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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.

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

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