Answer to Question #163076 in Statistics and Probability for Ajay

Question

The mean arrival rate to a service centre is 3 per hour. The mean service time is found 

to be 10 minutes foe service. Assuming Poisson arrival and exponential service time, 

find

 (i) the utilisation factor for this service facility, 

 (ii) the probability of two units in the system, 

 (iii) the expected number of units in the system, and 

 (iv) the expected time in hours that a customer has to spend in the system.

Answer 1

Average arrival rate $λ = 3$ per hour.

 Average service time $\dfrac{1}{\mu} = 10$ minutes, and hence $μ = 6/hour$

 (i) the utilization factor $\rho = \dfrac{λ}{\mu} = \dfrac{3}{6} = 0.5$

(ii) the probability that an arrival will find the place free

     $= p(0) = 1 –\rho = 1 – (0.5) = 0.5$

   $p (1) = \rho p(0) =0.5\times 0.5=0.25$

(iii) Average no. of customers in system, $L = \dfrac{p}{(1–p)} = \dfrac{0.5}{(1 – 0.5)} = 1$

(iv) Average waiting time in system $W = \dfrac{L}{λ} = \dfrac{1}{3} = 1/3 Hour = 20 mins$