Eric Slud Suggested Form of a Final Project on Queueing ============================================= A supermarket with 20 cash registers must decide how many cashiers to employ at various times of day (and night). Demand (ie number D(t) of customers wanting to check out per unit time) is known to vary by a factor of 5 during the day, somewhat cylclically with busy periods in the morning and lunchtime and an extra-busy time from 5 to 8pm. Assume that D(t) is given either analytically or as a data-determined sequence, say for t at 10-minute intervals, either assumed piecewise-linear or otherwise interpolated in-between. For each customer who begins service, assume first that the time S to complete service is (independently of all other customers) a random variable with P(S >t) = G(t) known to be of the form exp(-b*t) for appropriately chosen constant b. (You may choose some other G if you like: but random variables S are particularly easy to simulate.) You could make this more complicated by making the amount purchased by each customer a random variable A, in which case the service-completion rate b would be made to depend on A . Also consider the possibility that customers come in "classes" according to how many items they have in their baskets, either "many" (number of items > 15) or "few", and that the proportion of the customer population with "many" items at time t is also a function Q(t) which is given in data or hypothesized. Consider also the fraction F(t) who simply leave the store if they see the lines are longer than K people (say, K=8 or 10) or if they have actually been waiting longer than M minutes on line. Make some reasonable assumptions about the profit lost to the supermarket by such aborted purchases. Based on specific choices for these input functions [maybe more than one combination of such choices if you want to explore the robustness of your conclusions], the objective of the project is to develop a computational strategy to find the number of checkout clerks employed by the store as a function of time, so as to optimize the profit to the store. (You should also make reasonable assumptions about the labor costs.) The primary method of investigation in this project is stochastic simulation and numerical optimization, so that you can look at many trajectories of waiting-line and customer-purchases and can optimize the choice of how many checkers to employ by looking at the typical or average trajectories as well as the variability and the frequencies of extreme waiting lines. The general methodology would fall under `Operations Research', but you are urged to develop your own approach. If you can find real data to make your parameter choices realistic, so much the better. But after making some simulated whole-day waiting-line trajectories, you should justify your choices by the customer population and throughput you generate, by comparison with a realistic supermarket.