Is This a Good Start to A story?

Seems a good basis for a story. Just a couple of things I would like to suggest though. First thing is Mary wasn't laying in deserted alleys, plural, just an alley, singular. There seems to be a bit of tidying up needed between the first and second paragraphs. Something is missing at the end of the first paragraph...perhaps, her last thoughts drifted to her mother sitting at home watching the television and her little terrier waiting by the door for her return. In the first paragraph Mary is still alive and in the second, much later, she is dead. Also I wouldn't mention (twice) the make and model of the inspector's car as it really isn't necessary. These are only my suggestions though and I have no expertise in this area.