Question

Suppose that the customers arrive at a hamburger stand at an average rate of 49 per hour, and the arrivals follow a Poisson distribution. Joe, the stand owner, works alone and takes an average of 0.857 minutes to serve one customer. Assume that the service time is exponentially distributed.

a) What is the average number of people waiting in queue and in the system? (2 points)

b) What is the average time that a customer spends waiting in the queue and in the system? (2 points) Recently, Joe has started receiving complaints regarding long waiting time from customers who want faster service. In response, Joe is considering hiring Jim to work with him at the stand. Assume that Jim can also serve customers at the same rate as Joe. Compute the average time that a customer spends waiting in the queue and in the system for the following cases:

c) There is a single queue at the stand for both servers. (2 points)

d) There are two queues at the stand, each operating independently, and customers are served FCFS within each queue. Assume that the average arrival rate to each queue is half of that in the previous case, i.e., 49/2, and follows a Poisson distribution. (2 points)

e) From a customer’s perspective, which system is better in terms of waiting time in the system (a shared queue for the two servers as in Part c or dedicated queues for each server as in Part d; select from the list in the Answersheet)? (1 point)

Answer #1

This is a M/M/1 queue model

Arrival rate, a = 49 per hour

Service rate, s = 1/(0.857/60) = 70 per hour

a) Average number of people waiting in the queue (Lq) =
a^{2}/(s*(s-a)) = 49^{2}/(70*(70-49)) = 1.63

Average number of people in system (L) = a/(s-a) = 49/(70-49) = 2.33

b) Average time a customer spends in the queue (Wq) = Lq/a =
1.63/49 = **0.033** hours
= **2** minutes

Average time a customer spends in the system (W) = L/a = 2.33/49
= **0.0476** hours = **2.857**
minutes

c) With 2 servers, this is M/M/s queue model with s=2

Average time in queue (Wq) = **0.0020** hours =
**0.11966** minutes *(refer cell I13 of
above excel sheet)*

Average time in system (W) = **0.0163** hours =
**0.9768** minutes *(refer cell J13 of
above excel sheet)*

d) Average arrival rate, a = 49/2 = 24.5 per hour

Average time in queue (Wq) = a/(s*(s-a)) = 24.5/(70*(70-24.5)) =
**0.0077** hours = **0.4615** minutes

Average time in system (W) = 1/(s-a) = 1/(70-24.5) =
**0.022** hours = **1.32** minutes

e) From a customer perspective, shared queue for two servers is better, because it gives lower waiting time (Wq) and time in system (W)

Suppose that the customers arrive at a hamburger stand at an
average rate of 49 per hour, and the arrivals follow a Poisson
distribution. Joe, the stand owner, works alone and takes an
average of 0.857 minutes to serve one customer. Assume that the
service time is exponentially distributed.
a) What is the average number of people waiting in queue and in
the system? (2 points)
b) What is the average time that a customer spends waiting in
the queue...

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than can be handled by the Burger house food service staff. Thus,
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6. Consider a queueing system having two servers and no queue.
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Please answer the entire problem!
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than can be handled by the Burger Dome food service staff. Thus,
customers wait in line to place and receive their orders. Suppose
that Burger Dome analyzed data on customer arrivals and concluded...

Burger Dome sells hamburgers, cheeseburgers, French fries, soft
drinks, and milk shakes, as well as a limited number of specialty
items and dessert selections. Although Burger Dome would like to
serve each customer immediately, at times more customers arrive
than can be handled by the Burger Dome food service staff. Thus,
customers wait in line to place and receive their orders. Suppose
that Burger Dome analyzed data on customer arrivals and concluded
that the arrival rate is 33 customers per...

Problem 15-25 (Algorithmic)
Burger Dome sells hamburgers, cheeseburgers, French fries, soft
drinks, and milk shakes, as well as a limited number of specialty
items and dessert selections. Although Burger Dome would like to
serve each customer immediately, at times more customers arrive
than can be handled by the Burger Dome food service staff. Thus,
customers wait in line to place and receive their orders. Suppose
that Burger Dome analyzed data on customer arrivals and concluded
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system.

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n a barber the rate for the number of the customers is 3 per
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D) What is the probability that exactly 2 customers
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