Basic Concepts
We describe an M/M/1 queueing model where the customer base is partitioned into a finite number of priorities. A higher-priority customer gets serviced before a lower-priority customer. If there is more than one such customer, the first in the queue gets service. Once a customer is in the server, it won’t get bumped even if a higher-priority customer arrives.
The arrival rates for the various priority classes can be different, although the service rates will all be the same, denoted μ.
Suppose that you have m priority classes 1, 2, …, m with 1 higher priority than 2, etc. Suppose too that the arrival rates for these priority classes are λ1, λ2, …, λm. Thus, the total arrival rate is λ = λ1 + λ2 + … + λm, and so the utility is ρ = λ/μ. A steady state is achieved when ρ < 1.
Properties
For k = 1, 2, …, m
where
From these, we derive
Example
Example 1: Calculate L, Lq, W, and Wq for each of the priority classes where the arrival rates for each of the 3 priority classes and the service rate are as described in column B of Figure 1.
Figure 1 – M/M/1 with non-preemptive priority
The results are shown on the right side of Figure 1. Column J displays the formulas in column H. Note that λ = 1+1.5+2 = 4.5.
Worksheet Functions
Excel Function: The Real Statistics Resource Pack supports the following array function. Here, R1 is a row or column array or cell range that contains the arrival rates for the various priority classes.
MM1P(R1, μ, lab): returns an array with L, Lq, W, Wq, ρ for each priority class in an M/M/1 non-preemptive priority queueing model with the mean arrival rates in R1 and mean service rate of μ.
If lab = TRUE (default FALSE) then an extra column of labels is appended to the output.
You can obtain the values in Figure 1 via the array formula =MM1P(B4:B6,B2,TRUE).
Links
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
No longer available online
Hillier, F. S., Lieberman, G. J. (2010) Introduction to operations research. 9th Ed.