You are in charge of a quarry that supplies sand and stone aggregates to your co

Question

You are in charge of a quarry that supplies sand and stone aggregates to your company's construction sites. Empty trucks from construction sites arrive at the quarry's huge piles of sand and stone aggregates and wait in line to enter the station, which can load either sand or aggregate. At the station they are filled with material, weighed, checked out, and proceed to a construction site. currently, 8 empty trucks arrive per hour, on average. once a truck has entered a loading station, it takes 6 minutes for it to be filled, weighed, and checked out. concerned that trucks are spending too much time waiting and being filled, you are evaluating two alternatives to reduce the average time the trucks spend in the system.

The first alternative is to add side boards to the trucks, and to add a helper at the loading station, at a total cost of 50,000. The arrival rates of the trucks would change to 6 per hour, and the filling time would be reduced to 4 minutes. The second alternative is to add another loading station at a cost of 80,000. The trucks would wait in a common line and the truck at the front of the line would move to the next available station

. The average waiting time under the current system, and the two alternatives are how many minutes?

Explanation / Answer

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Current System: Waits for 24 minutes

Arrival rate = Lambda = (7+1) per hour

Service rate = Miu = 6 minutes per truck =

6 min – 1 truck

60 min – 1 * 60/6 = 10 trucks per hour

Utilization rate = Chi = Lambda/Miu

= (7+1)/10 = 0.7999999

Length of the system = Ls = Number of trucks waiting and the number of trucks being filled

Ls = Chi / (1-Chi) = 0.799999 / (1-0.799999) = 3.999975 = 4

Number of trucks waiting = Lq = Ls – Chi = 4 – 0.7999 = 3.2

Ws = time spent by a truck (both waiting and being filled) = Ls/Lambda = 4/ (7+1) = 0.5 hours = 0.5*60 = 30 minutes

Wq = time a truck waits for = Lq / Lambda = 3.2/(7+1) = 0.4 hours = 0.4*60 = 24 minutes

This matches with the service time of 30-24 = 6 minutes per truck – hence our calculations are confirmed to be correct

Hence under the current system, average waiting time = 24 minutes per truck

1st alternative: waits for 2.66667 minutes

Invest 50,000, add side boards to trucks, add a helper in the filling station:

Lambda = 6 trucks per hour

Miu = 4 minutes per truck

4 min – 1 truck

60 min – 1 * 60/4 = 15 trucks per hour

Chi = Lambda / Miu = 6/15 = 0.4

Ls = chi/(1-chi) = 0.4/(1-0.4) = 0.4/0.6 = 0.6667

Lq = Ls – Chi = 0.6667 – 0.4 = 0.26667

Ws = Ls/Lambda = 0.66667/6 = 0.111111667 hours = 0.1111111667 * 60 minutes = 6.6667 minutes

Wq = Lq/Lamda = 0.266667/6 = 0.044445 hours = 0.0444445 * 60 minutes = 2.66667 minutes

Service time = Ws – Wq = 6.66667 – 2.66667 = 4 minutes (matches with the value given in the question – double confirmed that we are correct)

This is a tremendous improvement over the past situation of 24 minutes, as in this 1st alternative, a truck waits only for 2.66667 minutes

2nd alternative: Waits for 1.1 minute

Extra server, hence C=2

Lambda = (7+1)
Miu = 10 trucks per hour

Chi = 0.7999999999

Using the QTP add on tool pack for excel, and using the built in function QTPMMS_Wq((7+1), 10, 2)

The syntax of QTPMMS_Wq is QTPMMS-Wq(Arrival rate, Service rate, number of servers)

QTPMMS_Wq((7+1), 10, 2) = 0.01905 hours = 0.01905 * 60 minutes = 1.143 minutes = 1.1 minute

QTPMMS_Ws((7+1), 10, 2) = 0.1191 hours = 0.1191 * 60 minutes = 7.143 minutes

service time = Ws – Wq = 7.143 – 1.143 = 6 minutes

Here a truck waits only for 1.1 minute – much quicker than the current system and the 1st alternative.

Hence the 2nd alternative is the quickest.

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