This paper presents a heavy traffic analysis of the behavior of multi-class acyclic queueing networks in which the customers have deadlines. We assume the queueing system consists of J stations, and there are K different customer classes. Customers from each class arrive to the network according to independent renewal processes. The customers from each class are assigned a random deadline drawn from a deadline distribution associated with that class and they move from station to station according to a fixed acyclic route. The customers at a given node are processed according to the earliest-deadline-first (EDF) queue discipline. At any time, the customers of each type at each node have a lead time, the time until their deadline lapses. We model these lead times as a random counting measure on the real line. Under heavy traffic conditions and suitable scaling, it is proved that the measure-valued lead-time process converges to a deterministic function of the workload process. A two-station example is worked out in details, and simulation results are presented to illustrate the predictive value of the theory. This work is a generalization of Doytchinov, Lehoczky and Shreve [5], which developed these results for the single queue case.

The fluid model has proven to be one of the most effective tools for the analysis of stochastic queueing networks, specifically for the analysis of stability. It is known that stability of a fluid model implies positive (Harris) recurrence (stability) of a corresponding stochastic queueing network, and weak stability implies rate stability of a corresponding stochastic network. These results have been established both for cases of specific scheduling policies and for the class of all work conserving policies. However, only partial converse results have been established and in certain cases converse statements do not hold. In this paper we close one of the existing gaps. For the case of networks with two stations we prove that if the fluid model is not weakly stable under the class of all work conserving policies, then a corresponding queueing network is not rate stable under the class of all work conserving policies. We establish the result by building a particular work conserving scheduling policy which makes the associated stochastic process transient. An important corollary of our result is that the condition ρ ∗ ≤ 1, which was proven in [10] to be the exact condition for global weak stability of the fluid model, is also the exact global rate stability condition for an associated queueing network. Here ρ ∗ is a certain computable parameter of the network involving virtual station and push start conditions. 1

This paper considers a GI/GI/1 processor sharing queue in which jobs have soft deadlines. At each point in time, the collection of residual service times and deadlines is modeled using a random counting measure on the right half-plane. The limit of this measure valued process is obtained under diffusion scaling and heavy traffic conditions and is characterized as a deterministic function of the limiting queue length process. As special cases, one obtains diffusion approximations for the lead time profile and the profile of times in queue. One also obtains a snapshot principle for sojourn times. 1. Introduction. Congestion

The simulation results of a comparative study of scheduling policies using four SEMATECH dataset models are reported. A new family of policies termed Discrete Proportional Processor Sharing (DPPS) is shown to exhibit excellent performance characteristics over a variety of fab situations. The DPPS policies have practical advantages over other scheduling policies including use of only local state information, allowance of flexible decision-making, and no use of job arrival information. The DPPS policies have also been shown mathematically to be throughput optimal. 1

I would like to express my sincere gratitude to my advisor, Professor Jim Dai for his direction, support and feedback. His genius, patience and deep insights make it a pleasure to work with him. I also acknowledge the help and support of the other members of my committee, Professors Leon McGinnis, Richard Serfozo, John Vande Vate and Yang Wang. I would also thank the Virtual Factory Lab for providing computing resources during my four years research. Particularly, I thank Dr. Douglas Bodner for his support. My appreciation goes out to the entire school of ISyE at Georgia Tech, students and faculty, for their support and help. I would especially like to thank Ki-Seok Choi for his willingness to help me. In particular, I owe much to Zheng Wang who had the substantial tasks of proof-reading a draft of this thesis. On a more personal level, I would like to thank my friends, Jianbin Dai and Sheng Liu, for their help during my study at Georgia Tech. I would like to thank the National Science Foundation, which has supported my research through grants DMI-9457336 and DMI-9813345. I also thank Brooks Automations Inc., AutoSimulations division for donating AutoSched AP software and providing technical support. I can hardly imagine how this research could be done without the AutoSched AP software. Finally, I thank my family for their love and support throughout. Particularly, I thank my wife Miao Liu for her continuous support and encouragement. iii Contents Acknowledgements iii

The fluid model has proven to be one of the most effective tools for the analysis of stochastic queueing networks, specifically for the analysis of stability. It is known that stability of a fluid model implies positive (Harris) recurrence (stability) of a corresponding stochastic queueing network, and weak stability implies rate stability of a corresponding stochastic network. These results have been established both for cases of specific scheduling policies and for the class of all non-idling policies. However, only partial converse results have been established and in certain cases converse statements do not hold. In this paper we close one of the existing gaps. For the case of networks with two stations we prove that if the fluid model is not weakly stable under the class of all non-idling policies, then a corresponding queueing network is not rate stable under the class of all non-idling policies. We establish the result by building a particular non-idling scheduling policy which makes the associated stochastic process transient. An important corollary of our result is that the condition ρ ∗ ≤ 1, which was proven in [12] to be the exact condition for global weak stability of the fluid model, is also the exact global rate stability condition for an associated queueing network. Here ρ ∗ is a certain computable parameter of the network involving virtual station and push start conditions. 1