topic Re: M/M/C Queue Theory Formula Implementation in DAX Commands and Tips

M/M/C Queue Theory Formula Implementation

Data_Struggles_ — Tue, 28 Feb 2023 20:45:20 GMT

I am attempting to implement an M/M/c Queuing Theory model in PowerBI using DAX(we previously used an Excel Erlang add-in but found it to be inaccurate at low volumes) and I'm running into issues. I am not a statistician by any means so I've used a GitHub doc as reference(link below; page 150 for M/M/c formulas, 145 for variable info) as I've built the formulas. Below is the DAX script for the table and the link to the reference doc:
Ref doc: https://yzr95924.github.io/pdf/book/Basic-Queueing-Theory.pdf

QTheoryTest3 = var Volume = 50 //in 30 min interval var IntervalSize = 60 //in minutes var AHT = 10 //in minutes var AHT_Adj = AHT / IntervalSize // AHT_Adj is S var ArrivalRate = Volume * (IntervalSize / 30) // ArrivalRate is lambda var ServiceRate = (1 / AHT) * IntervalSize var MinServers = 2 var MaxServers = 50 RETURN ADDCOLUMNS( ADDCOLUMNS( ADDCOLUMNS( ADDCOLUMNS( ADDCOLUMNS( SELECTCOLUMNS(GENERATESERIES(MinServers,MaxServers),"Val1",[Value]), //Val1 is c "Val1-1",[Val1] -1 ), "TrafIntensity", ArrivalRate * AHT_Adj //TrafIntensity is rho ), //The Summation piece of the Erlang C probability. Val2 is n "SubProb", SUMX(SELECTCOLUMNS(GENERATESERIES(0,[Val1-1]),"Val2",[Value]), DIVIDE(POWER([TrafIntensity],[Val2]),FACT([Val2])) + DIVIDE(POWER([TrafIntensity],[Val1]),FACT([Val1]))) ), "ErlangCProb", DIVIDE(DIVIDE(POWER([TrafIntensity],[Val1]),FACT([Val1])),(1 - DIVIDE([TrafIntensity],[Val1])) * [SubProb]) ), "AvgWait", DIVIDE([ErlangCProb] * AHT_Adj, [Val1] * (1 - DIVIDE([TrafIntensity],[Val1]))) * IntervalSize //converted back to minutes )

For this test table my desired output is the Average Wait Time (as I'll implement that in a faux for/while loop process), and as I looked through the formulas in the reference doc I found that I largely needed to use the Erlang C probability formula and the expected Wait time formula (both highlighted below) to achieve that. The calculation of these two formulas is what is being performed in my DAX script.

Unfortunately, I am receiving inaccurate results(see screenshot of table below). I have run through the setup compared with the formulas in the reference sheet and for the life of me I can't figure out what is wrong.

As I mentioned above, I'm not much of a statistician and so it is possible I missed something, so I am hoping someone may have some additional insight into where I'm going wrong.

Re: M/M/C Queue Theory Formula Implementation

Greg_Deckler — Tue, 28 Feb 2023 23:00:37 GMT

Data_Struggles_ Any chance you can post a link to your PBIX?

Re: M/M/C Queue Theory Formula Implementation

Data_Struggles_ — Wed, 01 Mar 2023 01:39:00 GMT

HI Greg_Deckler! Yes, I can't attach the pbix directly so here's a OneDrive link to my test pbix:
MMC_Queue_Theory_Test.pbix

Re: M/M/C Queue Theory Formula Implementation

Data_Struggles_ — Sat, 04 Mar 2023 01:25:21 GMT

As these things tend to go, it turns out I just misunderstood the Summation piece of the Erlang C probability formula. I was able to update my DAX to the following and it worked like a charm:

QTheoryTest3 = var Volume = 50 //in 30 min interval var IntervalSize = 60 //in minutes var AHT = 10 //in minutes var AHT_Adj = AHT / IntervalSize // AHT_Adj is S var ArrivalRate = Volume * (IntervalSize / 30) // ArrivalRate is lambda var ServiceRate = (1 / AHT) * IntervalSize var MinServers = 2 var MaxServers = 50 RETURN ADDCOLUMNS( ADDCOLUMNS( ADDCOLUMNS( ADDCOLUMNS( ADDCOLUMNS( SELECTCOLUMNS(GENERATESERIES(MinServers,MaxServers),"Val1",[Value]), //Val1 is c "Val1-1",[Val1] -1 ), "TrafIntensity", ArrivalRate * AHT_Adj //TrafIntensity is rho ), //The Summation piece of the Erlang C probability. Val2 is n "SubProb", SUMX(SELECTCOLUMNS(GENERATESERIES(0,[Val1-1]),"Val2",[Value]), DIVIDE(POWER([TrafIntensity],[Val2]),FACT([Val2]))) ), "ErlangCProb", DIVIDE(DIVIDE(POWER([TrafIntensity],[Val1]),FACT([Val1])), DIVIDE(POWER([TrafIntensity],[Val1]),FACT([Val1])) + ((1 - DIVIDE([TrafIntensity],[Val1])) * [SubProb])) ), "AvgWait", DIVIDE([ErlangCProb] * AHT_Adj, [Val1] * (1 - DIVIDE([TrafIntensity],[Val1]))) * IntervalSize //converted back to minutes )