# Stochastic Processes : M/M/s Queue, Stationary Distribution and Long-Run Proportion

Consider the M/M/s queue, with arrival rate...

5. Consider the M/M/s queue, with arrival rate Λ >0 and service rate μ >0.

(a) Find the condition involving Λ, μ and s that is necessary for there to exist a stationary distribution. Why does this condition makes sense?

(b) Find the long-run proportion of time that there are k customers in the system, for k = 0,1,...

See the attached file.

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#### Solution Preview

Please see the attached file for the complete solution.

(a) For system to be stationary, has to be less than 1, that is, .

This is because is the arrival rate of customers, which is the expected number of arrivals per hour, and ...

#### Solution Summary

M/M/s queue, stationary distribution and long-run proportion are investigated and discussed. The solution is detailed and well presented.