5. (35 points total) Consider a small temporary medical facility constructed to process incoming patients having probable COVID-19. The temporary structure is small and has four staffed tringe rooms and three rooms to accommodate patients who are waiting for a tringe room. Patient-arrival events follows a Poisson process with a rate of one arrival event per two hours. Each arrival event consists of the arrival of either a single patient, or two patients or three patients. Let py denote the probability that a given arrival event consists of the arrival of a total of 1 patents. Assume that i = 0.7. P2 = 0.2, and ps = 0.1. Upon arrival a patient is taken to an triage room if an tringe room is available. If a triage room is not available then the patient is taken to an available waiting room. The patients stay in the waiting room until a triage room becomes available at which time the patient is transferred to an triage room. If no waiting room is available for a particular patient arriving to the facility then that patient is diverted to another facility. If the facility is deemed to be "closed to accepting new patients" (See Policy One and Policy Two below) then arriving ambulances are directed to take their patients to an alternate hospital Patients' length of stay in a tringe room follows a negative exponential distribution with a mean of 2.5 hours This small temporary medical facility is considering two different policies of when to close down admittance due to capacity constraints. The facility is interested in studying the statistical performance of the facility with respect to closures caused by capacity limitation, starting from an epoch at which all rooms are empty. You will be asked to model the system two ways: 1) Policy One the facility remains open at all times as long as there are rooms of some sort available (current policy): and 2) Policy Two: the facility comes to new patients (does not admit any new patients) as soon as least one waiting room is filled Policy One Details: Here the facility remains open to admissions as long an there is at Jemat one waiting room still available. Assuming that all 4 triage rooms are occupied them 10 waiting rooms are occupied then tur facility can accommodate arrival events consisting of 1 or 2 or 3 patients in the arrival event. If the facility has just 1 waiting room occupied (and thus 2 waiting rooms available) then the facility will accommodate arrival events consisting of 1 or 2 or patients in the arrival event, and if there are 3 patients in the arrival event then the facility can receive 2 of the 3 patients and send one patient to an alternate facility, . If 2 waiting rooms are occupied and this I waiting room table) then the facility will accommodate arrival events consisting of patient in the arrival event, and if there are 2 or 3 patients in the arrival event the the facility can receive patient and und the other 1 or 2 patients to an alternate facility . Finally if there waiting rooms available the rivalent consisting of the arrival of either 1 or 2 of Jarvis patients the all of the arriving patients would be sent to an alternate facility, and there would not be change the state in the model of the tempor medical facility at all Notices If the state of the system. X). In the number of patients in the facility at the (o the states would be 0.1.2., 7)then if Xs then all the waiting vitable and . If X(t) = 4 then the rate of going from State 4 to State 5 is -pithe rate of going from State 4 to State 6 is up the rate of going from State 4 to State 7 is Ps. And . if X(t) = 5 then the rate of going from State 5 to State 6 is d-p and the rate of going from State 5 to State 7 is - (p2 + p) (and in this case one arriving patient is sent to another facility). And finally if X(t) = 6 then the rate of going from State 6 to State 7 is (Pi + P2 + ps) (and two arriving patients are sent to another facility). For both Policies, Assume that X(0) = 0. Questions for Policy One: () (5 points) Write the Rate Matrix (The transition dingram is too messy to draw with all of those ares) (1) (5 points) Compute the pmf of the system state at time=4 hours (Remember to whow your work, any formulated and/or MATLABS code), (e) (5 points) Compute (label all terms) the long-run fraction of time that the facility is closed to accepting new arrivals of any size. (Remember to show your work, any formulae used, and/or MATLAB code). (d) (5 points) Remembering that the initial state is 0 patients at the facility at time = 0 Compute the mean time until the facility is closed to arrivals of any size (Remember to show your work, any formule used, and/or MATLAB code), Policy Two Details: Here the facility remains open to admissions until there is one or more waiting rooms occupied. When at least one waiting room is occupied, the the facility is forced to close to all future admissions. For this policy the staff is only interested in the performance of the system up until the facility is closed to future adminions for the first time. Notice: If the state of the system is X(t) 4 then the facility will accommodate arrival events of any size. So, for example, if X(t) - 4 and we can have an arrival event of size 3. taking the number in the system up to 7, so out state space still needs to go up to State 7 If X(t) 25 then the facility is closed to accepting any sine arrival events, Hint: It may be helpful to think of the states that force a closure as states 50,60,70, thus one option for the state space might be {0,1,2,3,4,50,60,7C): Questions for Policy Two: () (5 points) Write the Rate Matrix in Standard Form (tholingram is too many to drow with all of those aros) (1) (s points) Let the initial state be patients at the facility at time - 0. Compute that mean time until the facility is closed to arrivals of any siau (Remember to show your work, any formule tused, and/or MATLAB code) 6) (5 point) Aame that X (0) = 0 and compute the inf of how many patients the hospital will send to an alternate facility when it is forced to close Remember to show your work my formed, and/or MATLAB code). 