Question

Burger Dome sells hamburgers, cheeseburgers, French fries, soft drinks, and milk shakes, as well as a limited number of specialty items and dessert selections. Although Burger Dome would like to serve each customer immediately, at times more customers arrive than can be handled by the Burger Dome food service staff. Thus, customers wait in line to place and receive their orders. Suppose that Burger Dome analyzed data on customer arrivals and concluded that the arrival rate is 30 customers per hour and 1 customer processed per minute.

Compare a multiple-server waiting line system with a shared queue to a multiple-server waiting line system with a dedicated queue for each server. Suppose Burger Dome establishes two servers but arranges the restaurant layout so that an arriving customer must decide which server's queue to join. Assume that this system equally splits the customer arrivals so that each server sees half of the customers. How does this system compare with the two-server waiting line system with a shared queue? Compare the average number of customers waiting, average number of customers in the system, average waiting time, and average time in the system. If required, round your answers to four decimal places.

Number of Customers waiting in a shared single and dedicated queue

Average number of customers in the system in a shared single and dedicated queue

Average waiting time shared single and dedicated queue in minutes

Average time in the system shared single and dedicated queue in minutes

Answer 1

In order to analyze this, we need to approach the problem with the multi-server queuing model. The formulas used are shown below.

Using the formula, we know that

Lambda = 30 per hour

Mu = 60/1 = 60 per hour

For the matter of speed, let’s calculate these values using excel.

In case of a shared queue and two server model we have c = 2 and thus the results are shown below.

In case there are two different queues for each server then their service capacity still remains the same. However, they will have separate arrival rate for customers. We can consider lambda = 15. The result is shown below. In this case we multiply the length of the queue and the system by 2.

Number of customer waiting in queue

Single shared queue = 0.03

Two dedicated queues = 0.83*2 = 0.166

Average number of customers in system

Single shared queue = 0.5333

Two dedicated queues = 0.0222*2 = 0.0444

Average waiting time in queue

Single shared queue = 0.0011 hour or 0.066 minutes

Two dedicated queues = 0.0055 hour or 0.33 minutes

Average time in the system

Single shared queue = 0.0178 hour or 1.068 minutes

Two dedicated queues = 0.0222 hour or 1.332 minutes