May I ask you to answer to these questions?

The emergency department (ED) of a hospital employs a team of porters throughout the day. A porter is required when a patient should be moved from one unit to another, e.g., from ED to MRI or ER to a hospital ward. It can be assumed that each such move requires an average of 20 minutes. The demand for the porters varies throughout the day, following the inflow rate of patients to the ED. The hourly average demands are given in the table below.

Hour of Day	Arrivals per hour	Hour of Day	Arrivals per hour	Hour of Day	Arrivals per hour
0:00	10	8:00	15	16:00	20
1:00	10	9:00	20	17:00	20
2:00	10	10:00	20	18:00	15
3:00	7	11:00	20	19:00	15
4:00	7	12:00	20	20:00	15
5:00	7	13:00	20	21:00	15
6:00	7	14:00	20	22:00	10
7:00	15	15:00	20	23:00	10

1. Assuming that each move requires an exponential service time with mean 20 minutes, and demand arrivals have a non-homogeneous Poisson process, find the required number of porters in each hour so that the probability that a patient will wait for at most 60 minutes is at least 90% [1].
2. Now assume that 20% of the demand is categorized as urgent, 30% as ASAP (as soon as possible) and the remaining 50% as routine. The service levels require that an urgent patient should be served within 20 minutes, an ASAP patient within 40 minutes and a routine patient within 60 minutes. With the same assumptions in part (1), can you find the required number of porters in each hour so that the service levels are met for all types of patients with a probability of at least 90%? Write down all the assumptions and simplifications you make. Compare these requirements with those of part (1). Comment on their similarities and differences.
3. Assume that the shifts of porters can start at midnight, 4:00, 8:00, 12:00, 16:00, and 20:00, and each shift is 8 hours long. Write down a mathematical program to find the minimum number of porters that satisfy all the requirements you found in part (1). Solve for this program.
4. Assume that the night shifts cost more, so that a porter who work on one of the shifts starting at midnight, 4:00, and 20:00 receives 1.5 more of the regular salary. Change the mathematical program in part (3) to minimize the overall labor cost and solve the new program. Compare the solutions of parts (3) and (4). Comment on their similarities and differences.

Performance Evaluation of a Hospital

KoKo Childrens Hospital is specialized in heart-related diseases. It has four main departments: (1) Emergency Department (ED), (2) Outpatient Care (OC), (3) Inpatient Care (IC) and (4) Intensive Care Unit (ICU).

Patients arrive at ED according to a Poisson process with rate l₁. OC is designed to have open access, meaning that patients are not given an appointment as they are always welcome. Accordingly, arrivals to OC can also be modeled as a Poisson process with rate l₂. Departments IC and ICU do not receive patients directly, instead they accept patients who referred from departments ED and OC. Let p_km denote the probability that a patient moves from department k to m, where k{1,2,3,4} and m{0,1,2,3,4} with m=0 meaning that the patient leaves the hospital. For simplicity, assume that a patient stays in department k for an exponential random time with rate m_k, and each department can be modeled as a single server.

The values of the parameters are given as follows:

Department	Arrival rate	Service rate	Routing probabilities
			0	1	2	3	4
1 (ED)	10	11	0.7	--	0	0.2	0.1
2 (OC)	30	32	0.9	0	--	0.1	0
3 (IC)	0	7	0.8	0	0	--	0.2
4 (ICU)	0	3	0.3	0	0	0.7	--

1. Formulate this system as a queueing network. Draw the corresponding picture and write down the flows.
2. Find the effective arrival rates for each node of the network.
3. Does this network have a steady-state probability distribution? Why or why not? Explain.
4. Find the steady-state probability distribution of the network, if it exists.
5. Can you calculate the expected waiting time of a patient in the ED? In the OC? Why or why not? Explain.
6. Can you calculate the expected waiting time of a patient in the IC? In the ICU? Why or why not? Explain. Think about the differences between the departments ED&OC from the departments IC&ICU?
7. Why do we need to assume that the departments operate as single servers? What can be the drawbacks of this assumption?