is thus critical for the performance measurement and resource plan of computer clusters. However, the estimating keeps to be a well-known open challenge for years, and only rough bounds for a limited range of load factors have been given. This paper developed a closed-form linear transformation technique for jointly-identical random variables: An order statistic can be represented by a linear combination of maxima. This brand-new technique is then used to transform the sojourn time of non-purging (n, k) fork-join queues into a linear combination of the sojourn times of basic (k, k), (k + 1, k + 1), ..., (n, n) fork-join queues. Consequently, existing approximations for basic fork-join queues can be bridged to the approximations for non-purging (n, k) fork-join queues. The uncovered approximations are then used to improve the upper bounds for purging (n, k) fork-join queues. Simulation experiments show that this linear transformation approach is practiced well for moderate n and relatively large k.
We consider the double queue that arises when arriving customers simultaneously place two demands handled independently by two servers. It is assumed that the customer arrivals form a Poisson process with mean 1, the servers have exponential service times with rate @math , and @math , which insures stability of the queue. Let @math be the respective lengths of the @math - and @math -queues, and @math at equilibrium. In a previous paper with the same title we obtained a formula for the generating function @math . We use this to derive the asymptotic behavior of @math as @math . The asymptotic results are employed to study the interdependence of @math . We derive limit laws for the expectation and distribution for either of these random variables conditioned on the other. In this article, we study the problem of finding tight bounds on the expected value of the kth-order statistic E [Xk:n] under first and second moment information on n real-valued random variables. Given means E [Xi] = mi and variances Var[Xi] = σi2, we show that the tight upper bound on the expected value of the highest-order statistic E [Xn:n] can be computed with a bisection search algorithm. An extremal discrete distribution is identified that attains the bound, and two closed-form bounds are proposed. Under additional covariance information Cov[Xi,Xj] = Qij, we show that the tight upper bound on the expected value of the highest-order statistic can be computed with semidefinite optimization. We generalize these results to find bounds on the expected value of the kth-order statistic under mean and variance information. For k < n, this bound is shown to be tight under identical means and variances. All of our results are distribution-free with no explicit assumption of independence made. Particularly, using optimization methods, we develop tractable approaches to compute bounds on the expected value of order statistics. Although order statistics have been studied for several decades, most of the results are based on the assumption of independent and identically distributed (i.i.d.) random variables. In the literature, how to compute the mth order statistics of n correlated random variables is still a problem. This article proposes a recursive algorithm based on statistical min max operations to compute order statistics for general correlated and not necessarily identically distributed random variables. The algorithm has an O(mn) time complexity and O(m p n) space complexity. A binary tree-based data structure is further developed to allow selective update of the order statistics with O(nm2) time. As a vehicle to demonstrate the algorithm, we apply it to the path selection algorithm in at-speed testing. A novel metric multilayer process space coverage metric is proposed to quantitatively gauge the quality of path selection. We then show that such a metric is directly linked to the order statistics, and our recursive algorithm can thus be applied. By employing a branch-and-bound path selection algorithm with these techniques, this article shows that selecting an optimal set of paths for a multimillion-gate design can be performed efficiently. Compared to the state of the art, experimental results show both the efficiency of our algorithms and better quality of our path selection. An approximation technique, called scaling approximation, is introduced and applied to the analysis of homogeneous fork join queuing systems consisting of K>or=2 servers. The development of the scaling approximation technique is guided by both experimental and theoretical considerations. The approximation is based on the observation that there exist upper and lower bounds on the mean response time that grow at the same rate as a function of K. Simple, closed-form approximate expressions for the mean response time are derived and compared to simulation results. The relative error in the approximation is less than 5 for K > 