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 $1.$Figure $1$: Depiction of the call centre.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 questions:$($a$)$ Formulate a continuous$-$time Markov chain for the part of the call centre consisting of

Trainee $1$ and their $3$ waiting slots, i$.$e$.$ the part enclosed by the red dashed lines in

Figure $1.$ Your formulation should include the definition of the states and the transition

rates between states. The transition rates should be expressed in terms p$,\backslash$lambda $,1$ and

u $1.$

$($b$)$ Write down the balance equations for the continuous$-$time Markov chain that you have

formulated.

$($c$)$ Derive the expressions for the steady state probabilities of the continuous$-$time Markov

chain that you have formulated.

$($d$)$ Assuming that p $=0\mathrm{.}45,\backslash$lambda $=5\mathrm{.}3$ and $1=3\mathrm{.}9$ and

u $1=0\mathrm{.}86,$ determine the probability

that a call which is dispatched to Trainee $1$ will be rejected.

$($e$)$ 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.

Note that Part $($d$)$ considers only queries that have been dispatched to Trainee $1$ but

Part $($e$)$ considers the whole call centre.

Hint:

$$ There is a mistake that some people may make regarding the calculation of the mean

waiting time in Part $($e$).$ We will not tell you exactly what the mistake is but the

following example of probability calculations will illustrate that. Let us assumed that

you have two coins, which we will refer to as Coin $1$ and Coin $2.$ Coin $1$ is a fair coin

and the mean number of heads you get is $0\mathrm{.}5.$ Coin $2$ is a biased coin and the mean

number of heads you can get is $0\mathrm{.}6.$ Let us say you do the following:

$$ You randomly pick one of the two coins with the probabilities of picking Coins $1$

and $2$ being, respectively, $0\mathrm{.}7$ and $0\mathrm{.}3.$ You toss the coin picked. You repeat this

many times.

You want to calculate the mean number of heads that you will get. A wrong answer is

$0\mathrm{.}55.$ The correct answer should be $0\mathrm{.}53.\backslash$geotriangle $\backslash$geotriangle $\backslash$geotriangle

Reminder: If you use a computer program to derive your numerical answers, you must

include your computer program in your submission. Do not forget to show us your steps to

obtain your answer.