Question

Timing-Waiting Lines

Joe Hammer is thinking about setting up a special counter for the do-it-yourself customers at which they can get, not only help where to find products in the store, but also some quick advice about the best way to handle their upcoming projects. Experience has taught Joe that six minutes is a good figure to allow for the average time required to serve a “do-it-yourselfer” and that these customers will arrive every 15 minutes throughout the day.

a.) If joe sets up the counter under these conditions, what operating characteristics might he expect?

b.) What might Joe do to avoid the costs of idleness?

c.) What is the likelihood(probability) that three or more customers will be at the counter, either waiting or being served, at any given time?

Calculate the Utilization rate, idleness rate, Average time in queue, Average time in system, Average number in queue, Average number in system, and probability that three or more customers will be in the counter system at the same time.

Answer 1

Given data:-

Average arrival rate, λ = 1 in 15 minutes = 4 per hour
Average service rate, μ = 1 in 6 minutes = 10 per hour

a)

Utilization rate, u = λ/μ = 4/10 = 40%

Idleness rate = 1 - u = 60%

Average time in queue, Wq = λ/{μ.(μ - λ)} = 4/(10*(10-4)) = 0.0667 hrs. = 4 minutes

Average time in the system, W = Wq + service time = 4+6 = 10 minutes

Average number in queue, Lq = Wq * λ = 0.0667*4 = 0.267

Average number in system, Ls = Lq + λ/μ = 0.267+0.4 = 0.667

b)

To reduce the idleness or to improve the utilization, more number of customers can be diverted to the special counter by making them aware just at the entry to the shop that such facility exists. Other method can be small discount to encourage the self-service behavior. But improving utlization will definitely increase the waiting time as well which also has a cost. So, the improvement in utilization should be done optimally i.e. by reducing the total cost which is the sum of the cost of waiting and cost of underutilization.

c)

Prob{3 or customers in the system} = Pn>3 = (λ/μ)^3+1 = 0.4^4 = 0.0256

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