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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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