managing same domain name with multiple sub domains with different physical servers

I am having one primary temp domain

www.product.com

For each client i need to have separate sub domain mapped to same server with same port(80) through virtual host.

www.client1.product.com
www.client2.product.com
www.clientn.product.com

what’s my problem is my server reaches maximum load, so i am unable to add additional clients.

my current setup

www.product.com is mapped with server 1 using only 1 ip address

server 1 using

www.client1.product.com
www.client2.product.com
.
.
.
www.client20.product.com

Here is my actual problem

now server1 reaches maximum utilization of physical resources

so i am buying another additional server from my datacenter

so in server2 i need to continue sub domain from

www.client21.product.com
.
.
.
www.clientn.product.com 

1) Is it possible?

2) Is there any solution for my problem , if yes please clarify me.

Note:

I am new to web hosting and newbie in managing dedicated servers.

Update:

More information about my application architecture

www.client1.product.com uses separate unique rails instance with separate mysql db 
www.client2.product.com uses separate unique rails instance with separate mysql db 

for more clarity of my question visit

http://stackoverflow.com/questions/12210495/phusion-passenger-nginx-virtual-host-sub-domain-configuration-for-ruby-on-rail

As I understood the question you want to host multiple subdomains on a www.product.com that too on multiple physical boxes. Your first server goes out of resources by adding clinet1-20..

You should configure the second box exactly the same that is your first box is, i.e. OS, web server,database server etc..

Then in web server configuration just add the virtual hosts same as you do with your first box.

After that all you have to do is you have to make changes in DNS of www.product.com

You have to add A record for the added subdomains for example if you added client22.product.com on the second server, you put in the code, you added the database, then add an A record as client22 and put in the IP address of the second server. wait for few minutes before DNS propages. Verify this by doing a ping test on client22.product.com to see if it resolves to the IP you added with the A record.

I hope it helps.

Check more discussion of this question.