Approach Controller voiced concern regarding the lack of appropriate speed control provided by ZME controllers; describing an overtake event that required immediate TRACON action to secure separation.

Narrative

I was working the east arrival position when I accepted two hand-offs on two air carrier aircraft; both aircraft on the HLI1 arrival. Just prior to my airspace boundary; the first aircraft slowed to an indicated 240 knots. About 8 miles in-trail; the second aircraft indicated 310 knots. Both aircraft were at 10;000 feet. Memphis Center called to tell me that the second aircraft was slowing to 250 knots; actual speed. As soon as the first aircraft entered my airspace; I descended him to 5;000 feet with an expeditious descent through 9;000. Memphis Center called again to say that the second aircraft had the other aircraft in sight and added that the first had slowed drastically. I answered that I had the first aircraft expediting through 9;000 to avoid any conflict. This is not the first time this has happened; and I believe it happens because Memphis Center issues a clearance that tells aircraft to maintain a certain speed to a point at our boundary and then resume normal speed. The Center Controller does not expect the aircraft to slow or at least not as drastically as some aircraft do.