For an M/M/1 queueing system with Poisson arrival with rate λ and exponential service time with mean
1/μ (rate μ), some customers may be dissatisfied with the service they received and must be served again. Suppose that on completion of service, a customer is dissatisfied with probability 1−α, for some 0 < α < 1, independently of whether that customer had been dissatisfied (one or more times) before. If customers are always satisfied (i.e., if α = 1), we have a standard M/M/1 system. See Figure 1 below for illustration. Subsequent service times on the same customer, if any, are also independent and exponentially distributed with mean 1/μ. Assume that dissatisfied customers are served again immediately in the server, until satisfied.
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