Call volume on phone system

[bnoon]

I have been out of school for quite some time and don't use this type math much anymore, so here goes. I need an equation to figure out how to get the same projected amount of calls delivered to a remote call center for Fridays. Our remote call center receives 20% of our calls on days Monday through Thursday, but on Friday they close 4 hours prior to our call center closing. By contract they need to be sent the same amount of calls daily as they would have had they been open the full day. Let's say 1,000 calls daily total, 20% goes to them for 200 calls. How do I get them 200 calls by 5 without just sending them 200 calls all at once or going over or under? It doesn't have to be 200 exactly, but the percentage of total calls should average out to be very near 200.

If this were a flat percentage, I would just write the equation for a 14 hour window at 20% = 10 hour window at x%, = 200 calls... but here's where it gets tricky... Call volumes are not flat throughout the day. It's a fairly consistent curve through the day when averaged over the last month or two of calls, peaking around noon, and tapering off later in the afternoon. I remember some of the things for figuring parabolic curves (which is what I think I need here), but I just can't remember how to plug this all in. See attached diagram. Help!

 

[bnoon]

I'm now wondering if this is more of a Calculus problem since the rate of calls can be watched/marked by hour and graphs can be drawn... dang... Does a moderator agree???

 

[JeffM]

I have been out of school for quite some time and don't use this type math much anymore, so here goes. I need an equation to figure out how to get the same projected amount of calls delivered to a remote call center for Fridays. Our remote call center receives 20% of our calls on days Monday through Thursday, but on Friday they close 4 hours prior to our call center closing. By contract they need to be sent the same amount of calls daily as they would have had they been open the full day. Let's say 1,000 calls daily total, 20% goes to them for 200 calls. How do I get them 200 calls by 5 without just sending them 200 calls all at once or going over or under? It doesn't have to be 200 exactly, but the percentage of total calls should average out to be very near 200.

If this were a flat percentage, I would just write the equation for a 14 hour window at 20% = 10 hour window at x%, = 200 calls... but here's where it gets tricky... Call volumes are not flat throughout the day. It's a fairly consistent curve through the day when averaged over the last month or two of calls, peaking around noon, and tapering off later in the afternoon. I remember some of the things for figuring parabolic curves (which is what I think I need here), but I just can't remember how to plug this all in. See attached diagram. Help!

I'm not sure it is a math problem at all. How does your call routing system work? Do calls normally go to the internal call center first and only go to the remote center when wait times exceed a critical value or queue length exceeds some target? If that is so, can you change the routing process on Friday so that the remote center is assigned more calls each hour. Instead of modelling the routing scheme mathematically, it may be more efficient to experiment with changing it or with changing the staffing levels in the internal center so that a higher percentage of the calls go outside early in the day.

And of course, you can always discuss changing the terms of your contract. If you have a good relationship with your vendor, there may be a way to alter terms in a way that benefits both of you if you explain what your problem is and how you will have to solve it under the current terms of the contract. They may prefer that you solve it a different way.

A long time ago, I found that solutions to practical problems often involve discovering what constraints do not really exist.