Comfort Zone Dec 23 Edit5

Slide 1:

Queues are definitely a waste of time when you have nothing to do but wait. But, have we ever thought of how to overcome queues or suggest a method to overcome queues? We did!

Creating Comfort Zone By Time Allocation:

Creating Comfort Zone By Time Allocation R. Anshul, Sri Sankara School, Adyar K. Jayaganesh , DAV Public School, Velachery AMTI 45 th Conference

Queue Everywhere!:

Queue Everywhere!

Slide 4:

Let us now analyze a real life situation

The Real Life Situation:

The Real Life Situation Let us consider the following situation: The restaurant works for ‘ x ’ hours a day. Net number of tables the waiters can serve is less than the total number of tables available. Number of customers arriving is greater than or equal to the number of tables available. Let us say that the restaurant is running at a loss.

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Since the restaurant is running at a loss, the owner gives an advertisement in the paper for a new temporary waiter. ‘ n ’ waiters have come seeking a temporary job. Now, he has to choose the waiter who can manage time better, do the best loss reduction and is economical too! This should also ensure that almost all people are attended to and there will not be any further mismanagement.

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Mr.A 1 Can serve T A 1 tables in ‘b’ minutes. Charges ` S A 1 every ‘b’ minutes.

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Mr.A 2 Can serve T A 2 tables in ‘b’ minutes. Charges ` S A 2 every ‘b ’ minutes.

Slide 10:

Similarly, there are ‘n’ waiters. The ‘n th ’ waiter is Mr. A n . Can serve T A n tables in ‘b’ minutes. Charges ` S A n every ‘b ’ minutes.

Slide 11:

Mr A 1 Charges ` S A 1 every ‘b’ minutes. Can serve T A 1 tables in ‘b’ minutes . Mr. A 2 Charges ` S A 2 every b minutes. Can serve T A 2 tables in b minutes . . . . Mr. A n Charges ` S A n every ‘b’ minutes. Can serve T A n tables in ‘b’ minutes.

Solution:

Solution Variable Meaning T Number of tables each waiter can serve W Number of waiters already working R Average earning per table S Salary for a waiter for ‘b’ minutes Note : These are the parameters of the existing waiters

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Earning till n th duration of ‘ b ’ minutes Loss till n th duration of ‘ b ’ minutes : Total income till n th duration of ‘ b ’ minutes : Few Preliminary Expressions

Strategy:

Strategy For convenience, let us re-arrange the applicants list in a way such that that they are sorted in the ascending order of their capacity ( T A 1 ≤ T A 2 ≤ … ≤ T A n ). Now, from the new sorted list, we have to choose a waiter A j (1 ≤ j ≤n) such that T A j is nearest (lesser) or equal to The reason will be explained consequently.

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Let us say we appoint some waiter Mr. P j Let us define the new total income as , the new cumulative earning as and the cumulative loss as In , since T A j is nearest to , we can ensure least loss. This is the reason for choosing such a T A j .

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Simplifying , we get Our aim is to get a higher income , that is, simplifying which we get : Thus, we can conclude that the waiter demanding the least S A j satisfying the above inequality is the best choice that can be made.

Slide 17:

Let us now look at a more formal approach to queues

Arrival Rate:

Arrival Rate Let us now see a few formal terms connected with queues : Arrival Rate : The number of resources entering a given system in unit time. Service Rate : The number of resources whose need has been fulfilled and is ready to exit the system in unit time. Here, “resources entering a given system” is analogous to “people entering a shop”. A Formal Approach

Waiting Time in a System:

Waiting Time in a System ( S > A ) Let A (arrival rate) be x resources in t seconds, S (service rate) be ( x+y ) resources in t seconds. S-A resources will be in the system for t seconds ( y resources will be idle in the system for t seconds). So, 1 resource will have to wait for at least

Waiting Time in a Queue:

Waiting Time in a Queue Total Time in the System = Waiting time in Queue + Service Time (for a resource) Waiting Time in Queue for 1 resource = Waiting time in Queue for all the resources =

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Now, let us see some applications of the formulae we have mentioned earlier .

Application 1:

Application 1 Choosing an Efficient Repairman

Slide 23:

Repairman A Services 4 ACs every hour Charges ` 250 per hour

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Repairman B Charges ` 340 per hour Services 6 ACs per hour

The Question:

The Question Let us take the following situation: ACs start malfunctioning at the average rate of 3 per hour. The malfunctioning of one AC is independent from the malfunctioning of another. Cost of Idle AC hour is Rs. 50 The office works for 8 hours a day. Which repairman should be hired?

Solution:

Solution Application Of Formula for Waiting Time in a System

Case I : Repairman A is hired:

Case I : Repairman A is hired The waiting time of 1 Idle AC in the system is Idle AC Hours cost : Repairman Charges = ` ( 250 x 8 ) = ` 2 000 Total Charges = ` 3 200.

Case II : Repairman B is hired:

Case II : Repairman B is hired The waiting time of 1 AC in the system is Idle AC Hours cost : Repairman Charges = ` ( 340 x 8) = ` 2720 Total Charges = ` 3 120

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The best choice to make is to hire Repairman B.

Application 2:

Application 2 Reducing The Size Of The Queue

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It costs the clinic ` 100 per patient. Each minute of decrease in the average service time of 10 minutes would cost the clinic ` 10 more per patient. What should be done by the clinic to decrease the average size of the queue to half ?

Solution:

Solution Application Of Formula for Waiting Time in a Queue

Slide 34:

Average Arrival Rate = A = Average Service Rate = S = We need to change the waiting time in the queue for ‘A’ people to Now, what we can do is alter the service rate. So, let it be

Final Steps:

Final Steps Thus, the clinic should attend 8 emergencies per hour (2 more) to reduce the average queue size to half.

Summary:

Summary In the presentation, we derived a condition for improving a restaurant running at a loss due to mismanagement of time and lack of supply for the high demand. Then, we saw few formal terms regarding queues – Arrival Rate, Service Rate, Waiting Time in a Queue, Waiting Time in a system, derived formulae for the last 2 terms and saw their simple applications.

Slide 37:

We hope you enjoyed our presentation

We would like to thank our mentor Shri Sadagopan Rajesh:

We would like to thank our mentor Shri Sadagopan Rajesh

Slide 39:

We thank the amti panel members For giving us this opportunity