MatLab:Generate N pseudo-random numbers with a Poisson distribution having mean M and total T where N,M, and T are user defined

I’d like to be able to generate in MatLab a sequence of N pseudo-random numbers with a Poisson distribution having mean M. The sum of the N numbers should be T. N, M, and T are always positive or zero and would be user specified parameters to any function.

Obviously, if T is small relative to N it is likely that there will be problems achieving a total of T. In that case the function could just return the values T and then N-1 zeros or an error code.

I have been trying variations based on the method of generating random numbers with a given distribution provided at http://matlabtricks.com/post-44/generate-random-numbers-with-a-given-distribution and trying various normalizations at each step but have not been successful.