Question

Customer arrives at a grocery store to checkout counter according to a Poisson process with rate per minute. Each customer carries a number of items that is uniformly distributed between 1 and 40. The store has 2 checkout counters, each capable of processing items at a rate of 15 per minute. To reduce the customer wait in queue, the store manager considers dedicating a one of the two counters to customers with x items or less and dedicating one of the two counters to customer x items. Write a small program to find the value of x that minimizes the average customer waiting time.

Answer 1

The Matlab program to plot and find solution to the above problem:

n = 40;
lamabda = 1;
r = 15;

W = zeros(14,1);

% Computing the mean waiting time for x = 21 to 34
for x = 21:34
% Counter 1
lambda1 = x/40;
ES1 = (1+x)/30;
ES1sq = sum((1:x).^2)/225/x;
rho1 = lambda1*ES1;
W1 = (lambda1 * ES1sq)/2/(1-rho1);
% Counter 2
lambda2 = (40-x)/40;
ES2 = (x+41)/30;
ES2sq = sum(((x+1):40).^2)/(9000-225*x);
rho2 = lambda2*ES2;
W2 = (lambda2 * ES2sq)/2/(1-rho2);
% average waiting time
W(x-20) = (x/40)*W1 + (40-x)/40*W2;
end

% plotting W(x) against x
plot(21:34,W,'r*','Markersize',6)
set(gca,'FontSize',12)
xlabel('x','FontSize',14)
ylabel('Meaning waiting time','FontSize',14)
grid

If any doubts please comment below.......

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