# Average time spent in the system for a customer

Customers who have purchased a laptop often calls a customer support center to get technical help. A call is handled by a regular service representative. If the problem cannot be handled by a regular service representative, the call is transferred to a specialist. $10\%$ of all calls are transferred to a specialist. On average, there are $40$ customers being served or waiting to be served by a regular representative. On average, there are $10$ customers being served or waiting to be served by a specialist. The average rate of incoming calls is $50$ per hour. There are $30$ regular representatives and $10$ specialists.

(a) What is the average time spent in the system for an arbitrary customer? State any assumptions needed to answer this question.

(b) What is the average time spent in the system for a customer who needs to talk to a specialist?

My attempt: (a) Assume no preference among which regular representative would pick up the call, and assume a call is handled by a regular service representative. Then by Little's law with $L= 40$, $W= 50$ and $c_1 = 30$, average time spent = $(40/50)*0.9/30 =$ $\fbox{0.024}$ hour.

(b) I could see that on average, the time spent in the system with a regular representative BEFORE being transferred to the specialist = $0.8*0.2 = 0.16$ hour. But the average arrival rate $\lambda_2 = 50*0.9 = 45$ and $c_2 = 10$, so shouldn't the answer be $10/(50*(1-0.1)*10) + 0.16 = 0.022+0.16 =$$\fbox{0.182}$ hour

My question: Could anyone please help verify if my solution above is incorrect? I'm quite skeptical on both parts actually, but I cannot see where I get wrong (if there is one). Any thought would be appreciated.

• Nobody wants to help me with the problem above?? – user177196 Jan 28 '17 at 19:28