Question

You are the manager of a local bank where three tellers provide services to customers. On​ average, each teller takes three minutes to serve a customer. Customers​ arrive, on​ average, at a rate of 43 per hour. Having recently received complaints from some customers that they waited a long time before being​ served, your boss asks you to evaluate the service system.​ Specifically, you must provide answers to the following​ questions:

a. What is the average utilization of the​ three-teller service​ system?

b. What is the probability that no customers are being served by a teller or are waiting in​ line?

c. What is the average number of customers waiting in​ line?

d. On​ average, how long does a customer wait in line before being​ served?

e. On​ average, how many customers would be at a​ teller's station and in​ line?

I would really appreciate it if all work is shown. Thank you for your time!

Answer 1

A bank with three tellers:

Given data:

Number of servers M = 3

Service time = 3 minutes

Service rate µ= 60/3 = 20 customers per hour

Arrival rate λ= 43 customers per hour

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This is M/M/S type of waiting line model

(a)

Average utilization of the system = (λ/Mµ) = (20/3*43) = 20/129 = 0.15503

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(b)

Probability of no customer is being served by a teller P0

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(c)

Average number of customers waiting in line:

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(d)

Average waiting time:

Wq = Lq/λ = 2.160410/43 = 0.0502420 hour = 0.0502420*60 = 3.0145255 minutes

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(e)

Average number of customers in the system:

Ls = Lq + (λ/µ) = 2.160410 + (43/20) = 2.160410 + 2.15 = 4.31041 customers

Hope the answer will help. Thank you.