I'm not sure that I understand your question because I don't know what you mean by "assigned." However, the time that it takes from when an offer is submitted to the seller's lender and the lender returns with a decision can vary significantly. In my experience, it can take anywhere from one to five months from when the lender first receives an offer.
There are many factors that cause this delay. First, if the seller's agent isn't experienced in short sales, they may not know the procedure for getting the process started with the bank. Also, if the agent isn't in constant contact with the lender, that can slow things significantly. In addition, if the lender is overwhelmed or unorganized, that can cause a big delay. Finally, there are many other forces such as investors, second lenders, etc that can work silently behind the scenes to cause big delays.
After all of this delay, the lender may decide to foreclose if they don't get an offer that they find to be acceptable.