It　is　interesting　but　difficult　and　challenging　to　study　Ethereum　with　multiple　mining　pools.　One　of　the　main　difficulties　comes　from　not　only　how　to　represent　such　a　general　tree　with　multiple　block　branches　(or　sub-chains)　related　to　the　multiple　mining　pools,　but　also　how　to　analyze　a　multi-dimensional　stochastic　system　due　to　the　mining　competition　among　the　multiple　mining　pools.　In　this　paper,　we　first　set　up　a　mathematical　representation　for　the　tree　with　multiple　block　branches.　Then　we　provide　a　block　classification　of　Ethereum:　Regular　blocks　(in　the　main　chain),　orphan　blocks,　uncle　blocks,　stale　blocks,　and　nephew　blocks,　and　give　some　key　ratios　and　probabilities　of　generating　the　different　types　of　blocks　by　applying　the　law　of　large　numbers.　Based　on　this,　we　further　discuss　the　growth　rate　of　blockchain　and　the　reward　allocation　among　the　multiple　mining　pools　through　applying　the　renewal　reward　theorem.　Finally,　we　use　some　simulation　experiments　to　verify　our　theoretical　results,　and　show　that　the　approximate　computation　approaches　developed,　such　as　the　key　ratios　and　probabilities,　the　long-term　growth　rate　of　blockchain,　and　the　long-term　reward　allocation　(rate)　among　the　multiple　mining　pools,　can　have　a　faster　convergence.　Therefore,　we　provide　a　powerful　tool　for　observing　and　understanding　the　influence　of　the　selfish　mining　attacks　on　the　performance　of　Ethereum　with　multiple　mining　pools.　We　believe　that　the　methodology　and　results　developed　in　this　paper　will　shed　light　on　the　study　of　Ethereum　with　multiple　mining　pools,　such　that　a　series　of　promising　research　can　be　inspired　potentially.　IEEE