How Can I Calculate Call Traffic for Optimal Caller and Receiver Allocation?

[kryten68]

Hi,

I'm in a bit of a fix with this problem and as such if someone were prepared to help me get to get to a point where I had a formula I can re-use I would be happy to pay a small bounty. Paypal donation, Amazon gift... something like that?

My problem is one of calculating call traffic. I need to be able to calculate/predict how many calls I will be able to process in a given period of time, where the calls will all last a specific duration. I have complete control over how many callers there can be and how many receivers there can be also.

Let me give an example:-

Lets say I wish to achieve 3200 calls in an hour. Each call will last for an average of 300 seconds once connected to the receiver. Each call will take on average 15 seconds from initiation to being connected to the receiver. So the call is actually 315 seconds in total duration as far as the caller is concerned. Each receiver is given one minute of break time between calls.

I need to achieve 3200 calls per hour. How do I calculate how many callers I need and how many receivers I need to get that call throughput? Eg would I need to have 200 callers and 400 receivers?

The values given above are really just examples. One day I may need to work out the values for 4000 calls per hour the next I may need to work out the values for 290 calls per hour... So what I need is some sort of re-usable formula. Once understood I can make the formula into a program to make life easier!

Would be extremely grateful for any [urgent] assistance anyone can offer me.

Many thanks,
Ron

 

[Euge]

Hi Ron,

I just want to make something clear here. This is a free site, so there's no need to offer a bounty. We all volunteer to give math help. (Bigsmile)

 

[kryten68]

Okay. Yes, I understand.. apologies if I broke any rules there.
To be honest, I'm just so desperate with this problem I hoped it might inspire someone to cut in quickly and try to help ... :(

Sorry about that.

 

[Prove It]

I think we need more information since you said what you posted was just one possible scenario.

What will be the same in every scenario? What is likely to change in each?

 

[Jameson]

Hi Ron. I think the easiest place to start would be to assume a steady influx of calls for 1 receiver and see how many that receiver can take per hour. Then we can scale the number of receivers up to the desired number of total calls.

There are 3,600 seconds in 1 hour. So, in general the average number of calls per receiver, per hour is $\displaystyle \frac{3600}{\text{Avg call time}+\text{Avg connection time}}$.

In your example this would be \(\displaystyle \frac{3600}{300+15}=11.4285714286\). Now if we want 3200 calls per hour, we can take $\displaystyle \frac{3200}{11.4285714286} \approx 280$, so you would need roughly 280 receivers at this pace. If you have random gaps where no one is calling or the average time goes up across the board during certain hours, then this model changes but this is a start.

 

[kryten68]

Thank you so much. I think that is on the right tracks.

So, the only two items that are variable in this problem are:-
The number of callers
The number of receivers.

I think in the example equation you kindly supplied Jameson, you reduced the call duration?

Would it be possible, please, to show how target call rates would be achieved by increasing the callers and receivers instead?

Apologies for not being so clear before. I think if there were an equation or formula which allowed me to play with those two values, callers and receivers, where the call duration 300s and the connection delay 15s are constant. That would be "it".

Thank you! I really appreciate this!

- - - Updated - - -

EDIT:

Wait.. I've woken up and had a couple of coffees and re-read your post and it makes much more sense now.

So basically I can take the seconds in the hour over (call duration + answering delay) and that gives me a number. In your example let's say 11.43. I can then take the desired number of calls per hour, 3200 say, and put that over the 11.43 which comes out as approx 280. Which is the number of receivers needed to get that call volume assuming the callers are constantly sending calls.

I can follow that.

Is there a formula that would describe the number of callers needed to 'balance' the other side? So if the number comes out at 280 receivers is it possible to calculate how many callers would need to be sending calls to maintain that throughput?

Many thanks!
Ron

 

[kryten68]

I think we need more information since you said what you posted was just one possible scenario.

What will be the same in every scenario? What is likely to change in each?

Hi,

Thanks for the reply.
The only things that will change are:-

Number of callers.
Number of receivers.

The call duration, answer delay and a short after call rest will all be constant and could be folded into a single time value such as 320 seconds (300s on a call, 15s answer delay, 5s after call gap). The period that is being measured is also constant: 1 hour.

What I need is a way to model the number of callers and receivers to predict the call volumes and throughout with those numbers. Eg If I were to throw 20 callers and 40 receivers at the scenario what would be the total number of calls over the hour.

Presumably if a formula is identified that works, changing the constants I've mentioned here would not break it? But that would only be needed later. I think once I understand the basic relationships I might be able to deal with it.

 

[kryten68]

Additionally, would it be reasonable to assume that the same formula describes the capacity for the callers?

So that would be 3600 over (total call duration + small inter call gap). To get the total number of calls that each caller can make in one hour. Would I then be able to take that number and divide the total number of calls I want to generate by it?

Eg if I want to generate 1400 calls per hour:

3600 / (15s + 300s + 5s) or 3600s / 320s = 11.25

So each caller can make 11.25 calls per hour at those specifications.

So, following on from Jamesons example, should I then do:-

1400 / 11.25 = 124.4

So would that mean I would need approximately 125 callers to achieve that volume over the hour?

Thanks, folks, I really appreciate the pointers and help here... as you can probably tell this is a pressing matter for me, to say the least!

 

[kryten68]

Anyone? Please?...

I think the information so far is very useful but I'm just trying to get to a point where I can throw different values at the number of callers and the number of receivers then hopefully with a formula work out how many calls per hour that will yield.

If no one can help any further would it be possible to point me somewhere else that might be able to help or perhaps a book recommendation?

Thanks
Ron

 

[Jameson]

Hi Ron,

Your calculations are correct but I would work it differently. I wouldn't say "each caller can make 11.25 calls per hour", rather "each receiver can take 11.25 calls per hour". We assume that the receiver is sitting down for the whole hour taking calls I think.

125 is the correct number of receivers needed for the volume you specified in the above post, but once again I should comment that this is assuming the flow of calls is steady with no gaps and that there are no periods where the average duration changes. Another common question might be "What is the probability that the number of calls per receiver is less than 9?" or something like that. If you have a certain tolerance that you can't dip below, then this might become important. However, this needs some more assumptions and some more math to model.

 

[kryten68]

Hi Jameson

Thanks again for your ongoing input.

The situation I have is that to all intents and purposes I have an unlimited number of receivers. Assume 12,000 all there 24/7 just waiting to take a call.

I therefore have two variables that seem to matter the most:-

The total number of calls I WANT to achieve in an hour.
The number of callers needed to achieve that number of calls in an hour.

I don't have unlimited callers, if I did I'd just use that number of callers from the word go and get my target on the first call from each caller!

I have to figure out how to select the right number of callers to achieve that hourly rate.

Eg if I need to achieve 4200 calls per hour .. How many callers do I need? Assume that I can bring 12,000 receivers online to cope with that portion of the call... It's how to calculate how many callers that matters most of all.

Thanks
Ron

 

[Jameson]

Hi Ron,

I think you have the formula down. Just plug the numbers into this: $\displaystyle \frac{3600}{\text{Avg call time}+\text{Avg connection time}}$.

Call that number $A$. Then take \(\displaystyle \frac{\text{Desired volume of calls}}{A}\) to get how many receivers you need.

Assuming you have some time to test and tweak your results, what I would do is solve for the number of receivers like you have already done and test it out for a week. Take note of the average call time and average connection time and note any irregularities to the model. If after a week the volume per hour is where it should be more or less, then awesome. If not, you can see where to tweak it.

What do you think?