(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.
There are some exact quantity analyses @cite_15 @cite_13 for purging @math fork-join queues, as it is equivalent to a single queue with @math times the service rate. @cite_15 gave comprehensive research on purging @math fork-join queues, with considerations on multi-class jobs, interferences from un-forked jobs and heterogeneous service time distributions. @cite_13 take the purging overheads into consideration, since the cancellation of running jobs typically incurs unnegligible delays in practice.
@cite_15 @cite_13
2015774390 2152474925 2094799119 618449037
In a Fork-Join (FJ) queueing system an upstream fork station splits incoming jobs into N tasks to be further processed by N parallel servers, each with its own queue; the response time of one job is determined, at a downstream join station, by the maximum of the corresponding tasks' response times. This queueing system is useful to the modelling of multi-service systems subject to synchronization constraints, such as MapReduce clusters or multipath routing. Despite their apparent simplicity, FJ systems are hard to analyze. This paper provides the first computable stochastic bounds on the waiting and response time distributions in FJ systems. We consider four practical scenarios by combining 1a) renewal and 1b) non-renewal arrivals, and 2a) non-blocking and 2b) blocking servers. In the case of non blocking servers we prove that delays scale as O(logN), a law which is known for first moments under renewal input only. In the case of blocking servers, we prove that the same factor of log N dictates the stability region of the system. Simulation results indicate that our bounds are tight, especially at high utilizations, in all four scenarios. A remarkable insight gained from our results is that, at moderate to high utilizations, multipath routing 'makes sense' from a queueing perspective for two paths only, i.e., response times drop the most when N = 2; the technical explanation is that the resequencing (delay) price starts to quickly dominate the tempting gain due to multipath transmissions. Fork-join structures have gained increased importance in recent years as a means of modeling parallelism in computer and storage systems. The basic fork-join model is one in which a job arriving at a parallel system splits into K independent tasks that are assigned to K unique, homogeneous servers. In the paper, a simple response time approximation is derived for parallel systems with exponential service time distributions. The approximation holds for networks modeling several devices, both parallel and nonparallel. (In the case of closed networks containing a stand-alone parallel system, a mean response time bound is derived.) In addition, the response time approximation is extended to cover the more realistic case wherein a job splits into an arbitrary number of tasks upon arrival at a parallel system. Simulation results for closed networks with stand-alone parallel subsystems and exponential service time distributions indicate that the response time approximation is, on average, within 3 percent of the seeded response times. Similarly, simulation results with nonexponential distributions also indicate that the response time approximation is close to the seeded values. Potential applications of our results include the modeling of data placement in disk arrays and the execution of parallel programs in multiprocessor and distributed systems. Approximation techniques are developed to evaluate the performance of symmetric fork-join synchronization delays for K M G 1 queues. For a server utilization spl rho , the mean response time for fork-join requests is expressed as the sum of the mean response time at one of the queues and the mean synchronization delay as follows: R sub K sup F 1 ( spl rho )=R sub 1 ( spl rho )+F sub K spl alpha sub K ( spl rho ) spl sigma sub 1 ( spl rho ), where F sub K is obtained from the previous equation at spl rho =0, R sub 1 ( spl rho ) and spl sigma sub 1 ( spl rho ) are the mean and the standard deviation of response time at any one of the queues, respectively, and spl alpha sub K ( spl rho ) is a low-degree service-time distribution dependent polynomial in spl rho , whose coefficients are determined from simulation results. We also use simulation results to show that when fork-join requests share the servers with local requests, a good approximation (and an upper bound) to the fork-join response time is obtained by treating the components of fork-join response time as independent, i.e., the mean fork-join response time can be approximated by the expected value of the maximum of the response times at the K queues. Fork-join queueing networks model a network of parallel servers in which an arriving job splits into a number of subtasks that are serviced in parallel. Fork-join queues can be used to model disk arrays. A response time approximation of the fork-join queue is presented that attempts to comply with the additional constraints of modelling a disk array. This approximation is compared with existing analytical approximations of the fork-join queueing network.