Using a slight modification of the framework in Bramson [7] and Williams [52], we prove heavy traffic limit theorems for six families of multiclass queueing networks. The first three families are single station systems operating under first-in first-out (FIFO), generalized head-of-the-line proportional processor sharing (GHLPPS) and static buffer priority (SBP) service disciplines. The next two families are re-entrant lines operating under first-buffer-first-serve (FBFS) and last-buffer-first-serve (LBFS) service disciplines; the last family consists of certain 2-station, 5-class networks operating under an SBP service discipline. Some of these heavy traffic limits have appeared earlier in the literature; our new proofs demonstrate the significant simplifications that can be achieved in the present setting.

We study the distribution of steady-state queue lengths in multiclass queueing networks under a stable policy. We propose a general methodology based on Lyapunov functions, for the performance analysis of infinite state Markov chains and apply it specifically to Markovian multiclass queueing networks. We establish a deeper connection between stability and performance of such networks by showing that if there exist linear and piecewise linear Lyapunov functions that show stability, then these Lyapunov functions can be used to establish geometric type lower and upper bounds on the tail probabilities, and thus bounds on the expectation of the queue lengths. As an example of our results, for a re-entrant line queueing network with two processing stations operating under a work-conserving policy we showthat E[L] =O 1 (1; ) 2 � where L is the total number ofcustomers in the system, and is the maximal actual or virtual traffic intensity inthenetwork. In a Markovian setting, this extends a recent result by Daiand Vande Vate, which states that a re-entrant line queueing network with two stations is globally stable if < 1: We also present several results on the

This paper proves that the stability region of a 2-station, 5-class reentrant queueing network, operating under a non-preemptive static bu#er priority service policy, depends on the distributions of the interarrival and service times. In particular, our result shows that conditions on the mean interarrival and service times are not enough to determine the stability of a queueing network, under a particular policy. We prove that when all distributions are exponential, the network is unstable in the sense that, with probability one, the total number of jobs in the network goes to infinity with time. We show that the same network with all interarrival and service times being deterministic is stable. When all distributions are uniform with a given range, our simulation studies show that the stability of the network depends on the width of the uniform distribution. Finally, we show that the same network, with deterministic interarrival and service times, is unstable when the it is operated under the preemptive version of the static bu#er priority service policy. Thus, our examples also demonstrate that the stability region depends on the preemption mechanism used.

For multiclass queueing networks, dispatch policies govern the assignment of servers to the jobs they process. Production policies perform the analogous task for queueing networks whose servers are subject to switch-over delays or setups, a model we refer to as setup networks.

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

In a batch processing network, multiple jobs can be formed into a batch to be processed in a single service operation. The network is multiclass in that several job classes may be processed at a server. Jobs in di#erent classes cannot be mixed into a single batch. A batch policy specifies which class of jobs is to be served next. Throughput of a batch processing network depends on the batch policy used. When the maximum batch sizes are equal to one, the corresponding network is called a standard processing network, and the corresponding service policy is called a dispatch policy. There are many dispatch policies that have been proven to maximize the throughput in standard networks. This paper shows that any normal dispatch policy can be converted into a batch policy that preserves key stability properties. Examples of normal policies are given. These include static bu#er priority (SBP), first-in--first-out (FIFO) and generalized round robin (GRR) policies. Keywords: batch processing, multiclass queueing networks, stability, throughput, fluid model, semiconductor wafer fabrication, furnace operation. 1

One of the primary tools in establishing the stability of a fluid network is to construct a Lyapunov function. In this paper, we establish the sufficiency in the use of a Lyapunov function. Specifically, we show that a necessary and sufficient condition for the stability of a generic fluid network (GFN) is the existence of a Lyapunov function for its fluid level process. Then by applying this result to various specific fluid networks, including a fluid network under all work-conserving service disciplines, a fluid network under a priority service discipline, and a fluid network under a FIFO service discipline, we establish the existence of a Lyapunov function for their fluid level processes is a necessary and sufficient condition for their stabilities. The result is also applied to various fluid limit models and a linear Skorohod problem.

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