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In this paper, we define and study a new class of capacity planning models called MAP queueing networks. MAP queueing networks provide the first analytical methodology to describe and predict accurately the performance of complex systems operating under bursty workloads, such as multitier architectures or storage arrays. Burstiness is a feature that significantly degrades system performance and that cannot be captured explicitly by existing capacity planning models. MAP queueing networks address this limitation by describing computer systems as closed networks of servers whose service times are Markovian Arrival Processes (MAPs), a class of Markov-modulated point processes that can model general distributions and burstiness. In this paper, we show that MAP queueing networks provide reliable performance predictions even if the service processes are bursty. We propose a methodology to solve MAP queueing networks by two state space transformations, which we call Linear Reduction (LR) and Quadratic Reduction (QR). These transformations dramatically decrease the number of states in the underlying Markov chain of the queueing network model. From these reduced state spaces, we obtain two classes of bounds on arbitrary performance indexes, e.g., throughput, response time, and utilizations. Numerical experiments show that LR and QR bounds achieve good accuracy. We also illustrate the high effectiveness of the LR and QR bounds in the performance analysis of a real multitier architecture subject to TPC-W workloads that are characterized as bursty. These results promote MAP queueing networks as a new class of robust capacity planning models.