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We consider two scaling regimes for the tail event in which we demonstrate the existence of a restricted large deviations principle and identify the unique rate function associated with these asymptotics. Our results identify when the maxima of both coordinates are typically attained by two different versus the same index, and how this depends on the correlation between the coordinates of the bivariate Gaussian random vectors. Our results complement a growing body of work on the extremes of Gaussian processes. The results are also relevant for steady-state performance and simulation analysis of networks of infinite server queues.\n<\/jats:p>","DOI":"10.1007\/s11134-019-09632-z","type":"journal-article","created":{"date-parts":[[2019,10,15]],"date-time":"2019-10-15T19:52:18Z","timestamp":1571169138000},"page":"333-349","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Large deviations of bivariate Gaussian extrema"],"prefix":"10.1007","volume":"93","author":[{"ORCID":"http:\/\/orcid.org\/0000-0003-1331-9697","authenticated-orcid":false,"given":"Remco","family":"van der Hofstad","sequence":"first","affiliation":[]},{"given":"Harsha","family":"Honnappa","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2019,10,15]]},"reference":[{"issue":"5","key":"9632_CR1","doi-asserted-by":"publisher","first-page":"746","DOI":"10.1007\/BF01040651","volume":"8","author":"AA Borovkov","year":"1967","unstructured":"Borovkov, A.A.: On limit laws for service processes in multi-channel systems. 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