I'm looking for kinds of models able to describe a situation such as the one described below (what I'm actually dealing with is load balancing but this is a more concrete example):
We've a got a supermarket with cashiers. The more cashiers there is, the more throughput $T$ the supermaket is able to handle. The manager also somehow can boost a cashier to reduce his response time $\tau$ (which is the time it takes to handle fully a customer).
However, both adding cashiers or boosting a cashier have costs which depend on the type of the action taken. Let's say for instance that when a cashier is added, the throughput is totally blocked for 30 seconds.
My goal is to dynamically make the right decisions to either boost the throughput (when there is a peak of customers in the shop) or reduce the response time (when there are little customers).
My current approach is based both on monitoring (being able to react to high peaks or idle periods with fixed thresholds to carry out the appropriate decision) and forecasts (using double seasonal exponential smoothing on time series) based on recorded data on which there is both a daily and weekly patterns.
My questions are :
- How can I somehow include the costs of the decisions in my model ?
- Which variable should I monitor ? I do want to affect the throughput or the response time, but if I take one of those as the "input" variable, my time series will not actually reflect how many customers were actually there. My model will be absolutely wrong as it won't make the difference between the actual load and what was made possible thanks to previous decisions carried out (if I'm not clear, I can explain further). The one lead I have is that I should also monitor the amount of people queueing.
- My current approach is based on univariate time series modeling. Would it be a good idea to consider multivariate time series modeling (for instance a vector with the throughput and the response time which could very well be correlated) ?