Question

Benny the Barber owns a one-chair shop. At Barber College, they told Benny that his customers would exhibit a Poisson arrival distribution and that he would provide an exponential service distribution. His market survey data indicate that customers arrive at a rate of two per hour. It will take Benny an average of 20 minutes to give a haircut. Based on these figures, find the following:

a. The average number of customers waiting.

Short answer

Assume customer arrival to be Poisson distributed and service to be exponentially distributed.

Step-by-step solution

Customer time

Client Number implies the system-generated, or another recognizing number, allotted by the division to each individual conducting commerce with the division . The client number of a private person is for the most part the person’s driver's permit or non-operating recognizable proof permit number.

The average number of customers waiting

Arrival rate, = 2 per hour

Service rate, µ = 1 in 20 min = 3 per hour

The average number of customers waiting

$\left({\mathrm{L}}_{\mathrm{q}}\right)=\frac{{\mathrm{\lambda }}^2}{\mathrm{\mu }\times \left(\mathrm{\mu }-\mathrm{\lambda }\right)}$

Here,

$\mathrm{\lambda }=\mathrm{Rate}\mathrm{of}\mathrm{customer}\mathrm{arival}$

$\mathrm{\mu }=\mathrm{rate}\mathrm{of}\mathrm{service}\mathrm{delivery}$

Substitute the above values in the equation

(1) Proceed as follows: Average number of customers waiting,

$\left({\mathrm{L}}_{\mathrm{q}}\right)=\frac{{\left(2\right)}^2}{3\times \left(3-2\right)}$

= 1.333 customers

Hence, the average number of customers waiting 1.333.