# Resturant Problem

Suppose that a fast-food restaurant sells salad and burger. There are two cashiers. With cashier 1, the num-
ber of seconds that it takes to complete an order of salad is uniformly distributed in f55; 56; : : : ; 64; 65g; and
the number of seconds it takes to complete an order of burger is uniformly distributed in f111; 112; : : : ; ; 129; 130g.
With cashier 2, the number of seconds that it takes to complete an order of salad is uniformly distributed in
f65; 66; : : : ; 74; 75g; and the number of seconds it takes to complete an order of burger is uniformly distributed
in f121; 122; : : : ; ; 139; 140g.
Assume that the customers arrive at random times but has an average arrival rate of r customers per
minute. Consider two dierent scenarios.
Customers wait in one line for service, and when either of two cashiers is available, the rst customer
in the line goes to the cashier and gets serviced. In this scenario, when a customer arrives at the
restaurant, he either gets serviced if there is no line up, or wait at the end of the line.
Customers wait in two lines, each for a cashier. The rst customer in a line will get serviced if and only
if the cashier for his line becomes available. In this scenario, when a customer arrives at the restaurant,
he joins the shorter line. In addition, we impose the condition that if a customer joins a line, he will
not move to the other line or to the other cashier when the other line becomes shorter or when the
other cashier becomes free.
In both scenarios considered, a cashier will only start serving the next customer when the customer
he is currently serving has received his ordered food. (That is the point we call \the customerâ€™s order is
completed".)

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