Question

State Bank always has two tellers on duty. Customers arrive to receive service from a teller at a mean rate of 40 per hour. A teller requires an average of 2 minutes to serve a customer. When both tellers are busy, an arriving customer joins a single line to wait for service. Experience has shown that customers wait in line an average of 1 minute before service begins.

(a) Describe why this is a queueing system.

(b) Determine the basic measures of performance—Wq, W, Lq, and L for this queueing system. (Hint: We don’t know the probability distributions of interarrival times and service times for this queueing system, so you will need to use the relationships between these measures of performance to help answer the question.)

Answer #1

. A bank has 4 tellers with different service rates, i.e.,
teller 1 averages 10 customers/hour, teller 2 averages 15
customers/hour, teller 3 averages 12 customers/hour, and teller 4
averages 10 customers/hour (all with exponential service times).
Assume that there are 20 seats in the lobby, and any customer
arriving to a fully-occupied lobby leaves.
a. If the system is modeled as a Markov chain, what is the total
number of states?
b. What is the average time between departing...

LF Bank has 3 tellers serving customers through a shared line
where the next customer in line goes to the first available teller.
Each teller can serve 30 customers per hour with exponential
service times and the customers arrive at a rate of 47 per hour,
according to a Poisson distribution. On average, how many customers
are in the system? (Note: carry all your results to at least 4
decimal places)
Round your final answer to 2 decimal places.

Willow Brook National Bank operates a drive-up teller window
that allows customers to complete bank transactions without getting
out of their cars. On weekday mornings, arrivals to the drive-up
teller window occur at random, with an arrival rate of 30 customers
per hour or 0.5 customers per minute. In the same bank waiting line
system, assume that the service times for the drive-up teller
follow an exponential probability distribution with a service rate
of 36 customers per hour, or 0.6...

A large bank branch employs 5 tellers to serve its customers.
Customers arrive according to a Poisson process at a mean rate of 3
per minute. The transaction time between the teller and customer
has an exponential distribution with a mean of 1 minute. The
management has established the following guidelines for the
satisfactory level of service to customers:
The average number of customers waiting in line to begin
service should not exceed 1.
At least 95% of the time,...

Problem 15-3 (Algorithmic) Willow Brook National Bank operates a
drive-up teller window that allows customers to complete bank
transactions without getting out of their cars. On weekday
mornings, arrivals to the drive-up teller window occur at random,
with an arrival rate of 18 customers per hour or 0.3 customers per
minute. In the same bank waiting line system, assume that the
service times for the drive-up teller follow an exponential
probability distribution with a service rate of 30 customers per...

At a Lebanese bank in downtown Beirut, people arrive randomly at
a bank teller at an average rate of 30 an hour. If the teller takes
an average of 0.5 minutes to serve each customer, what is the
average number of customers in the queue and how long do they wait
to be served?
Q2.1 What happens if the average service time increases to 1
minute?
Q2.2 What happens if the average service time increases to 2
minutes?

The First American Bank of Rapid City has one outside drive-up
teller. It takes the teller an average of 4 minutes to serve a bank
customer. Customers arrive at the drive-up window at a rate of 12
per hour. The bank operations officer is currently analyzing the
possibility of adding a second drive-up window, at an annual cost
of $20,000. It is assumed that arriving cars would be equally
divided between both windows. The operations officer estimates that
each minute’s...

Customers arrive at bank according to a Poisson process with
rate 20 customers per hour. The bank lobby has enough space for 10
customers. When the lobby is full, an arriving customers goes to
another branch and is lost. The bank manager assigns one teller to
customer service as long as the number of customers in the lobby is
3 or less. She assigns two tellers if the number is more than 3 but
less than 8. Otherwise she assigns...

The National Bank of Germany is opening a new bank teller.
Management estimates that customers will arrive at the rate of 10
per hour. The teller who will staff the window can serve customers
at the rate of one every four minutes.
Assuming Poisson arrivals and exponential service, find:
Average number of customers in the system.
Average time in the system.
Average waiting time in the line.
Average number of customers in the line.
Utilization of the teller.

Four cashiers are on duty in a bank where customers may be
assumed to arrive independently and at random, at an average rate
of 60 per hour. If a cashier is free, then an arriving customer
receives immediate attention; otherwise a central queue is formed.
The service time for each cashier may be assumed to be
exponentially distributed with mean 2 minutes. The traffic
intensity is .
Assume that the queue is in equilibrium
What is the probability that at any...

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