Newell and Jeff are the two barbers in a barber shop they own and operate. They provide two chairs for customers who are waiting to begin a haircut, so the number of customers in the shop varies between 0 and 4. For n = 0, 1, 2, 3, 4, the probability Pn that exactly n customers are in the shop is
(a) Calculate L. How would you describe the meaning of L to Newell and Jeff?
(b) For each of the possible values of the number of customers in the queueing system, specify how many customers are in the queue. Then calculate Lq. How would you describe the meaning of Lq to Newell and Jeff?
(c) Determine the expected number of customers being served.
(d) Given that an average of 4 customers per hour arrive and stay to receive a haircut, determine W and Wq. Describe these two quantities in terms meaningful to Newell and Jeff.
(e) Given that Newell and Jeff are equally fast in giving haircuts, what is the average duration of a haircut?
The Correct Answer and Explanation is :
It seems I can’t access the image directly, but I can certainly help you with the problem you’ve described based on the information you provided. Here’s how I would approach each part:
(a) Calculate L. How would you describe the meaning of L to Newell and Jeff?
To calculate ( L ), we use the expected number of customers in the system, which is given by the formula:
[
L = \sum_{n=0}^{4} n \cdot P(n)
]
Where:
- ( n ) represents the number of customers in the shop.
- ( P(n) ) represents the probability that there are exactly ( n ) customers in the shop.
For example, if the probabilities ( P(0) = 0.1, P(1) = 0.2, P(2) = 0.3, P(3) = 0.2, P(4) = 0.2 ), then you would calculate ( L ) as:
[
L = 0 \cdot 0.1 + 1 \cdot 0.2 + 2 \cdot 0.3 + 3 \cdot 0.2 + 4 \cdot 0.2 = 0 + 0.2 + 0.6 + 0.6 + 0.8 = 2.2
]
Meaning of L to Newell and Jeff:
L represents the average number of customers in the shop, which includes both those waiting and those receiving a haircut. It gives Newell and Jeff an understanding of how busy their shop is on average.
(b) For each of the possible values of the number of customers in the queueing system, specify how many customers are in the queue. Then calculate Lq. How would you describe the meaning of Lq to Newell and Jeff?
To calculate ( L_q ), the average number of customers in the queue, we need to consider the situation when customers are waiting. For example, if there are 0 or 1 customers in the shop, no one is in the queue, but if there are 2, 3, or 4 customers, the queue will have customers.
So, for each state of the shop:
- ( n = 0 ): 0 customers in queue.
- ( n = 1 ): 0 customers in queue.
- ( n = 2 ): 0 customers in queue.
- ( n = 3 ): 1 customer in queue.
- ( n = 4 ): 2 customers in queue.
Then, ( L_q ) is calculated as:
[
L_q = \sum_{n=0}^{4} \text{(customers in queue)} \cdot P(n)
]
(c) Determine the expected number of customers being served.
The expected number of customers being served is the number of customers in the system minus those in the queue. Mathematically:
[
L_s = L – L_q
]
Where ( L_s ) is the expected number of customers being served.
(d) Given that an average of 4 customers per hour arrive and stay to receive a haircut, determine W and Wo.
( W ) is the average time a customer spends in the system (waiting + receiving a haircut), and ( W_0 ) is the average time a customer spends waiting in the queue.
These can be calculated using the Little’s Law:
- ( L = \lambda \cdot W ), where ( \lambda ) is the arrival rate.
- ( L_q = \lambda \cdot W_0 ).
Given ( \lambda = 4 ) customers per hour, you can solve for ( W ) and ( W_0 ).
(e) Given that Newell and Jeff are equally fast in giving haircuts, what is the average duration of a haircut?
If Newell and Jeff are equally fast, you can compute the average haircut duration by dividing the time spent per customer by the number of barbers. You can use the information about the number of customers served and the total time spent to calculate this.
Let me know if you need help with further specific calculations or clarifications!