Question

(16 marks) 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 and the average number of customers waiting in line

b) The average time a customer spends in the system

c) Probability that exactly 2 customers are waiting in line

d) What observation can you make from the calculations above?

Need answers for a,b,c,d.

Answer 1

Answer:

Given,

Average arrival rate = 6 for every 20 minutes

= 18/hr

Average service rate μ = 1 in 3 minutes

= 20 per hr

a)

Utilization, u = / μ

substitute values

= 18/20  

= 90%

Average number of customers waiting = Lq

= ^2 / μ(μ - )

substitute values

= 18^2 / (20(20-18))

= 8.1

b)

Average time Ws = 1 / (μ - )

= 1 / (20 - 18)

= 1 / 2

= 30 minutes

c)

P(X = 2) = (/μ)^3*(1 - /μ )

= (0.9^3)*(1 - 0.9)

= 0.073

d)

The usage is exceptionally high (90%) and still the holding up line is very long (~ 8 individuals).

The time has come to improve the administration rate by making a few upgrades.

Something else, utilize two servers rather than one.