Waiting Lines Decision Making Theory PDF

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Waiting Lines Decision Making Theory...

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Name : Reza Novita Sari ID

: 1711011054

Assignment Waiting Lines and Queuing Theory

From historical data, Harry’s Car Wash estimates that dirty cars arrive at the rate of 10 per hour all day Saturday. With a crew working the wash line, Harry figures that cars can be cleaned at the rate of one every 5 minutes. One car at a time is cleaned in this example of a single-channel waiting line.  Assuming Poisson arrivals and exponential service times, find the 1) average number of cars in line 2) average time a car waits before it is washed 3) average time a car spends in the service system 4) utilization rate of the car wash 5) probability that no cars are in the system. 

Answer :

The cars cleaned at the rate one of every 5 minutes. So in one hour, there are 12 cars cleaned.

The Formula!:!!𝜆!=!10 (Arrival rate of cars in one hour) !

!

𝜇!= 12 (Number of cars served in one hour)

Open the Microsoft Excel. Click the Excel QM buton.

Click the Excel QM then select “Waiting Lines” and “Single-Server Model”

The spreadsheet initialization will appear, fill in the title with “Harry’s Car Wash”, the sheet with “1”, and select “probabilities” only because we will not analyze the cost. Then click OK

The spreadsheet will appear. Fill in the arrival rate with “10” and the service rate with “12”. The solution automatically appear.

From the solution using Excel QM, we know that : •

Average number of cars in line is 4.166666667

•

Average time a car waits before it is washed is 0.416666667

•

Average time a car spends in the service system is 0.5

•

Utilization rate of the car wash is 0.833333333

•

Probability that no cars are in the system is 0.166666667

Testing the Solution •

The average number of cars in line Formula :

Solution : Lq =

#$% #&'(#&)#$)

= 4.166666667

•

Average time a car waits before it is washed Formula :

Solution : #$

Wq = #&'(#&)#$) = 0.416666667

•

Average time a car spends in the service system Formula :

W=

# #&)#$

= 0.5

•

Utilization rate of the car wash Formula :

Solution : 𝜌=

#$ #&

= 0.833333333

•

Probability that no cars are in the system

Solution : P0 = 1 -

#$ #&

= 0.166666667

The manual solution and the solution using Excel QM showing the same number. It means the methods and the solutions from Excel QM is correct.

•

Average number of cars in line is 4.166666667. It means there are 4 cars waiting in the queue.

•

Average time a car waits before it is washed is 0.416666667. It means cars need to wait 0.416666667 before being washed in the queue.

•

Average time a car spends in the service system is 0.5.

•

Utilization rate of the car wash is 0.833333333. It shows the probability that the service facility is being used.

•

Probability that no cars are in the system is 0.166666667

The waiting line model was valuable in predicting potential waiting times, queue lengths, idle times, and so on to make the better services of Harry’s Car Wash. If once Harry wants to enhancing their services, he can use this number as the consideration....