danryan.us

Suppose total staff is $S = F + V$ and that the rate at which staff can serve people in either queue is one per time period. This means, for example, that $Q1_{t+1}= Q1_t + arrivals - F_t$. And how should we allocate staff? One way would be to make the staff at each station proportional to the number of people in that queue. Thus, the number of staff at the sign in table should be $F_{t+1} = \frac {Q1_t} {(Q1_t + Q2_t}$. Thus:

$Q1_{t+1}= Q1_t + arrivals - F_t$

$Q2_t+1 = Q2_t + F_t - V_t$

$F_{t+1} = \frac {Q1_t} {(Q1_t + Q2_t)}$

$V_{t+1} = \frac {Q2_t} {(Q1_t + Q2_t)}$

Note that these equations would need some tweaking in practice to deal with situations where one or the other queues would be empty to avoid silly things like negative queues and dividing by zero.