I propose to investigate four important problems for stochastic networks:
1. Performance of processor sharing networks;
2. performance of processor sharing queues with deadlines;
3. the waiting time distribution in ( exhaustive ) polling systems with heavy tailed service time and/or inter arrival time distributions; and
4. dynamic control problems related to the scheduling of jobs in large scale processing computers. The first two problems represent logical directions in which to build on the techniques developed during my dissertation research. The objective is to obtain heavy traffic limit theorems for certain measure valued processes which describe the performance of processor sharing networks.
For problem 1, the networks considered will vary in complexity, beginning with a singe processor sharing queue with multiple input classes, progressing to feed forward networks of processor sharing queues, and ultimately including more general networks that are not necessarily feed forward.
For problem 2, a single processor sharing queue is considered, but with service deadlines attached to each job entering the system. The objective for problem 3 is to prove a heavy traffic limit theorem for the waiting time distribution in the above polling system. This will require entirely different techniques than needed for problems 1 and 2. The objective for problem 4 is to find good scheduling policies for parallel processing facilities which can choose how to a lot their resources (how many processors to assign) to incoming jobs. As there is little theoretical knowledge for this system, the investigation will begin with a performance analysis of some policies commonly in use.