If the average arrival rate in a queue is 13/hr and the average service rate is 20/hr, then the average number of customers in the line (including the customer being served) will be:

Concept:

The probability that there are k customers in the system = (ρ)k(1 - ρ) = \(\frac{λ }{μ }\left( {\frac{1}{{μ - λ }}} \right)\)

No. of customers in the  system or length of the system  \({L_S} = \frac{λ }{{μ - λ }}=\frac{\rho}{1~-~\rho}\)

No. of customers in the queue or length of the queue \({L_q} = \frac{{{λ ^2}}}{{μ \left( {μ - λ } \right)}}\)

Average arrival time and the time spent in the queue (before being served) (Waiting time in queue) = \({W_q} = \frac{λ }{μ }.\frac{1}{{μ - λ }}\)

Calculation:

Given:

average arrival rate, λ = 13/hr, average service rate, μ = 20 / hr

No. of customers in the  system or length of the system \({L_S} = \frac{λ }{{μ - λ }}=\frac{\rho}{1~-~\rho}\)

\(\rho = \frac{13}{20}=0.65\)

\(L_s=\frac{0.65}{1~-~0.65} = 1.857\)