(a) First, determine the Queuing Models

1. Queuing Models Poisson Arrivals Standard (Infinite Queue) Exponential Service Times Single Server (M/M/1)

Standard M/M/1 Model
P0 = 1 – P
(Probability of 0 customer in the system)

P (n ≧k ) = ρk
(Probability of

Pn = P0ρn
(Probability of exactly n customers in the system)

Ls = λ / ( μ – λ )
(Mean no. of customers in the system)

Lq = ρλ / ( μ – λ )
(Mean no. of customers in queue)

Lb = λ / ( μ – λ )
(Mean no. of customers in queue for a busy system)

Ws = 1 / ( μ – λ )
(Mean time customer spends in the system)

Wq = ρ / ( μ – λ )
(Mean time customers spends in the queue)

Wb = 1 / ( μ – λ )
(Mean time customers spends in queue for a busy system)

Now, in our case,
Λ (mean arrival rate) = 15/hr μ (mean service rate per busy server)
Old machine’s mean service time= 3mins, 60/3 =20/hr
New machine’s mean service time= 2mins, 60/2=30/hr

Wq =ρ/ ( μ – λ )
(Mean time customers spends in the queue),
Where ρ = λ / μ

Old machine: (15/20)/ 20-15 = 0.1 hr = 9mins
New machine: (15/30)/ 30-15= 0.03333 hr = 2mins

According to the case, the average wage of the people who bring the documents to be copied is $8/hr,

If the company rent the old machine and there are 15 arrivals/hr,
There total mean time spend in the queue is ( 15 x 9 )mins = 135mins = 2.25hr,
The total wage = $8 x 2.25 = $18

If the company rent the new machine and there are 15 arrivals/hr,
The total mean time spend in the queue is ( 15x2)mins = 30mins = 0.5hr,
The total wage = $8 x 0.5 =$4
Added the additional cost of renting the new machine is $10/hr
The total cost is $4 +$10 = $14/hr

Consider the lost of productive time of employees as time spent waiting in queuing, the new one can reduce from 9mins to 2mins.
The old machine needs 135mins to finish the 15 arrivals’ copies while the new machine only takes 30mins.
Moreover, the new machine can reduce the total wage significantly from $18 to $4.
Although the additional renting cost added, the new machine’s total cost each hour is $14 while the old one is $18.
Therefore, the company should rent a new machine to replace the old one.

(b) For the old copying machine, what is the probability when a person arrives that he or she will encounter people already waiting in line for service? (Be careful to identify properly the number of customers who might be present for this situation to arise.)

In this situation, there should be 1 person is copying, at least 1 person is waiting and 1 person has just arrived.
Therefore, The number of people in the system (n) = 3.

Pn = P0ρn
(Probability of exactly n customers in the system)

P3 = P0ρ3
Where P0 = ( 1-ρ )
= (1- (15/30)) (15/30)3
=0.0625

Therefore, the probability when a person arrives that he/she will encounter people already waiting in line for service is 0.0625.

(c) Suppose now that the company plans to rent one more (old) machine with an additional cost $15 per hour. Is it a good decision? Which method is better in terms of saving the cost, renting a new copying machine or renting two old copying machines? Explain your answer with the assumption that there is only one single waiting line and no balking or reneging.

(i) If the company rent one more (old) machine, the Queuing Model should be changed to standard M/M/c (Multiple Service) Model.
Now, creates an M/M/2 system in this case, the arrival rate is 15/hr.
Because calculations for the M/M/c model are so tedious, I use from our book Appendix C, “Values of Lq for the M/M/c Queuing Model,” to solve for Lq.
For this problem, in which c=2 and ρ=(15/30) =0.5,
Find Lq= 0.033,
Thus, Ls = Lq + ρ
= 0.033 + 0.5
= 0.533
According to Standard M/M/c model, using the equation,
Ws = Ls/ λ
= 0.533/ 15
=0.0355hr, = 2.132mins

(ii) If the company rent a new machine,
Ws = 1 / ( μ – λ ) (Mean time customer spends in the system)
= 1/ (30-15)
=0.0667hr, = 4mins
Options:
(a) 2 old machines: Mean time customer spends in the system: 2.132mins,
Additional cost: $15/hour
(b) 1 new machine: Mean time customer spends in the system: 4mins
Additional cost: $10/hour
According to the mean arrival rate is 15/hour,
Option (a) can handle the 15 arrivals in 32 minutes while option (b) can handle these in 60 minutes. As their average wage of the people who bring the documents to be copied is $8 an hour, I think option (b) is a better decision since this method not only better in terms of saving the cost but also fully utilized. Although option (a) is faster, but this method is more expensive and sometimes the servers are not in used, it wastes the resources.

(d) Assume that the company selects to use the best method in part c. Because of the limited space of the copying area, the company states that not more than six people would be allowed in the copying area. Determine the mean number of the people in the queue.

Assume that the company selected to rent a new machine.
According to Standard M/M/1 Model,
Lq = ρλ / ( μ – λ )
(Mean no. of customers in queue)

Lq = (6/30)(6) / (30-6)
=0.05 people

(a) Analyze Bob’s letter and indicate the strengths and weaknesses of this letter.
Bob’s letter is on behalf of Fast& Good Hotel in response to the complaint from Barbara and apologized to the customer in order to mollify her frustrations. Besides apology, Bob also explains why Barbara’s room had not been cleaned during the day while she was out and the actions which the hotel was taken after receiving Barbara’s complaint.
There are some strengths and weaknesses in Bob’s letter.
Strengths are: (1) Apology
In Bob’s letter, he apologized twice genuinely. The hotel that responds and apologizes may mollify the complaint the complaining customer. (2) Explanation
In Bob’s letter, he explained the hotel’s current situation in detail that why the cleaner are inexperienced and hope that the customer could understand.
Moreover, Bob’s letter also mentions the action was taken to the receptionist. It shows the hotel has sincerity to fix the problem.
However, it is not the only problem, the inexperienced cleaner is another problem that should be fixed and Barbara mentioned this problem more than once. Not fixing the problem is risky, though, and may backfire on the company. There are still have other weaknesses in Bob’s letter below: (1) Reimbursed
As Barbara mentioned in her letter, her stay was expensive but was provided poor customer service, the hotel may consider to give some reimburse like anther free stay. It not only may pour oil on troubled water but also create a chance to re-establish confidence of Fast &Good Hotel from Barbara’s by providing excellent customer. (2) Assurance
Bob’s letter did not give any assurance that the hotel would try their best to improve and fix the problems that Barbara had mentioned. It is important since that is a way to increases customer’s confidence in the hotel. (3) A “Thank you”
Bob’s letter had not thanked for Barbara’s complaint. A complaint is a gift to a business since the customer use their time to complain and remind the business what dissatisfied the customers and what should be improved. Therefore, the hotel should thanks for Barbara’s complaint which benefits the business since not everyone would do this when dissatisfied but just move to other alternatives.

Write an “improved-version” of the response letter. Fast& Good Hotel 88 Camp Fire Street Heat Point, Ocean Country March 30, 2012
Ann Chan
1220 Water Way Dr.
Riverdale, Ocean Country
Dear Ms. Chan:
On behalf of Fast& Good Hotel, I am writing in response to your letter of 5th March. I am terribly sorry for what had happened to you at Fast& Good Hotel. Our hotel would like to express our apologies to you.
With regard to your complaint on the front desk staff, after the investigation, I have already rebuked the receptionist at the front desk at that time and issued her a warning letter stating that she will be fired if similar case happens again. I am really sorry for her impolite attitudes and brought you any unhappy feelings.
Moreover, concerning your complaint on room cleaning service, since the economy in Heat Point is very good and the employment rate maintains at a very low level in these few years so it is not easy for us to hire experienced cleaning staff. Hence we need to employ inexperienced staff and it affects our daily operations. So, hopefully, you can understand our difficult situation and forgive the staffs who provided the cleaning services. Our hotel would consider to train up them as soon as possible. Sorry for bringing you any inconvenience.
In order to reimburse your last unhappy stay and thanks for your valuable advises, we would like to invite you to have another stay in the future for free. Please give us a chance to provide you an outstanding customer service and re-establish your confidence in our hotel. We assure that we would improve our service quality and try our best to satisfy every customer.
On behalf of Fast& Good Hotel, I would like to apologize to you again and hope you will be satisfied with my explanations. Finally, thank you for your letter and we will be very glad if you will visit Fast& Good Hotel some day.

Sincerely,
Bob Lee
Manager, Fast&Good Hotel