Problem 10. (10 Points) Maggie is considering opening a neighbor grocery store.

Question

Problem 10. (10 Points) Maggie is considering opening a neighbor grocery store. She has estimated that customers arrive at the stand at a mean rate of 30 por hour. If she hires a full-time cashier, the service time is estimated to be exponentially occasional long lines and complaints from customers. Because of hiring another there is no room for a second checkout stand, Maggie is considering the alternative person to help the cashier by bagging the groceries. This help would reduce time required to process a customer to 1 minute, but the distribution still would be expected exponential Maggie would like to have the of time that there are more than two customers at the ) percent she would alsoTike-to have,acustomer penti omaverage finishing service, Which approach would satisfy Maggie's . checkout stand to be below (25 no more than 25 minutes before w 1.5 en 49 4 yoo ?-?

Explanation / Answer

So the objectives of <2.5 mins as Wq is not satisfied.

And 2 people in queue is at 42%.

New system:

So the objectives of <2.5 mins as Wq is satisfied as Wq = 1 min

And 2 people in queue is at 12.5% probability.

Current system Kendall's notation M/M/1 Comment Arrival rate A 30 per hour Service rate S 40 60/1.5 U Utilization ratio U=A/S 0.75000 <1 Length of queue diminishing Ls Expected number of people in system Ls A/(S-A) 3.00 Ws Avg waiting time in the system Ws 1/(S-A) 0.10000 hours 6.0000 mins Wq Avg waiting time in the Line Wq A/S(S-A) 0.07500 hous 4.5000 mins Lq Avg no of customer waiting in line Lq A^2/S(S-A) 2.2500 Po Prob of 0 units in system Po 1-(A/S) 0.25000 25.00% Pn Prob of k units in system (A/S)^(K+1) 0.42188 K = 2

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