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NadineSch

Member

Registered: 2016-04-24

Posts: 13

Word problem differential equation

Hello dear Mathlovers (I wish I can soon say I'm one of you ;-))

I have an exam upcoming and part of it is about differential equations. We got some exercises to study with along with solutions. But: It's only the actual solution, so not showing how to get there. I have one exercise in particular that I struggled with, because I made calculations and they seem to make perfect sense. But as I compared it to the solutions at the end: Not the same So I was hoping that you could help me out here. Especially with the calculations so that I can follow/compare/see where I went wrong etc.

Here it is:
On normal weekdays on average 18 customers per minute want to pay at the cash desk in a supermarket. At a cash desk it takes on average 2 minutes to serve a customer. To avoid long waiting times, one cash desk is opened per 10 customers in the queue.

a) Construct a differential equation that models the evolution of the number of customers queuing for the cash points in time.

b) Derive a function that describes the number of queuers for the cash desks in the supermarket at each time of the day. When the supermarket opens in the morning, there are no customers waiting at the cash desks, of course.

c) What is the maximal number of cash desks that simultaneously will be operational in the course of the day ?

d) In the long run, what will be the (average) waiting time for a customer at the cash desks ?

Thank you so very much indeed.

Nadine

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Word problem differential equation

Hi;

I thought they would use a Poisson distribution for a queuing problem? Please let me see your DE and whatever else you have.

On normal weekdays on average 18 customers per minute

To even construct a simulation I think I need to know how the customers are distributed. Do they arrive like a Normal distribution with mean 18 and sd of what?

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Word problem differential equation

It is not clear whether any cash desk is open until the first 10 people queue up.

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Word problem differential equation

Anyway taking x and no. of desks as continuous variable

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

NadineSch

Member

Registered: 2016-04-24

Posts: 13

Re: Word problem differential equation

For everyone that helped: I have passed my exam!! Thank you very much. I only found it polite to come back to you with this.