Project
Continuous Time Markov Chains: Pharmacy Windows Design
You are hired to design pick-up windows for a pharmacy. Currently there are 3 pharmacists hired in this store. Each worker has been cross trained to perform all functions. Two window designs are under consideration. The first proposal (called PR1) has each worker serves one window. When a patient approaches a window, all services (order, picking, consultation, and payment) are provided. Three windows are available. Customers have equal chance to choose any open window.
The second proposal (PR2) divides the windows into two categories: drop-off (order and picking) and check-out (consultation and payment). Assuming there are 3 windows total, two windows are devoted to drop off and one is devoted to check out. In this case, a patient will approach one of the available drop-off windows first, the worker at that window will work on this order (i.e. order and picking) but passes the finished order to the check-out window for consultation and payment. The worker at the check-out window is only responsible for the check-out functions (i.e. consultation and payment).
The pharmacy estimates that the average number of patient arrivals is 20 per hours during regular hours from 7 am to 7 pm. Each order takes on the average of 6 minutes for filling the order for proposal PR1. For proposal PR2, the average time for order and picking is about 4 minutes and the average check-out time is about 2 minutes once a patient is called. There are 5 chairs in the waiting room. Patients leave if there are already 5 patients in the waiting room. Once a patient arrives and decides to stay, he/she will take a number which is used as the order number. A small number represents the higher priority for the drop-off window if patients are waiting for dropping off their orders.
The primary performance measure is twofold: the patient waiting time (i.e. customer satisfaction) and worker occupancy time (i.e. productivity).
Please provide a contrast analysis between PR1 and PR2 for the questions listed below.
Modeling
1. Design continuous stochastic processes for PR1 and PR2. Clearly define its variables and its state space. Use proper notations.
2. Prove that the PR1 stochastic process is CTMC based on a few assumptions. List these assumptions for each model.
3. Draw the rate diagrams for both proposals.
Computations (56 points; 8 points each)
4. Define both the generator and rate matrices Q and R for PR1 and PR2.
5. Use small time increments to calculate the transient probabilities for each state the first 12 hours assuming there is no patient at time for PR1 and PR2.
6. Plot the results on XY plot (X is the time in hours and Y is the probability) from Q5. Identify the cut-off time into steady state process for each proposal. Discuss and compare both proposals.
7. Provide a steady state solution of ap. Please interpret the assumptions and result. Show all work.
8. Calculate the expected first passage time when all three workers are busy for PR1 and two drop-off/picking workers in PR2. Interpret the meaning of this statistic and compare both proposals.
9. Calculate the expected number of steps to first passage (hint: think DTMC) when all workers for PR1 (and 2 drop-off/picking workers for PR2) are occupied. Discuss the differences of solutions between Q8 and Q 9.
10. Calculate the expected transition time and variance from no patient to a full room of patients for both PR1 and PR2. How would this information help the decision making process?
Analysis and Decision Making
11. Generate meaningful statistics so that you can evaluate PR1 and PR2 (put detailed calculation and work in an appendix) and choose either PR1 or PR2. Please discuss your recommendation. (Hint: think bulk rates, patient waiting time, and station utilization rate, etc.)
12. Please also discuss the transient state solutions and the steady state solutions for the recommended solution in Q11. Would the transient state probabilities affect your decision making? Explain.
13. Based on the recommendation from Q11, how would you improve the recommended operation model (PR1 or PR2) further without sacrificing worker productivity too much?