With　an　increasing　number　of　machine-type　communication　devices　(MTCDs),　machine-to-machine　(M2M)　communications　have　attracted　great　attentions　from　both　academia　and　industry.　Different　from　traditional　communication　networks,　the　data　connections　with　M2M　communications　are　typically　small-sized　but　with　high　frequency,　necessitating　the　efficiency　optimization　of　both　energy　consumption　and　computation.　In　this　paper,　we　introduce　mobile　edge　computing　(MEC)　into　virtualized　cellular　networks　with　M2M　communications,　to　decrease　the　energy　consumption　and　optimize　the　computing　resource　allocation　as　well　as　improve　computing　capability.　Moreover,　based　on　different　functions　and　quality　of　service　(QoS)　requirements,　the　physical　network　can　be　virtualized　into　several　virtual　networks,　and　then　each　MTCD　selects　the　corresponding　virtual　network　to　access　through　the　embedded-SIM　(eSIM)　technology.　Meanwhile,　the　random　access　process　of　MTCDs　is　formulated　as　a　partially　observable　Markov　decision　process　(POMDP)　to　minimize　the　system　cost,　which　consists　of　both　the　energy　consumption　and　execution　time　of　computing　tasks.　Furthermore,　to　facilitate　the　network　architecture　integration,　software-defined　networking　(SDN)　is　introduced　to　deal　with　the　diverse　protocols　and　standards　in　the　networks.　Extensive　simulation　results　with　different　system　parameters　reveal　that　the　proposed　scheme　could　significantly　improve　the　system　performance　compared　to　the　existing　schemes.