Fresh Sub hires three workers during the peak hours. When customers arrive, one worker is dedicated to order taking and preparing the bread and the meat; then the customer is passed to the second worker, who asks the customer about the vegetable selection and specializes in assembly; finally the customer is passed to the third worker, who is a cashier. The service time for the first worker (order taker) follows a normal distribution with mean equals to 3 minutes and a standard deviation equals to 1 minutes. The service time for the second worker (assembler) is uniformly distributed between 2.4 minutes to 2.6 minutes. The service time for the third worker (cashier) is exactly 2.4 minutes. Customers arrive following a Poisson process with a rate of 15 customers per hour (i.e., the inter-arrival time is exponential).
a. Does Fresh Sub have enough capacity to serve the customers?
b. What are the utilization rates of the three workers?
c. On average, how many customers will wait for the first worker (order taker)? And how long on average will a customer wait?
d. As the customers finish order taking and bread/meat selection and move onto the second worker (assembler), from the assembler’s view, does the arrival rate still follows a Poisson process?
e. Will customers ever wait for the second worker (assembler)? Will customers ever wait for the third worker (cashier)?
Did you find an answer to this question? I'm definitely in the same 311 class and couldn't figure this out.
To determine if Fresh Sub has enough capacity to serve the customers, we need to calculate the total service time and compare it to the inter-arrival time.
a. Total service time:
The first worker (order taker) follows a normal distribution with a mean of 3 minutes and a standard deviation of 1 minute.
The second worker (assembler) has a service time uniformly distributed between 2.4 minutes to 2.6 minutes.
The third worker (cashier) has a service time of exactly 2.4 minutes.
To calculate the total service time, we need to consider the time spent with each worker. Let's denote the service time for the first worker as T1, the service time for the second worker as T2, and the service time for the third worker as T3.
The total service time for each customer can be calculated as:
Total service time = T1 + T2 + T3
b. Utilization rates:
The utilization rate for each worker can be calculated as:
Utilization Rate = (Total Service Time)/(Average Inter-arrival Time)
c. Average number of customers waiting and average wait time for the first worker:
Using Little's Law, we can find the average number of customers waiting and the average wait time for the first worker.
Average Number of Customers Waiting = Utilization Rate * Average Wait Time
Average Wait Time = Utilization Rate * Total Service Time
d. Arrival rate for the second worker:
The arrival rate for the second worker will still follow a Poisson process if the service time for the first worker is exponentially distributed. Since the service time for the first worker follows a normal distribution, the arrival rate for the second worker will not follow a Poisson process.
e. Waiting times for the second worker (assembler) and third worker (cashier):
Since the second worker's service time is only 0.2 minutes (2.6 - 2.4) and the third worker's service time is fixed at 2.4 minutes, it is highly unlikely that customers will wait for the second worker (assembler) or the third worker (cashier).
I will now calculate the answers using the given information.
To answer these questions, we need to analyze the system and calculate various performance measures. Let's go step by step:
a. To determine if Fresh Sub has enough capacity to serve the customers, we need to calculate the total service time and compare it to the arrival rate. The total service time is the sum of the service times of all workers. The arrival rate is given as 15 customers per hour, so we need to convert it to minutes: $\lambda = \frac{15}{60} = 0.25$ customers per minute.
The total service time = service time of order taker + service time of assembler + service time of cashier
= mean service time of order taker + mean service time of assembler + mean service time of cashier
= 3 + (2.4 + 2.6) / 2 + 2.4
= 8.2 minutes
Since the arrival rate is lower than the total service time, Fresh Sub has enough capacity to serve the customers.
b. The utilization rate of each worker can be calculated as the ratio of their mean service time to the average time between consecutive customer arrivals. The average time between consecutive customer arrivals is the reciprocal of the arrival rate, i.e., $\frac{1}{\lambda}$. The utilization rate of each worker is therefore:
Utilization rate of order taker = mean service time of order taker / (1 / arrival rate)
= 3 / (1 / 0.25)
= 0.75
Utilization rate of assembler = mean service time of assembler / (1 / arrival rate)
= (2.4 + 2.6) / 2 / (1 / 0.25)
= 1
Utilization rate of cashier = mean service time of cashier / (1 / arrival rate)
= 2.4 / (1 / 0.25)
= 0.6
c. To calculate the average number of customers waiting for the first worker (order taker), we can use Little's Law, which states that the average number of customers in the system (including both waiting and being served) is equal to the average arrival rate multiplied by the average time a customer spends in the system.
For the first worker, the average service time is the mean service time, which is 3 minutes. So, the average time a customer spends with the first worker is 3 minutes.
Using Little's Law: average number of customers in the system = arrival rate * average time in the system
Average number of customers waiting for the order taker = arrival rate * average time in the system - 1 (since 1 customer is being served)
= 0.25 * 3 - 1
= 0.75 - 1
= -0.25 customers
The negative result means that, on average, there will be no customers waiting for the order taker.
To calculate the average waiting time for a customer, we can use the formula: average waiting time = average number of customers waiting / arrival rate.
Average waiting time for the first worker = -0.25 / 0.25
= -1 minute
Again, the negative result indicates that, on average, a customer will not wait for the order taker.
d. From the assembler's view, the arrival rate does not follow a Poisson process. This is because the second worker's service time is fixed and not random. When customers finish with the first worker and move to the assembler, they spend a fixed amount of time before moving to the next worker.
e. Customers will never wait for the second worker (assembler) because the service time for the assembler is fixed and lower than the arrival rate. The assembler can serve customers faster than they arrive.
Customers will also never wait for the third worker (cashier) because the service time for the cashier is fixed and lower than the arrival rate. The cashier can serve customers faster than they arrive.
To summarize:
a. Yes, Fresh Sub has enough capacity to serve the customers.
b. Utilization rates: Order taker -> 0.75 (75%), Assembler -> 1 (100%), Cashier -> 0.6 (60%).
c. On average, no customers will wait for the order taker. The average waiting time for a customer is -1 minute.
d. No, the arrival rate doesn't follow a Poisson process from the assembler's view.
e. Customers will never wait for the second worker (assembler) or the third worker (cashier) due to their fixed service times.