Question

Coffee Talk is a local café on the waterfront. There is counter service only with one clerk working the counter. On an average Friday (6 hour operation), 6 customers arrive at the counter every 20 minutes; on average, it takes 3 minutes to serve a customer. Find the following measures of performance of the café:
a) The average utilization of the counter clerk
b) The average number of customers waiting in line
c) The average time a customer spends in the system
d) Probability that exactly 2 customers are waiting in line
e) What observation can you make from the calculations above?

Answer 1

Average arrival rate, λ = 6 per 20 minutes = 18 per 60 minutes = 18 per hr.
Average service rate, μ = 1 in 3 minutes = 20 per 60 minutes = 20 per hr.

Assume Poisson arrival and Exponential service delivery

a)

Utilization, u = λ/μ = 18/20 = 90%

b)

The average number of customers waiting, L_q = λ² / {μ.(μ - λ)} = 18^2/(20*(20-18)) = 8.1 customrs

c)

The averaeg time in the system, W_s = 1 / (μ - λ) = 1 / (20 - 18) = 1/2 hrs = 30 minutes

d)

Prob{exactly 2 customers in waiting line} = Prob{exactly 3 customers in system} = (λ/μ)³*(1 - λ/μ) = (0.9^3)*(1 - 0.9) = 0.073

e)

Teh utilization is very high (90%) and still the waiting line is quite long (~ 8 people). It is time to improve the service rate by making some improvements (e.g. using technology). Otherwise, use two servers instead of one.