A call centre has $2$ trainees to deal with customer enquires. The centre has a dispatcher to

direct the calls automatically to one of the trainees. The dispatcher does not contain any

queueing facilities. At each trainee$$s terminal, there is a facility to queue up to $3$ calls. The

queueing network at the call centre is depicted in Figure $1.$The centre receives on average $\backslash$lambda queries per hour. The arrivals can be modelled by using

the Poisson distribution.

When a query arrives at the dispatcher, it will send the query to Trainee $1$ with a probability of p and to Trainee $2$ with a probability of $1$ p$.$ Note that the dispatcher does not

communicate with the trainee$$s terminals, so it is possible that the dispatcher sends a query

to a terminal that has a full queue. You can assume that the dispatcher takes a negligible

time to perform its work and no queries will be dropped at the dispatcher.

Nominally, Trainee $1$ can complete or service on average $1$ queries per hour. This service

rate applies when the number of customers waiting in their queue is $2$ or less. However, when

there are $3$ customers waiting in the queue, Trainee $1$ feels the pressure of the full queue

and their service rate slows down to

u $11$ where

u Trainee $2$ performs in exactly the

same way except that their service rate is $2$ and their service rate when their queue is full

is

u $22$ where

u You can assume all the service times are exponentially distributed and

independent of each other.

When a query arrives at a staff$$s terminal, it will be answered straight away if the staff is

not busy. Otherwise, the terminal will place the call in its queue if the queue is not full. If a

call arrives when the queue is full, then the call is rejected.Answer the following question:Assuming that p $=0\mathrm{.}45,\backslash$lambda $=5\mathrm{.}3,1=3\mathrm{.}9,$

u $1=0\mathrm{.}86,2=3\mathrm{.}5$ and

u $2=0\mathrm{.}8,$ determine

the mean waiting time of the queries that have not been rejected by the call centre.