We apply our new approach to modeling uncertain probabilities to queuing theory and the optimal design of web servers. This involves using fuzzy, finite, regular Markov chains to determine the fuzzy steady state probabilities and then computing the fuzzy numbers for system performance. We first ignore revenues and costs in determining an optimal system and then we incorporate these factors for optimal design. Then we add two new phenomena associated with the web in our optimization models: ''burstiness'' and ''long tailed distributions''.

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We start by determining fuzzy numbers ( ) for the arrival (service) rate for a fuzzy queuing system. These fuzzy numbers are then used to compute the fuzzy steady state probabilities which in turn are employed to calculate fuzzy numbers for system performance. These fuzzy numbers for system performance are inputs into optimization models for web planning. We then introduce a new way to perform (restricted) fuzzy arithmetic. We show how to use a standard (crisp) simulation package to obtain «-cuts of the fuzzy numbers for system performance from «-cuts of and .This implies that crisp simulation might be used for other fuzzy computations.

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