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Data_Struggles_'s avatar
Data_Struggles_
Frequent Visitor
3 years ago
Solved

M/M/C Queue Theory Formula Implementation

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.

  • Data_Struggles_'s avatar
    Data_Struggles_
    3 years ago

    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
    )

3 Replies

    • Data_Struggles_'s avatar
      Data_Struggles_
      Frequent Visitor

      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
      )