Question

For a travel agency clerk, it is expected to take 20 minutes to deal with a customer. On a weekend, an average of two customers per hour arrive at the agency. Both the customer arrival times and clerk’s servicing times are modelled by an exponential distribution.

a. What is the arrival rate per minute?

b. What is the servicing rate per minute?

c. What is the servicing rate per hour?

d. What is the traffic intensity?

e. What is the expected number of orders in the waiting line?

f. What is the expected number of orders in the system (waiting and serviced)?

g. What is the expected time for an order to spend in the system (waiting and serviced)?

h. What is the expected time for an order to spend in the waiting line?

i. What is the expected time for an order to spend in the servicing channel?

Answer 1

In this problem we will consider a M/M/1 queuing model .

a> Average rate per minute = 2/60 = 1/30 i.e, 0.0333 customers arrive per minute

b> Servicing rate per minute = 1/20 i.e,  0.05 customers are serviced per minute.

c> Servicing rate per hour = 60/20 =3 i.e., 3 customers are serviced per hour.

d> Mean arrival rate () = 2 customers per hour.

Mean service rate ( ) = 3 customers per hour.

Traffic intensity ( ) =   / = 2/3 = 0.6667

e> Expected number of orders in the waiting line = ² / (1-) = 1.3336

f> Expected number of orders in the system =   /(1-) = 2.0003

g> Expected time for an order to spend in the system =   /(1-) = 1.00015

h> Expected time for an order to spend in the waiting line = ²/(1-) = 0.6668

i>  Expected time for an order to spend in the servicing channel = 1.00015 - 0.6668 = 0.33335