Abstract
We consider an M/M/∞ model with m primary servers and infinitely many secondary ones. An arriving customer takes a primary server, if one is available. We derive integral representations for the joint steady state distribution of the number of occupied primary and secondary servers. Letting ρeλ/μ be the ratio of arrival and service rates (all servers work at rate μ), we study the joint distribution asymptotically for ρ→∞. We consider both meO(1) and m scaled to be of the same order as ρ. We also give results for the marginal distribution of the number of secondary servers that are occupied.
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