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What are the arrival and services rates for the cocktail bar? d the following measures of performance of the cocktail bar: What is the average utilization of the cocktail bar employee? On average, how many customers are waiting in line? What is the average time (in minutes) a customer spends waiting for service at the cocktail bar? =) How many customers are likely to be in the system' at any one time? ) What is the average time (in minutes) a customer spends 'in the system'? Vino Inc. is a local company that produces and bottles its own brand of wine using local raspberries. The company sells its wine by the case (12 bottles per case) to national and international markets by the. Demand in these markets have been increasing this year, with the following data for the past three months: Month Demand (bottles) April 6,500 May 7,000 June 8,500 Current production capacity to produce the wine is 100 cases (12 bottles per case) per day; the facility operates 7 days a week. However, due to scheduled maintenance of their equipment, management feels that they can realistically produce no more than 6,960 bottles per week. Last week, two employees were absent several hours each and only 6,000 bottles were produced. Vino also operates a cocktail bar and sells its wine by the glass. Revenue and expenses from the cocktail bar are kept separate from the production facility operations. The price of a glass of wine is $5.75; the cost of purchasing the wine from the production facility is $10.00 per bottle (4 glasses per bottle). The cocktail bar is open every evening from 4-8 pm with one server working. The fixed costs of the cocktail bar (salaries, rent, utilities, etc.) total $40,000 per year. At the cocktail bar, previous experience is that 18 customers per hour arrive to the bar (customer arrivals can be described as a Poisson distribution). On average, it takes 2 minutes to serve each customer (the service rate can be described as an exponential distribution).

Answer :

At the cocktail bar, on average, it takes 2 minutes to serve each customer (the service rate can be described as an exponential distribution). The average time a customer spends in the system is zero minutes.

Average customers per hour (arrival rate) = λ = 18 per hour

Service rate = µ = 2 minutes

The Poisson distribution can be used to find the arrival rate for the Cocktail Bar.

1. Average utilization of the Cocktail bar employee:

The average utilization can be calculated as the average service rate divided by the arrival rate:

Average utilization = Average service rate / Arrival rate

Average service rate = µ = 2 minutes

Average customers per hour = λ = 18 per hour

Average utilization = Average service rate / Arrival rate = µ / λ = 2 / 18 = 0.11 or 11%

2.Average number of customers waiting in the queue:

We can use Little’s law, which states that the average number of customers equals the average arrival rate multiplied by the average time spent in the system.

Average number of customers = Average arrival rate * Average time spent in the system

Average arrival rate = λ = 18 per hour

Average time spent in the system = Average waiting time + Average service time

Since the service rate (µ) is 2 minutes, the average service time is 1 / service rate = 1 / 2 = 0.5 minutes.

Calculating the average number of customers:

Average number of customers = λ^2 / (µ(µ - λ)) = 18^2 / (2(2 - 18)) = 18^2 / (-32) = -81 / 4.5 = -18

The negative answer indicates that there is no waiting time and, hence, on average, no customers in line. This result is due to the arrival rate being smaller than the service rate (µ < λ).

3. Average time a customer spends waiting for service at the cocktail bar:

The average waiting time can be calculated using the formula:

Average waiting time = Average number of customers waiting / Average arrival rate

Since the average number of customers waiting is 0 (as calculated above) and the average arrival rate is λ = 18 per hour:

Average waiting time = 0 / 18 = 0 minutes

Hence, the average waiting time for a customer at the cocktail bar is zero minutes. This indicates that customers are served as soon as they arrive.

4. Number of customers likely to be in the system at any one time:

The expected number of customers in the system can be found using the formula:

Expected number of customers in the system = Arrival rate / (Service rate - Arrival rate)

Arrival rate = λ = 18 per hour

Service rate = µ = 2 minutes

Expected number of customers in the system = Arrival rate / (Service rate - Arrival rate) = λ / (µ - λ) = 18 / (2 - 18) = 18 / (-16) = -1.125

This negative answer means that no customer will be in the system, resulting in zero customers.

5. Average time a customer spends in the system:

The average time a customer spends in the system can be calculated using the formula:

Expected time in the system = Expected number of customers in the system / Arrival rate

Arrival rate = λ = 18 per hour

Expected number of customers in the system = 0 (as calculated above)

Expected time in the system = Expected number of customers in the system / Arrival rate = 0 / 18 = 0 hours or 0 minutes.

Therefore, The average duration a customer spends in the system is 0 minutes.

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