Question

In a single server, infinite population queuing model. Arrivals follow a Poisson distribution with mean $\lambda =4$ per hour. The service times are exponential with mean service time equal to 12 minutes. The expected length of the queue will be

• A. 24.3
• B. 4
• C. 3.2
• D. 1.25

Answer 1

The correct option is C 3.2

$\text{Arrival rate (Poisson distribution),}\lambda =4\text{per hr.,}$

$\text{Service rate (Exponential distribution)}$

$\mu =5perhr.,\rho =0.8$

Expected length of queue or average number of customers in the queue,

$L_q=\frac{{\rho }^2}{1-\rho }=\frac{\left(\frac{\lambda }{\mu }\right)}{1-\left(\frac{\lambda }{\mu }\right)}=\frac{{\left(\frac{4}{5}\right)}^2}{1-\left(\frac{4}{5}\right)}=3.2$

$\text{Points to Remember:}$

• Average number of customers in the queue doesn't count the customers being served.
• Average number of customers in the system tells on an average total number of customers in the system and it improves both the customers waiting in the queue, along with being served.