5. (35 points total) Consider a small temporary medical facility constructed to process incoming patients having probable COVID-19. The temporary structure is small and has four staffed tringe rooms and three rooms to accommodate patients who are waiting for a tringe room. Patient-arrival events follows a Poisson process with a rate of one arrival event per two hours. Each arrival event consists of the arrival of either a single patient, or two patients or three patients. Let py denote the probability that a given arrival event consists of the arrival of a total of 1 patents. Assume that i = 0.7. P2 = 0.2, and ps = 0.1. Upon arrival a patient is taken to an triage room if an tringe room is available. If a triage room is not available then the patient is taken to an available waiting room. The patients stay in the waiting room until a triage room becomes available at which time the patient is transferred to an triage room. If no waiting room is available for a particular patient arriving to the facility then that patient is diverted to another facility. If the facility is deemed to be "closed to accepting new patients" (See Policy One and Policy Two below) then arriving ambulances are directed to take their patients to an alternate hospital Patients' length of stay in a tringe room follows a negative exponential distribution with a mean of 2.5 hours This small temporary medical facility is considering two different policies of when to close down admittance due to capacity constraints. The facility is interested in studying the statistical performance of the facility with respect to closures caused by capacity limitation, starting from an epoch at which all rooms are empty. You will be asked to model the system two ways: 1) Policy One the facility remains open at all times as long as there are rooms of some sort available (current policy): and 2) Policy Two: the facility comes to new patients (does not admit any new patients) as soon as least one waiting room is filled Policy One Details: Here the facility remains open to admissions as long an there is at Jemat one waiting room still available. Assuming that all 4 triage rooms are occupied them 10 waiting rooms are occupied then tur facility can accommodate arrival events consisting of 1 or 2 or 3 patients in the arrival event. If the facility has just 1 waiting room occupied (and thus 2 waiting rooms available) then the facility will accommodate arrival events consisting of 1 or 2 or patients in the arrival event, and if there are 3 patients in the arrival event then the facility can receive 2 of the 3 patients and send one patient to an alternate facility, . If 2 waiting rooms are occupied and this I waiting room table) then the facility will accommodate arrival events consisting of patient in the arrival event, and if there are 2 or 3 patients in the arrival event the the facility can receive patient and und the other 1 or 2 patients to an alternate facility . Finally if there waiting rooms available the rivalent consisting of the arrival of either 1 or 2 of Jarvis patients the all of the arriving patients would be sent to an alternate facility, and there would not be change the state in the model of the tempor medical facility at all Notices If the state of the system. X). In the number of patients in the facility at the (o the states would be 0.1.2., 7)then if Xs then all the waiting vitable and . If X(t) = 4 then the rate of going from State 4 to State 5 is -pithe rate of going from State 4 to State 6 is up the rate of going from State 4 to State 7 is Ps. And . if X(t) = 5 then the rate of going from State 5 to State 6 is d-p and the rate of going from State 5 to State 7 is - (p2 + p) (and in this case one arriving patient is sent to another facility). And finally if X(t) = 6 then the rate of going from State 6 to State 7 is (Pi + P2 + ps) (and two arriving patients are sent to another facility). For both Policies, Assume that X(0) = 0. Questions for Policy One: () (5 points) Write the Rate Matrix (The transition dingram is too messy to draw with all of those ares) (1) (5 points) Compute the pmf of the system state at time=4 hours (Remember to whow your work, any formulated and/or MATLABS code), (e) (5 points) Compute (label all terms) the long-run fraction of time that the facility is closed to accepting new arrivals of any size. (Remember to show your work, any formulae used, and/or MATLAB code). (d) (5 points) Remembering that the initial state is 0 patients at the facility at time = 0 Compute the mean time until the facility is closed to arrivals of any size (Remember to show your work, any formule used, and/or MATLAB code), Policy Two Details: Here the facility remains open to admissions until there is one or more waiting rooms occupied. When at least one waiting room is occupied, the the facility is forced to close to all future admissions. For this policy the staff is only interested in the performance of the system up until the facility is closed to future adminions for the first time. Notice: If the state of the system is X(t) 4 then the facility will accommodate arrival events of any size. So, for example, if X(t) - 4 and we can have an arrival event of size 3. taking the number in the system up to 7, so out state space still needs to go up to State 7 If X(t) 25 then the facility is closed to accepting any sine arrival events, Hint: It may be helpful to think of the states that force a closure as states 50,60,70, thus one option for the state space might be {0,1,2,3,4,50,60,7C): Questions for Policy Two: () (5 points) Write the Rate Matrix in Standard Form (tholingram is too many to drow with all of those aros) (1) (s points) Let the initial state be patients at the facility at time - 0. Compute that mean time until the facility is closed to arrivals of any siau (Remember to show your work, any formule tused, and/or MATLAB code) 6) (5 point) Aame that X (0) = 0 and compute the inf of how many patients the hospital will send to an alternate facility when it is forced to close Remember to show your work my formed, and/or MATLAB code)