At Tony and Cleo’s bakery, one kind of birthday cake is offered. It takes 15 minutes to decorate this particular cake, and the job is performed by one particular baker. In fact, it is all this baker does. What mean time between arrivals (exponentially distributed) can be accepted if the mean length of the queue for decorating is not to exceed five cakes?

Explanation:

Average length of queue, Lq = λ^2/μ/(μ-λ)

μ is service rate, λ is arrival rate

Mean time to service = 15 minutes  (decoration)

Service rate, μ = 60 / 15 = 4 per hour

It is given that the mean length of the queue for decorating is not to exceed five cakes, Lq = λ^2/μ/(μ-λ) = 5

λ^2/4/(4-λ) = 5

λ = 3.4164 per hour

Mean time between arrivals = 60 / 3.4164 = 17.56 minutes

The minimum acceptable mean time between arrivals = 17.56 minutes

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Explanation: