Question

RentAPhone is a new service company that provides European mobile phones to American visitors to Europe. The company currently has 90 phones available at Charles de Gaulle Airport in Paris. There are, on average, 25 customers per day requesting a phone. These requests arrive uniformly throughout the 24 hours the store is open. (Note:This means customers arrive at a faster rate than 1 customer per hour.) The corresponding coefficient of variation is 1.

Customers keep their phones on average 84 hours. The standard deviation of this time is 72 hours.

Given that RentAPhone currently does not have a competitor in France providing equally good service, customers are willing to wait for the telephones. Yet, during the waiting period, customers are provided a free calling card. Based on prior experience, RentAPhone found that the company incurred a cost of $1 per hour per waiting customer, independent of day or night.

(a) What is the average number of telephones the company has in its store? (Round your answer to nearest whole number.)

The average number of telephones are?

(b) How long does a customer, on average, have to wait for the phone? (Round your answer to two decimal places.)

The average waiting time is ___ Hours?

(c) What are the total monthly (30 days) expenses for telephone cards? (Round your answer to two decimal places.)

The total monthly expenses are ?

(e) How would waiting time change if the company decides to limit all rentals to exactly 84 hours? Assume that if such a restriction is imposed, the number of customers requesting a phone would be reduced to 15 customers per day. (Round your answer to five decimal places.)

The waiting time is ___ hours?

Answer 1

Answer a) The average number of telephones

Let,

m = No. of phones available with the company = 90,

a = Average inter arrival time = 24 hours / 25 customers = 0.96 Hours

p = Average time of customers keeping their phones = 84 hours

∴ Utilization rate = p / (m X a) = 84 / (90 X 0.96) = 84 / 86.4 = 0.9722 = 97.22 %

This means that = 90 X 97.22 % = 90 X 0.9722 = 87 phones are in use, and 90 - 87 = 3 phones are available on an average

Answer b) We are given CVa = Coefficient of variation for interarival.

Further CVp = Co-efficient of variation for processing time = Standard Deviation / p

= 72 / 84

CVp = 0.8571

Now, apply the formula of Expected Waiting TIme:

Tq = Expected Waiting Time =

By replacing the appropriate values, we get:

Expected waiting time =

= (0.9333) X (0.7039 / 0.0278) X (1.7346 / 2)

(Rounded to 2 decimal places)

Answer c) Total monthly expenses for telephone cards:

= Cost paid for each customer waiting in the queue X Avg No. of Customers in the queue X No. of hours in a day X No. of days in a month

Where

Cost paid for each customer waiting in the queue = 1 $

Avg No. of Customers in the queue = Tq X ( 1 / a) = 20.50 X (1 / 0.96) = 21.32 Customers,

No. of hours in a day = 24 hours, and

No. of days in a month = 30

Hence,

Total monthly expenses for telephone cards = 1 X 21.32 X 24 X 30 = 15350 $

Answer e)

Change in the expecteed waiting time =:

=

Where p = 84, m = 90,

Utilization = 84 / [(24 / 15) X 90] =   0.5833,

Hence, change in expected waiting time = 0.9333 X 0.0286 X 0.5

0.00133 Hours

(PyUtilization V2m+1)-1 1- Utilization -XVa+CV?

2 y (1)2 + (0.8571) 90411-0.9722)X(- x0.9722V2(90+1)-1

Tq = Expected Waiting Time = 0.93 X 25.32 X 0.87 = 20.50 Hours

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