We explore a non-stationary outer gap scenario for gamma-ray emission process in pulsar magnetosphere. Electrons/positrons that migrate along the magnetic field line and enter the outer gap from the outer/inner boundaries activate the pair-creation cascade and high-energy emission process. In our model, the rate of the particle injection at the gap boundaries is key physical quantity to control the gap structure and properties of the gamma-ray spectrum. Our model assumes that the injection rate is time variable and the observed gamma-ray spectrum are superposition of the emissions from different gap structures with different injection rates at the gap boundaries. The calculated spectrum superposed by assuming power law distribution of the particle injection rate can reproduce sub-exponential cut-off feature in the gamma-ray spectrum observed by Fermi-LAT. We fit the phase-averaged spectra for 43 young/middle-age pulsars and 14 millisecond pulsars with the model. Our results imply that (1) a larger particle injection at the gap boundaries is more frequent for the pulsar with a larger spin down power and (2) outer gap with an injection rate much smaller than the Goldreich-Julian value produces observe $>10$GeV emissions. Fermi-LAT gamma-ray pulsars show that (i) the observed gamma-ray spectrum below cut-off energy tends to be softer for the pulsar with a higher spin down rate and (ii) the second peak is more prominent in higher energy bands. Based on the results of the fitting, we describe possible theoretical interpretations for these observational properties. We also briefly discuss Crab-like millisecond pulsars that show phase-aligned radio and gamma-ray pulses.
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