The response time of jobs with multiple parallel tasks  is a critical performance metric in many systems, including MapReduce systems, coded data storage systems, etc. However, tight analytical characterizations of the response time of such jobs are largely unknown except for highly degenerate cases. The difficulty is rooted in the fact that a job with multiple tasks is considered complete only when all of its tasks complete processing; i.e., the job response time is the maximum of the response times of its tasks  , which is hard to analyze since these task response times are generally not independent. In this paper, we approach this problem by studying when the response times of a job's tasks are close to being independent. We consider a limited fork-join model with n servers, where each job consists of k ^{(n)} %DIFDELCMD < \le %%% ntasks. Upon arrival, each jobchooses k^{(n) under the condition k^{(n)} = o(n^{1/4}) , the steady state response times at any k^{(n) greatly generalizes the asymptotic-independence type of results in the literature where asymptotic independence is shown only for a fixed constant number of queues. We then further show that the job response time converges to the maximum of independent task response times, in a proper sense. This gives the first asymptotically tight analytical characterization of the response time of a multi-task parallel job. To complement the asymptotic independence result, we also show that when k^{(n) =\Theta(n), any number of multiple queues are not asymptotically independent. Analysis for the regime of k}  ^{(n)} between o(n^{1/4) and \Theta(n) remains open} \le  .
We study delay of jobs that consist of multiple parallel tasks , which is a critical performance metric in a wide range of applications such as data file retrieval in coded storage systems and parallel computing. In this problem, each job is completed only when all of its tasks are completed, so the delay of a job is the maximum of the delays of its tasks . Despite the wide attention this problem has received, tight analysis is still largely unknown since analyzing job delay requires characterizing the complicated correlation among task delays, which is hard to do. We first consider an asymptotic regime where the number of servers, n , goes to infinity, and the number of tasks in a job, k^{(n) ^{(n)} %DIFDELCMD < \le %%% queues under the condition k^{(n)} = o(n^{1/4}) . This greatly generalizes the asymptotic-independence type of results in the literature where asymptotic independence is shown only for a fixed constant number of queues. As a consequence of our independence result, the job delay converges to the maximum of independent task =\Theta(n), any number of multiple queues are not asymptotically independent. Analysis for the regime of k} delays. We next consider the non-asymptotic regime. Here we prove that independence yields a stochastic upper bound on job delay for any n and any k^{(n) ^{(n)} ) and \Theta(n) remains open} \le n. The key component of our proof is a new technique we develop, called "Poisson oversampling". Our approach converts the job delay problem into a corresponding balls-and-bins problem. However, in contrast with typical balls-and-bins problems where there is a negative correlation among bins, we prove that our variant exhibits positive correlation .
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The response time of jobs with
We study delay of jobs that consist of
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many systems, including MapReduce systems, coded data storage systems, etc. However, tight analytical characterizations of the response time of such jobs are largely unknown except for highly degenerate cases. The difficulty is rooted in the fact that a job with multiple tasks is considered complete
a wide range of applications such as data file retrieval in coded storage systems and parallel computing. In this problem, each job is completed
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complete processing; i.e., the job response time
are completed, so the delay of a job
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response times
delays
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. Despite the wide attention this problem has received, tight analysis is still largely unknown since analyzing job delay requires characterizing the complicated correlation among task delays, which is hard to do. We first consider an asymptotic regime where the number of servers, n
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which is hard to analyze since these task response times are generally not independent. In this paper, we approach this problem by studying when the response times of a job's tasks are close to being independent. We consider a limited fork-join model with n servers, where each job consists of k
goes to infinity, and the number of tasks in a job, k^{(n)
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ntasks. Upon arrival, each jobchooses k^{(n)
queues
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, the steady state response times at any k^{(n)
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We then further show that the job response time
As a consequence of our independence result, the job delay
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response times, in a proper sense. This gives the first asymptotically tight analytical characterization of the response time of a multi-task parallel job. To complement the asymptotic independence result, we also show that when k^{(n)
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delays. We next consider the non-asymptotic regime. Here we prove that independence yields a stochastic upper bound on job delay for any n and any k^{(n)
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n. The key component of our proof is a new technique we develop, called "Poisson oversampling". Our approach converts the job delay problem into a corresponding balls-and-bins problem. However, in contrast with typical balls-and-bins problems where there is a negative correlation among bins, we prove that our variant exhibits positive correlation