Nginx: increase the number of allowed connections from the same browser - ajax

Question

I need your help on tuning a bit Nginx and my server.
I've got a website which is hosted on my server, and an API, also hosted on this server.
The website sends Ajax requests to the API. The problem is, once I've opened a page, which sends Ajax requests to the API, I cannot browse the website anymore using this browser because, I've reached the number of allowed connection for this browser on the website's server.
How can I change Nginx or the server configuration in order to increase the number of connections allowed per browser?
Thanks for your help.
EDIT
Server conf
worker_processes 4;
worker_connections 1024;
multi_accept on;
keepalive_timeout 65;
Server block
server {
listen 80;
server_name mywebsite.com;
return 301 https://$server_name$request_uri;
}
server {
listen 443;
ssl on;
ssl_certificate /path/to/pem;
ssl_certificate_key /path/to/key;
# Set the server name
server_name mywebsite.com;
# add Strict-Transport-Security to prevent man in the middle attacks
add_header Strict-Transport-Security "max-age=31536000";
root /path/to/root/folder;
index index.php;
# Logs
error_log /path/to/error.log;
access_log /path/to/access.log;
# strip app_test.php/ prefix if it is present
rewrite ^/index\.php/?(.*)$ /$1 permanent;
location / {
index index.php;
try_files $uri #rewriteapp;
}
location #rewriteapp {
rewrite ^(.*)$ /index.php/$1 last;
}
# pass the PHP scripts to FastCGI server from upstream phpfcgi
location ~ ^/(index)\.php(/|$) {
fastcgi_pass unix:/var/run/php5-fpm.sock;
fastcgi_split_path_info ^(.+\.php)(/.*)$;
include fastcgi_params;
fastcgi_param SCRIPT_FILENAME $document_root$fastcgi_script_name;
fastcgi_param HTTPS on;
fastcgi_read_timeout 43200;
internal;
}
# Define error pages
# Error pages are located in the app folder
error_page 403 403.html;
location = 403.html {
root /path/to/error-pages;
internal;
}
error_page 404 404.html;
location = 404.html {
root /path/to/error-pages;
internal;
}
error_page 500 502 503 504 50x.html;
location = 50x.html {
root /path/to/error-pages;
internal;
}
}