Basic Concepts
The M/M/1/N queueing model is the same as the M/M/1 model except that now the population of customers is finite with N members.
Properties
Using the notation found in Queueing Theory, we now present formulas for the key properties of this queueing model once a steady state is reached.
Clearly pn = 0 for n > N, but for n ≤ N, we have
The formulas for L and Lq are
where the obtained mean arrival rate is
We calculate W and Wq using Little’s Law as follows
Example
Example 1: Calculate L, Lq, W, Wq, and pn for the M/M/1/N model with λ, μ, and N specified in column B of Figure 1.
The values for L, Lq, W, Wq, ρ, and λ-bar are calculated on the right side of Figure 1. Column K contains a list of the formulas used in column G. The formula in cell K2 references the formula for p0 shown in Figure 2.
Figure 1 – M/M/1/N queueing model (part 1)
Figure 2 shows how to calculate the pn = the probability that n customers are in the system and Pn = the probability that at most n customers are in the system.
We calculate p0 in cell N3 via the formula =1/(FACT(B4)*SUM(R3:R23)). Here cell R3 contains the formula =IF(Q3<=B$4,G$6^Q3/(FACT(B$4-Q3)*B$4^Q3),0).
The other formulas in column R are inserted by highlighting range R3:R23 and pressing Ctrl-D. The other formulas in column N are inserted in a similar fashion by highlighting N4:N23 and pressing Ctrl-D.
Note that column R contains values for n = 0 to 20 (only values n = 0 to 12 are displayed). If N > 20 then we need to expand range R3:R23 to at least n = N.
Figure 2 – M/M/1/N queueing model (part 2)
We fill in the other pn values in column N by inserting the formula =N$3*R4*FACT(B$4) in cell N4, highlighting the range N4:N23 and pressing Ctrl-D (rows 16 through 23 are not displayed, although the values in column N are all zero).
Worksheet Functions
Excel Functions: The Real Statistics Resource Pack supports the following array function.
MM1N(λ, μ, nn, lab): returns a column array with the L, Lq, W, Wq, λ-bar in the steady state for the M/M/1/N queueing model with exponential arrival and service rates with mean arrival rate λ and mean service rate μ, and where the population size is nn.
If lab = TRUE (default FALSE) then an extra column of labels is appended to the output. In addition, the following non-array function is also supported.
MM1NPn(λ, μ, nn, n, cum) = pn for the M/M/1/N queueing defined by nn, λ and μ if cum = FALSE (default) and Pn otherwise.
We can obtain the values in range F2:G6 of Figure 1 via the array formula =MM1N(B2,B3,B4,TRUE). We can obtain the values in Figure 2 by inserting =MM1Pn(B$2,B$3,B$4,M3) in cell N3 and =MM1Pn(B$2,B$3,B$4,M3,TRUE) in cell O3, highlighting N3:O15, and pressing Ctrl-D.
Examples Workbook
Click here to download the Excel workbook with the examples described on this webpage.
References
Ross, S. M. (2014) Introduction to probability models, 11th Ed. Academic Press
https://ebin.pub/introduction-to-probability-models-11nbsped-0124079482-9780124079489.html
Sztrik, J. (2021) Basic queueing theory
https://irh.inf.unideb.hu/~jsztrik/education/16/SOR_Main_Angol.pdf
Shores, T. S. (2017) Queueing theory basics and models
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