Recent　studies　have　suggested　the　necessity　to　incorporate　traffic　dynamics　into　the　process　of　epidemic　spreading　on　complex　networks,　as　the　former　provides　support　for　the　latter　in　many　real-world　situations.　While　there　are　results　on　the　asymptotic　scope　of　the　spreading　dynamics,　the　issue　of　how　fast　an　epidemic　outbreak　can　occur　remains　outstanding.　We　observe　numerically　that　the　density　of　the　infected　nodes　exhibits　an　exponential　increase　with　time　initially,　rendering　definable　a　characteristic　time　for　the　outbreak.　We　then　derive　a　formula　for　scale-free　networks,　which　relates　this　time　to　parameters　characterizing　the　traffic　dynamics　and　the　network　structure　such　as　packet-generation　rate　and　betweenness　distribution.　The　validity　of　the　formula　is　tested　numerically.　Our　study　indicates　that　increasing　the　average　degree　and/or　inducing　traffic　congestion　can　slow　down　the　spreading　process　significantly.　(C)　2012　American　Institute　of　Physics.　[http://dx.doi.org/10.1063/1.4772967]