M/M/s Queue Example

Example

Consider an s queue with $s=2$ servers, arrival rate $\lambda =3$ customers per hour, and service rate $\mu =2$ customers per hour per server.

Step 1: Check Steady-State Condition

Traffic intensity per server: $\rho =\frac{\lambda }{s\mu }=\frac{3}{2\times 2}=\frac{3}{4}<1$ ✓

Step 2: Auxiliary Quantities

${\theta }_0=1$ ${\theta }_1=\frac{\lambda }{\mu }=\frac{3}{2}$ ${\theta }_2=\frac{(\lambda /\mu {)}^2}{2!}=\frac{(3/2{)}^2}{2}=\frac{9/4}{2}=\frac{9}{8}$ ${\theta }_j=\frac{(\lambda /\mu {)}^s}{s!}{\left(\frac{\lambda }{s\mu }\right)}^{j-s}=\frac{9}{8}{\left(\frac{3}{4}\right)}^{j-2},j\ge 2$

Sum of ${\theta }_j$:

${\sum }_{j=0}^{\infty }{\theta }_j=1+\frac{3}{2}+\frac{9}{8}{\sum }_{j=2}^{\infty }{\left(\frac{3}{4}\right)}^{j-2}=1+\frac{3}{2}+\frac{9}{8}\cdot \frac{1}{1-3/4}=1+\frac{3}{2}+\frac{9}{8}\cdot 4=1+\frac{3}{2}+\frac{9}{2}=7$

Step 3: Steady-State Probabilities

${\pi }_0=\frac{1}{7},{\pi }_1=\frac{3/2}{7}=\frac{3}{14},{\pi }_2=\frac{9/8}{7}=\frac{9}{56}$

Step 4: Performance Metrics

Average queue length $L_Q$:

$L_Q={\sum }_{j=s+1}^{\infty }(j-s){\pi }_j={\sum }_{j=3}^{\infty }(j-2)\frac{9}{56}{\left(\frac{3}{4}\right)}^{j-2}$

Let $k=j-2$, then:

$L_Q=\frac{9}{56}{\sum }_{k=1}^{\infty }k{\left(\frac{3}{4}\right)}^k=\frac{9}{56}\cdot \frac{3/4}{(1-3/4{)}^2}=\frac{9}{56}\cdot \frac{3/4}{1/16}=\frac{9}{56}\cdot 12=\frac{27}{14}\approx 1.93$

Wait in queue: $W_Q=L_Q/\lambda =\frac{27/14}{3}=\frac{9}{14}\approx 0.643$ hours

Time in system: $W=W_Q+\frac{1}{\mu }=\frac{9}{14}+\frac{1}{2}=\frac{9}{14}+\frac{7}{14}=\frac{16}{14}=\frac{8}{7}\approx 1.14$ hours

Customers in system: $L=\lambda W=3\times \frac{8}{7}=\frac{24}{7}\approx 3.43$

Verify: $L=L_Q+\lambda /\mu =\frac{27}{14}+\frac{3}{2}=\frac{27}{14}+\frac{21}{14}=\frac{48}{14}=\frac{24}{7}$ ✓

NOTE

With 2 servers, the average queue length is about 1.93 customers, and customers wait about 39 minutes in queue before starting service. Compare this to an M/M/1 with the same total service capacity: if $\mu =4$ with 1 server, $\rho =3/4$ and $L=3$ customers. The multi-server system has slightly higher $L$ but distributes the load.

Related

• s Queueing System
• 1 Queueing System
• Queueing Formula L = λW