4. A call centre provides local customer support for a widely used software suite. Calls arrive at the call centre accor

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4. A call centre provides local customer support for a widely
used software suite. Calls arrive at the call centre according to a
Poisson process with an average of 60 calls per hour.
Calls are categorised into 3 different levels of complexity: easy,
normal, and hard. It is known that 80% of the calls that arrive to
the call centre are easy, 12% are normal and the remaining calls
are hard.
Conversely, there are 3 skill levels of operators in the call
centre. Junior operators, senior operators, and expert operators.
When a call arrives, it is randomly routed to one of the currently
free operators, or put on hold until an operator becomes available.
It has been observed that when a call first arrives at the call
centre each operator is equally likely to pick it up.
• Junior operators can only handle easy calls and forward all calls
they pick up that are not easy to a senior or expert operator, each
of whom is equally likely to pick it up.
• Senior operators can only handle easy and normal calls and
forward all calls they pick up that are hard to an expert
operator.
• Expert operators can handle any type of call.
(a) Each of the following random variables has a named
distribution. State the named distri-
bution and its parameter(s).
i. Let K be the number of calls that arrive to the call centre in a
1 minute interval.
ii. Consider an operator at the junior skill level. Let Z be the
number of calls they have to forward before they get a call that
they can handle.
iii. Let V be the number of calls that an operator at the standard
skill level has to pass on before they receive a call they have to
handle.
iv. As an incentive scheme to motivate junior operators to try
harder the junior operators get a small bonus for every 10 calls
they handle themselves rather than passing on to a more skilled
operator. Let X be the number of calls that a junior operator has
to forward to a more skilled operator before they have managed to
handle 10 calls on their own.
(b) Find the variance of K and Z, and expected value of X and V
.