From the customer's point of view, I would certainly appreciate a frank explanation as to how or why the service may be delayed, and I have been very annoyed with people who gave me the insincere 'song and dance' without producing the timely service.
This is also what many published articles recommend:
https://www.google.co.in/search?q=be+honest+with+customer
To extract just one relevant quote,
If you pretend that nothing is wrong, your customers will continue to be in the dark and speculate the reason for the poor service. They may begin to think the worst- that you were not doing your job! When you are honest, and tell them the actual reason (...) then you open the door for sympathy and understanding.
Source: http://www.principledprofit.com/ethics-articles/5-reasons-to-stay-honest-with-your-customers
Even at risk of losing some business in the short term, it is important to maintain credibility by being tactfully honestmaintain credibility by being tactfully honest with the customer. You need not go into the specific dynamics of how your company operates, as long as you can make the customer understand that
the delay in responding is directly related to your current workload (where you can make the 'feast or famine' point to emphasize that response time will vary depending on workload)
you value their business and shall attend to their request as early as possible
you are fair in assigning priority to work based on the established business model of the company (if not necessarily on a first-come, first-served basis).
Above all, it is necessary to inform the customer of the possible duration of the delay, so that they wouldn't later feel that you wasted their time with false assurances of priority and prevented them from getting prompt service from another company.