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.

 Once again, 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.

Links

↑ Queueing theory

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