dbtem on "WordPress Time Out and 100% Processor Usage Issue"

I am encountering a recurrent issue that I have run out of ideas on how to solve. I have a website running on Apache 2.2.14, PHP 5.3.2, MySQL 5.6.13, WordPress 3.8.3 using Gantry and Ximenia.

The problem is that it sometimes goes to 100% processor usage, and the site times out, and any communications with the server become extremely sluggish.

I have had this happen on two different physical machines now (with mirror copies of the same site) (both with traffic), so I know that the problem has to be related to one of the items mentioned above.

The server logs do not mention anything related to this issue.

I cannot determine if there are any plugins or themes that are causing it. Every plugin that is currently installed is being used.

The following plugins are used:

Allow PHP in Posts and Pages
Arconix FAQ
Category Post List Widget
Contact Form 7
Custom Permalinks
E-MAILiT Social Media Sharing Widget
Executable PHP Widget
Gantry Template Framework
Google Analytics for WordPress
Google Language Translator
JS & CSS Script Optimizer
NextScripts: Social Networks Auto-Poster
Page View
Popular Widget
Really Simple Popup
ReFlex Gallery
Register Plus Redux
RokAjaxSearch
RokBox
RokCommon Library
RokGallery
RokSocialButtons
RokSprocket
SEO Friendly Images
Simple Tags
Twitter Widget Pro
WooCommerce
WooCommerce Beanstream Payment Gateway
WP-Filebase
WP-Print
WP Tell a Friend Popup Form

I tried creating a mirror site, but the problem appears (guessing) to appear when there is traffic to the site. The problem does not appear on the mirror copy of the site that has no traffic.

If anyone could assist me, I would greatly appreciate it.