In a previous paper, hereafter referred to as I, we have analyzed the 7 TeV LHC data on W + jets events from the standpoint of IR-improved DGLAP parton shower effects, using the IR-improved Herwiri1.031 parton shower MC in comparison with the Herwig6.5 parton shower MC in the context of the exact $O(alpha_s)$ matrix element matched parton shower framework provided by MG5_aMC@NLO. In the current paper, we extend this analysis to the LHC 8 and 13 TeV data to investigate the energy dependence of the results obtained in I. Specifically, W~+ jet events are generated in the MADGRAPH5_aMC@NLO framework and showered by HERWIG6.521 and HERWIRI1.031 with $mathtt{PTRMS}=2.2$ and 0 GeV, respectively. The differential cross sections are reported as functions of jet multiplicity, transverse linear momenta ($P_{T}$), the jet pesudo-rapidity ($eta$) and the scalar sum of jet transverse momenta ($H_{T}$) for different jet multiplicities 1--3. The dijet cross sections as functions of transverse linear momenta, invariant mass of the dijet and the jet separation are shown as well. Distributions of angular correlations between the jets and the muon are examined as well and the corresponding cross sections are presented. The respective measured cross sections are compared with the exact next-to-leading-order (NLO) matrix element matched parton shower theoretical predictions provided by MADGRAPH5_aMC@NLO/HERWIRI1.031~($mathtt{PTRMS}=0)$ and MADGRAPH5_aMC@NLO/HERWIG6.521~($mathtt{PTRMS}=2.2~mathrm{GeV})$ and the phenomenological consequences are discussed with an eye toward their energy dependence.
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