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Two employees of a brokerage firm receive calls from customers regarding the purchase or sale of mutual funds. When their telephone lines were independent, each was busy a time of exponential expectancy $1/4$ hour with each client, during which time a new call is rejected, then a waiting time of exponential expectancy one hour before receiving the next call. Since the reorganization of the service, when a call is received by an employee who is employed, he was transferred to another employee if it is free, if the appeal is dismissed. Determine the generator to the number of employed workers: (a) before the reorganization of the service, and (b) after the reorganization of the service.

I did the part (a) and I got the generator

\begin{bmatrix} -2 & 2 & 0 \\ 4 & -5 & 1 \\ 0 & 8 & -8 \end{bmatrix}

Here the states of the number of employees are $0$, $1$ and $2$

Is anyone could explain to me how to obtain the part (b)? The dependence between the two employees is causing me a lot of problems.

1 Answers1

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For (b) the generator is almost the same:

\begin{bmatrix} -2 & 2 & 0 \\ 4 & -6 & 2 \\ 0 & 8 & -8 \end{bmatrix}

The only difference is the rate of going from state $1$ to $2$ equals $2$: the same rate as going from $0$ to $1$ because in both cases an incoming call to either employee will increment the state by $1$.

Mick A
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