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We study a queue of multi-skilled servers, serving multi-type customers. A bipartite graph describes which pairs of server-customer types are compatible. Servers and customers are matched according to the service policy FCFS-ALIS. For exponential services and Poisson arrivals, we show that the stationary queue length distribution has a product-form solution. Based on Little's distributional law, this product-form yields an exact expression for the Laplace transform of the waiting time for each customer type. This expression has an interesting probabilistic interpretation. For non-exponential services, we propose approximations for the mean waiting time by exploiting this interpretation.