Question

Question: Burrito King (a new fast-food franchise opening up nationwide) has successfully automated burrito production for its drive-up fast-food establishments. The Burro-Master 9000 requires a constant 45 seconds to produce a batch of burritos. It has been estimated that customers will arrive at the drive-up window according to a Poisson distribution at an average of one every 50 seconds. To help determine the amount of space needed for the line at the drive-up window, Burrito King would like to know the expected average time in the system, the average line length (in cars), and the average number of cars in the system (both in line and at the window).

Short answer

Answer

Queueing waiting line models are utilized to consider and dissect the distinctive lining designs taken after by organizations. Beneath these models, framework productivity is measured and considered concurring to the entry design and benefit patterns.

Step-by-step solution

The following are the determinations of the quick nourishment establishment "BK”

Arrival time = 50 seconds

Service time = 45 seconds

This is the case of a single channel with a constant service pattern, where the arrival is in the Poisson pattern.

Compute the arrival and service rate per minute

$$\begin{gathered} Servicerate=\frac{1minute}{servicetime} \\ =\frac{60}{45} \\ \mu =1.3customersperminute \end{gathered}$$

role="math" localid="1650603726907" $$\begin{gathered} Arrivalrate=\frac{1minute}{Arrivaltime} \\ =\frac{60}{50} \\ \lambda =1.2customersperminute \end{gathered}$$

 Calculate the average length of line (in cars) using the formula given

$$\begin{gathered} L_q=\frac{{\lambda }^2}{2\mu \left(\mu -\lambda \right)} \\ =\frac{{\left(1.2\right)}^2}{2\times 1.2\left(1.3-12\right)} \\ =\frac{1.44}{0.26} \\ =5.54cars \end{gathered}$$

Therefore, the average line length is 5.54 cars

The average number of cars in the system using the formula

$$\begin{gathered} L_s=L_q+\frac{\lambda }{\mu } \\ =5.54+\frac{1.2}{1.3} \\ =5.54+0.92 \\ =6.64cars \end{gathered}$$

Therefore, the average number of cars in the system is 6.46 cars

Calculate the Expected average time in the system using the formula given.

$$\begin{gathered} W_s=\frac{L_s}{\lambda } \\ =\frac{6.46}{1.2} \\ =5.38 \end{gathered}$$

Hence, the expected average time in the System is 5.38 minutes .