Question

Again, recall Little’s Law. An escalator can be used two ways: A person can stand and let the escalator take her to the other end. Or the person can walk the escalator, reaching the other end faster. Suppose by walking a person reaches the other end in 60% of the time she takes just standing. However, when all walk the escalator, they need more space between them to avoid bumping into each other. Thus, the number of persons who can be on the escalator at the same time reduces by 50%. Which method (everyone standing vs. everyone walking) allows escalator to be used by most people per unit of time? Explain briefly. For full credit, you must use Little's Law to prove it.

Answer 1

As per your question, First I will explain the little's law next what it stand for in mathematical expressions further I will provide you the solution and proof. Please bear with me.

# Little’s Law: The Little’s Law is a very practical mathematical theorem that smartly able to determine the average number of objects in a stationary queuing system, by considering the average waiting time of an object within a system and the average number of objects arriving at the system per unit of time.

Since this theorem is mathematical, there is a precise expression available for this:-

So, Let's assume L is the average number of objects in a queuing system. Further, is the average number of objects arriving at the system per unit of time and is the average waiting time for an object or time spends in a queuing system. Thus,

L =  

Means, the average number of objects in a queuing system per unit time = the average number of objects arriving at the system per unit of time X the average waiting time for an object.

Now coming to your question,

Let's assume the average time required/spend when person is standing on the escalator is per unit time. and the average number of persons arriving at the escalator is .

So, the average number of person in the escalator(L) when everyone standing per unit time=

Now as per your question, By walking a person reaches the other end in 60% of the time when person just standing. Means, X 60/100 is the average waiting time when walking.

And, When walking the number of persons who can be on the escalator at the same time reduces by 50%. Means, - X 50/100 = X 50/100.

Thus,

the average number of person in the escalator(L1) when everyone walking per unit time= X 50/100 X X 60/100 = 3   / 10

or 0.3.

When everyone standing, the average number of person in the escalator(L) per unit time is  .

When everyone walking, the average number of person in the escalator(L1) per unit time is 0.3.

Hence, When everyone standing,  the escalator to be used by most people per unit of time.

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