Você deve poder examinar os arquivos de log do roteador. Isso mostrará eventos de conexão e desconexão. Algo parecido com isto:
14:53:04, 18 May. (12682426.850000) PPPoE is up - Down Rate=79903Kbps, Up Rate=19999Kbps; SNR Margin Down=12.2dB, Up=15.2dB
14:53:04, 18 May. (12682426.840000) CHAP authentication successful
14:53:04, 18 May. (12682426.760000) CHAP Receive Challenge
14:53:04, 18 May. (12682426.760000) Starting CHAP authentication with peer
14:53:04, 18 May. (12682426.760000) PPP LCP Receive Configuration ACK
14:53:04, 18 May. (12682426.750000) PPP LCP Send Configuration Request
14:53:04, 18 May. (12682426.750000) PPP LCP Receive Configuration Reject
14:53:04, 18 May. (12682426.750000) PPP LCP Send Configuration ACK
14:53:04, 18 May. (12682426.750000) PPP LCP Receive Configuration Request
14:53:04, 18 May. (12682426.750000) PPP LCP Send Configuration Request
14:52:13, 18 May. (12682375.760000) PTM over DSL is up
14:48:33, 18 May. (12682155.340000) PTM over DSL is down after 211367 minutes uptime
14:48:33, 18 May. (12682155.340000) PPPoE is down after 211367 minutes uptime [Waiting for Underlying Connection (WAN Ethernet 2 - Down)]