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### 6.6. Two Variations on the M/M/1 Queue

#### 6.6.1. The M/M/8 Queue: A Responsive Server

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We now briefly consider two variations on the M/M/1 queue, essentially to give insight into how one proceeds with the analysis of a queueing system.

Suppose that a provider of service capacity brings on a new server to serve every arriving customer. This is like a private bank where new agents are brought on to provide individual attention to each customer when she or he arrives. This system can be modeled as a queue with an infinite number of servers, though, at any time, the number of servers is finite.

We can model and analyze this queue by using the same techniques as with an M/M/1 queue. We start with the state-transition-rate diagram shown in Figure 6.7. Note that µ_{j}, the rate of departure from the *j*th queue, is *j*µ, which models the fact that when there are *j* customers, there are *j* servers. From the diagram, we can directly invoke Equation 6.29 to write down , the stationary probability of being in state *j*, as