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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
A call centre has 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 trainees terminal, there is a facility to queue up to calls. The
queueing network at the call centre is depicted in Figure Figure : Depiction of the call centre.The centre receives on average 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 with a probability of p and to Trainee with a probability of p Note that the dispatcher does not
communicate with the trainees 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 can complete or service on average queries per hour. This service
rate applies when the number of customers waiting in their queue is or less. However, when
there are customers waiting in the queue, Trainee feels the pressure of the full queue
and their service rate slows down to
u where
u Trainee performs in exactly the
same way except that their service rate is and their service rate when their queue is full
is
u where
u You can assume all the service times are exponentially distributed and
independent of each other.
When a query arrives at a staffs 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 continuoustime Markov chain for the part of the call centre consisting of
Trainee and their waiting slots, ie the part enclosed by the red dashed lines in
Figure Your formulation should include the definition of the states and the transition
rates between states. The transition rates should be expressed in terms plambda and
u
b Write down the balance equations for the continuoustime Markov chain that you have
formulated.
c Derive the expressions for the steady state probabilities of the continuoustime Markov
chain that you have formulated.
d Assuming that p lambda and and
u determine the probability
that a call which is dispatched to Trainee will be rejected.
e Assuming that p lambda
u and
u 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 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 and Coin Coin is a fair coin
and the mean number of heads you get is Coin is a biased coin and the mean
number of heads you can get is Let us say you do the following:
You randomly pick one of the two coins with the probabilities of picking Coins
and being, respectively, and 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
The correct answer should be geotriangle geotriangle 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.
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