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 30 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 45 seconds. To help determine the amount of space needed for the line at the drive-up window A. What is the average line length (in cars)?

B. What is the average number of cars in the system (both in line and at the window)?

C. What is the expected average time in the system?

(A)0.6600 (B) 1.325 (C) 0.997 or 1 minute

Explanation:

Solution

Given that:

The constant rate = 30 seconds

The arrival rate according to Poisson distribution is = 45 seconds

Now,

(A) We solve for the average length line of cars

The formula is given below:

Lq = λ²/ 2μ ( μ -λ)

Here,

λ = this is the mean time of arrival rate

μ = This is the mean service rate

Thus we compute for the mean time arrival rate which is given below:

The mean arrival rate λ = arrival rate/ 60 seconds

= 60/45

= 1.33 customer per minute

Then we solve for the means service rate which is given below

The mean service rate μ = 60 seconds/ mean rate

= 60/30 = 2 customer per minute

We will now solve for the average line length in cars which is shown below:

Lq = λ²/ 2μ ( μ -λ)

Lq = 1.33²/2*2 (2-1.33)

Lq = 1.7689/4 (0.67)

Lq = 1.7689/2.68

Lq = 0.6600

Therefore the average length in line for cars is 0.6600 cars

(B) We solve for the average number of cars in the system

Ls =Lq + λ /μ

Ls =0.600 + 1.33/2

Ls =0.6600 + 0.665

Ls = 1.325

(C) Finally we need to find the expected average time in the system which is shown below:

Ws = Ls/λ

Ws= 1.325/1.33 = 0.997 or 1.00

The expected time average  in the system is 0.997 or 1.00 minutes.

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