Tuesday, December 6, 2016

Queue sizes and yields

Real quick, I teveiw some sloppy work I ws doing on two Poisson queues, people waiting in line.

I had customers waiting in line and clerks waiting in line, and as the total number of people went large, the queues looked like they obey Newton grammar. In this model, we can look up on wiki the probability distribution of the number of people at the counter, a negative number was a clerk waiting for a customer, a loss; a positive number was a customer waiting for a clerk, a gain.

If we meet Ito's condition, we can do loss and gain over time.  Or we can look at mean to variance on the net distribution, where we find high queuelengths of customers and clerks results in a wide variation of profits and losses.  If the store owner is dealing with chaotic customers, he has to keep extra clerks at the counters, a loss.  The yield can be approximated by mean/variance on the net distribution.

 What does the store owner do? He re-orders the graph.  She constructs the 10 items or less and 5 or less checkouts, re-orders the clerk work schedule, and orders inventory to match long and short sequence shoppers.  He gets a probability graph such that the nodes are checkout stands, and the queues of customers, at each node,  seldom more than three. He has discovered the theory of the customer, almost, and he has an upward sloping yield curve, with relative latency on the x axis, he can synchronize to the outside traffic and get time; Ito is happy.

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