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<p><b>Abstract</b>—A new analysis technique, dynamic-bubblesort analysis, is introduced for general <it>K</it>-queue first-in-first-out HFJ (homogenous fork/join queuing) systems of <tmath>$K \geq 2$</tmath>. The dynamic-bubblesort model dynamically sorts the branches of the queues based on the number of the tasks waiting for synchronization in each branch. Jobs arrive with mean rate <tmath>$\lambda$</tmath> and a general arrival distribution. Upon arrival, a job forks into <it>K</it> tasks. Task <it>k</it>, <tmath>$k = 1, 2,\ldots, K$</tmath>, is assigned to the <it>k</it>th queuing system, which is a first-in-first-out server with a general service distribution and an infinite capacity queue. A job leaves the HFJ system as soon as all its tasks complete their service. In other words, tasks corresponding to the same job are joined before departing the HFJ system. We obtain a general and simple hybrid solution which combines analysis and simulation for the mean response time that we denote by <tmath>$T_K$</tmath>. We obtain a very simple (as a function of <tmath>$T_1$</tmath> and <tmath>$T_2$</tmath> only) and general upper bound expression for <tmath>$T_K$</tmath> and we get an exact relationship between the cases for <tmath>$K =2$</tmath> and 3. We evaluate our results by simulating <tmath>$2, 3, \ldots, 99$</tmath>, and 100 queues for <tmath>$\rho = 0.1, 0.2,\ldots, 0.8$</tmath>, and 0.9, each for four <b>different</b> HFJ cases, where <tmath>$\rho =\lambda/\mu$</tmath> and <tmath>$\mu$</tmath> is the average task service rate for a server. The maximum absolute offset for our hybrid solutions from <b>all</b> the simulations is less than 0.33 percent (1/300), which is a reasonable error ratio for simulation. The maximum offset for our upper bounds over <b>all</b> the simulations is 21 percent. Also, we compare our results with three recent papers [<ref rid="bibL082919" type="bib">19</ref>], [<ref rid="bibL082920" type="bib">20</ref>], [<ref rid="bibL082922" type="bib">22</ref>].</p>
Fork/join queuing system, modeling and analysis, dynamic-bubblesort analysis, upper bounds, parallel queues, synchronization.

R. J. Chen, "A Hybrid Solution of Fork/Join Synchronization in Parallel Queues," in IEEE Transactions on Parallel & Distributed Systems, vol. 12, no. , pp. 829-845, 2001.
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