# Total hourly profit for a single-server food stand

Customers arrive to a single-server food stand according to a Poisson process with rate $20$ per hour. The time to serve a customer is exponentially distributed with a mean of $2$ minutes.

(a) The food stand offers that any customer waiting more than $6$ minutes receives a $\$5$discount off their order. The average customer spends$\$10$ for lunch, of which $\$4$is profit to the stand. What is the hourly profit of the food stand? (b) What is the approximate arrival rate (to the nearest multiple of$5$per hour) that maximizes the hourly profit to the food stand? My attempt: (a) First, using Little's Law for continuous-time markov chain to compute the average number of people in the system$= L = \frac{x}{1-x}$where$x =\frac{\frac{1}{3}}{\frac{1}{2}},$so$L = 2$people. Now, I compute$P(\text{waiting time}> 6 \ mins) = xe^{-\frac{1}{2}(1-\frac{1/3}{1/2})6} = \frac{2e^{-1}}{3}$. Thus, the expected profit of the stand from an average customer in this system$= (4-5)2\frac{e^{-1}}{3} + 4(1-2\frac{e^{-1}}{3}) = 4 - \frac{10e^{-1}}{3}.$Since the average number of people arrive in the system$= 20$per hour, we get the total hourly profit$= \fbox{$(4 - \frac{10e^{-1}}{3}$)20}$. (b) Using exactly the same method as above, but rewrite the final total hourly profit in terms of a function of arrival rate$\lambda$. Then use Calculus to solve for the maximum value. My question: Could someone please help review my solution to part (a) above? I'm skeptical on my last step, as it seems to me I only need to multiply$(4 - \frac{10e^{-1}}{3})$by$20$? I got stuck on part (b), because I used the same way as part (a) and ended up with solving: max$\ f(\lambda)$over$\lambda>0$where$f(\lambda) = 4 - \frac{5\lambda}{6}e^{-3+\frac{\lambda}{10}}\$. But this function has no critical point, there must be something wrong with my formulation. I could not find where is the mistake though.

• you should probably tag this as self-study – kjetil b halvorsen Feb 12 '17 at 21:09
• @kjetilbhalvorsen: could you please help me with this problem? I'm getting stuck on this for several days;p – user177196 Feb 13 '17 at 3:36