Email connectivity issue

Hi all, this one of my first posts, generally we're able to resolve our issues by asking our provider very nicely for help, or searching this forum, but I'm a little stumped with this one! A client has their emails on one of our VPS', they can send and receive fine from webmail and through outlook on their PC's. However they have a system that is hosted on a server in their office which sends out emails automatically, its been fine for months but recently has had issues sending those emails - normally out of hours (after 6pm) and at weekends. Some will work, others will fail. Our VPS has no special/unusual configurations made by us. This is an example of the error found in exim_mainlog on our VPS: 2016-09-29 22:04:36 SMTP connection from [213.123.218.193]:2514 (TCP/IP connection count = 1) 2016-09-29 22:04:56 SMTP connection from host213-123-218-193.in-addr.btopenworld.com [213.123.218.193]:2514 lost The company that mange the system have checked their logs, when it fails they can connect but get no response, they see a Receivesmtp timeout, which tries a few times then appears to give up: 29/09/16 22:02:19 SMTPclient: ReceiveSmtp start iTimeout= 20000 msec 29/09/16 22:02:39 SMTPclient: ReceiveSmtp select timed-out after 20000 msec 29/09/16 22:02:39 SMTPclient: No response from SMTP gateway We have no idea if this issue is our end, or there end... the fact our server isn't sending a response is a bit concerning, but I don't know where else to look. I've got no reports of anything failing, crashing etc - I rebooted exim yesterday just in case but the issue occured again last night. Any pointers on where to look or what might be going on would be greatly appreciated! Thanks!!! Zoee