With　the　explosive　growth　of　machine-type　communication　devices　(MTCDs),　both　academia　and　industry　are　paying　more　attention　to　M2M　communications.　In　this　article,　we　introduce　mobile　edge　computing　into　virtualized　cellular　networks　with　M2M　communications　to　decrease　energy　consumption　and　optimize　computing　resource　allocation.　Moreover,　based　on　different　functions　and　QoS　requirements,　the　physical　network　can　be　virtualized　into　several　virtual　networks;　then　each　MTCD　selects　the　corresponding　virtual　network　to　access　through　the　embedded　subscriber　identity　module　(eSIM)　technology.　Meanwhile,　the　random　access　process　of　MTCDs　is　formulated　as　a　partially　observable　Markov　decision　process　to　minimize　the　system　cost,　which　consists　of　both　energy　consumption　and　execution　lime　of　computing　tasks.　Furthermore,　to　facilitate　the　network　architecture　integration,　software-defined　networking　is　introduced　to　deal　with　the　diverse　protocols　and　standards　in　M2M　networks.　Extensive　simulation　results　with　different　system　parameters　reveal　that　the　proposed　scheme　could　significantly　improve　the　system　performance　compared　to　the　existing　schemes.