On the Performance of Large Loss Systems with Adaptive Multiserver Jobs
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In this paper, we study systems where each job or request can be split into a flexible number of sub-jobs up to a maximum limit. The number of sub-jobs a job is split into depends on the number of available servers found upon its arrival. All sub-jobs of a job are then processed in parallel at different servers leading to a linear speed-up of the job. We refer to such jobs as adaptive multi-server jobs. We study the problem of optimal assignment of such jobs when each server can process at most one sub-job at any given instant and there is no waiting room in the system. We assume that, upon arrival, a job can only access a randomly sampled subset of k(n) servers from a total of n servers, and the number of sub-jobs is determined based on the number of idle servers within the sampled subset. We analyze the steady-state performance of the system when system load varies according to λ(n) =1 - β n^-α for α∈ [0,1), and β≥ 0. Our interest is to find how large the subset k(n) should be in order to have zero blocking and maximum speed-up in the limit as n →∞. We first characterize the system's performance when the jobs have access to the full system, i.e., k(n)=n. In this setting, we show that the blocking probability approaches to zero at the rate O(1/√(n)) and the mean response time of accepted jobs approaches to its minimum achievable value at rate O(1/n). We then consider the case where the jobs only have access to subset of servers, i.e., k(n) < n. We show that as long as k(n)=ω(n^α), the same asymptotic performance can be achieved as in the case with full system access. In particular, for k(n)=Θ(n^αlog n), we show that both the blocking probability and the mean response time approach to their desired limits at rate O(n^-(1-α)/2).
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