M/M/s Queueing Model

Basic Concepts

The M/M/s queueing model describes the same queueing process as the M/M/1 queueing model except that s servers are used where s is a positive integer. When s = 1, this model is equivalent to the M/M/1 model.

 agOnceain, customer arrivals follow an exponential distribution at the rate of λ. Servicing also follows an exponential distribution with a service rate of μ for each server. We assume that when more than one server is available, the customer chooses randomly.

This time we set τ = λ/μ and ρ = λ/(μs). We will assume that ρ < 1 since, in this case, the process arrives at a steady-state (as defined in Queueing Theory).

Balance Equations

The steady-state equations for the M/M/s model are as follows.

μp₁ = λp₀

(λ + nμ)p_n = λp_n-1 + (n + 1)μp_n+1 for n < s

(λ + sμ)p_n = λp_n-1 + sμp_n+1 for n ≥ s

Number of customers in the system

The probability p_n that n customers are present in an M/M/s model in a steady-state can be calculated by the following formulas

We can use the notation P_n = p₀ + p₁ + ⋅⋅⋅ p_n. In particular, we use the notation C as the probability that when a new customer arrives it finds all the servers occupied and so the customer must queue. It turns out that

Key properties

The following mean values hold for M/M/s models in a steady state:

The corresponding variances are

where if τ ≠ s – 1

while if τ = s – 1

Distribution of w and w_q

The following formulas hold for M/M/s models in a steady state.

When τ ≠ s – 1

while if τ = s – 1

Example

Example 1: Calculate the various parameters for an M/M/s queueing model with 3 servers where λ = 1.5 and μ = 2.

The results are shown in Figures 1 and 2.

Figure 1 – M/M/s queueing model (part 1)

Columns M and N of Figure 2 are used to calculate p₀. Cell N3 contains the formula =G$8^M3/FACT(M3). The other values in column N are filled in by highlighting range N3:N12, and pressing Ctrl-D.  The value for p₀. (in cell N16) is calculated via the formula 

=1/(SUM(OFFSET(N3,0,0,B4,1))+OFFSET(N3,B4,0,1,1)/(1-G6))

For Example 1 with s = 3, this formula is equivalent to =1/(SUM(N3:N5)+N6/(1-G6). The advantage of the above formula is that it works with any value of s (although the values in column N need to be extended for s > 12).

Figure 2 – M/M/s queueing model (part 2)

Note that cell Q4 contains the formula

=Q$3*IF(P4<B$4,N4,G$8^P4/(FACT(B$4)*B$4^(P4-B$4)))

and R4 contains =R3+Q4, and similarly for the other formulas in columns Q and R.

Cell U3 contains the formula

=EXP(-B$3*T3)*(1+G$10*IF(G$9=0,B$3*T3,(1-EXP(-B$3*T3*G$9))/G$9))

and cell V3 contains the formula

=G$10*EXP(-B$4*B$3*T4*(1-G$6))

The other formulas in columns U and V are similar.

Worksheet Functions

Excel Functions: The Real Statistics Resource Pack supports the following array function.

MMs(λ, μ, s, lab): returns a column array with the L, L_q, W, W_q, ρ for the M/M/s queueing model with s servers and exponential arrival and service rates with mean arrival rate λ and mean service rate μ.

If lab = TRUE (default FALSE) then an extra column of labels is appended to the output. In addition, the following non-array functions are also supported.

MM1Pn(λ, μ, s, n, cum) = p_n for the M/M/1 queueing defined by s, λ, and μ if cum = FALSE (default) and P_n otherwise.

MM1W(λ, μ, s, t) = P(w > t) for the M/M/1 queueing defined by s, λ and μ

MM1Wq(λ, μ, s, t) = P(w_q > t) for the M/M/1 queueing defined by s, λ and μ

You can obtain the values in range F2:G6 of Figure 1 via the array formula =MMs(B2,B3,TRUE). You can obtain the values in Figure 2 by inserting =MMsPn(B$2,B$3,M3) in cell N3 and =MMsPn(B$2,B$3,M3,TRUE) in cell O3, highlighting N3:O15, and pressing Ctrl-D. Finally, you can obtain the values in Figure 3 by inserting =MMsW(B$2,B$3,Q3) in cell R3 and =MMsWq(B$2,B$3,Q3) in cell S3, highlighting R3:S15, 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
No longer available online

Wikipedia (2023) M/M/c queue
https://en.wikipedia.org/wiki/M/M/c_queue