# Queuing Theory and Operational Analysis

Queuing Theory and Operational Analysis

The initial step in the process is to construct a model from the information provided. The notations for the model are as follows:

Average of number of customers in the system = L

Average of size of the queue = Lq

Probability of no customers in the system = P_{0}

Probability of given number of customers being present in the system = P_{n}

Average of waiting time in the system = W

Average waiting time in the queue = Wq

Customer arrival per unit of time = λ

Number of customers served per unit of time = µ

Time spent serving the customer = P, which = λ/ µ

**Question 1:**

Based on the information offered, it is assumable that the average waiting time would be given by customer arrival per unit divided by the product of the number of customers served per unit of time and the difference between the number of customers served per unit of time and the number of arrivals per unit of time as demonstrated.

**Question 2:** The daily average loss of sales is determined using the queue with discouraged arrivals. In this, it is assumed that a number of the customers in the system would change their mind and opt to order something different or seek an alternative. From the case, the total cost of losses incurred by the organization is the number of customers that move to University Falafel after waiting for more than 20 minutes.

**Question 3** When a second server is added, it is logical that the time of wait would be reduced but not significantly due to the fact that, with an infinite number of customers in the system, only the income of the business is likely to change and not the average time spent by customers in the queue.

**Question 4**

To calculate the loss to the organization, the probability of waiting longer than expected is multiplied by the cost of loss of business to the competition:This would give 0.8 by 0.533 by $10 which would result in $4.264

**Question 5** From the calculations, the amount of savings from protection of income is significant for the organization. This is noted by the reduction of losses from loss of customers from $35.21 to $4.26. However, this saving should be compared with the cost of securing the income which is $75.

**Question 6**

- The time taken to serve a customer is 10 minutes per hour and there are only five orders. As such, there is likelihood that all 5 customers would be served within the hour and there is high probability than there are no waiting times. However, when the service time is increased, the probability of waiting more than 20 minutes in the system is increased. The probability of the delays of 5 minute increase in service time is 0.52.