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In the queuing literature Buzen's algorithm is sometimes also referred to as the '''Convolution algorithm'''.
==Problem setup==
Consider a closed queueing network with M service facilities and N circulating customers. Let <math>n_i</math> represent the number of customers present at the i-th facility, with <math>\sum_{i=1}^M n_i = N</math>. We assume that the service time for a customer at the i-th facility is given by an exponentially distributed random variable with mean <math>1/\mu_i</math> and that after completing service at the i-th facility a customer will proceed to the j-th facility with probability <math>p_{ij}</math>.
It follows from the [[Gordon-Newell theorem]] that the equilibrium distribution of customers in the network is given by:
:<math>P(n_1,n_2,\cdots,n_M) = \frac{1}{G(N)}\prod_{i=1}^M \left( X_i \right)^{n_i}</math>
where the <math>X_i</math> are found from the equations:
:<math>\mu_j X_j = \sum_{i=1}^M \mu_i X_i p_{ij}\,</math> for <math>j=1,\cdots,N</math>
and <math>G(N)</math> is a normalizing constant such that the above probabilities sum to 1. <ref>Buzen: op.cit.</ref>
The purpose of the Buzen algorithm is to numerically compute values of G.
==Derivation==
''G''(''N'') = ''g''(''M'', ''N'')
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